Is the Continuum Hypothesis a definite ... - Logic at Harvard
Is the Continuum Hypothesis a definite ... - Logic at Harvard
Is the Continuum Hypothesis a definite ... - Logic at Harvard
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<strong>Is</strong> <strong>the</strong> <strong>Continuum</strong> Hypo<strong>the</strong>sis a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>ical problem?DRAFT 9/18/11For: Exploring <strong>the</strong> Frontiers of Incompleteness (EFI) Project, <strong>Harvard</strong> 2011-2012Solomon Feferman[t]he analysis of <strong>the</strong> phrase “how many” unambiguously leads to a <strong>definite</strong> meaning for<strong>the</strong> question [“How many different sets of integers do <strong>the</strong>ir exist?”]: <strong>the</strong> problem is tofind out which one of <strong>the</strong> s’א is <strong>the</strong> number of points of a straight line… Cantor, afterhaving proved th<strong>at</strong> this number is gre<strong>at</strong>er than א 0 , conjectured th<strong>at</strong> it is א 1 . An equivalentproposition is this: any infinite subset of <strong>the</strong> continuum has <strong>the</strong> power ei<strong>the</strong>r of <strong>the</strong> set ofintegers or of <strong>the</strong> whole continuum. This is Cantor’s continuum hypo<strong>the</strong>sis. …But, although Cantor’s set <strong>the</strong>ory has now had a development of more than sixty yearsand <strong>the</strong> [continuum] problem is evidently of gre<strong>at</strong> importance for it, nothing has beenproved so far rel<strong>at</strong>ive to <strong>the</strong> question of wh<strong>at</strong> <strong>the</strong> power of <strong>the</strong> continuum is or whe<strong>the</strong>r itssubsets s<strong>at</strong>isfy <strong>the</strong> condition just st<strong>at</strong>ed, except th<strong>at</strong> … it is true for a certain infinitesimalfraction of <strong>the</strong>se subsets, [namely] <strong>the</strong> analytic sets. Not even an upper bound, howeverhigh, can be assigned for <strong>the</strong> power of <strong>the</strong> continuum. It is undecided whe<strong>the</strong>r thisnumber is regular or singular, accessible or inaccessible, and (except for König’s neg<strong>at</strong>iveresult) wh<strong>at</strong> its character of cofinality is.Gödel 1947, 516-517 [in Gödel 1990, 178]Throughout <strong>the</strong> l<strong>at</strong>ter part of my discussion, I have been assuming a naïve and uncritical<strong>at</strong>titude toward CH. While this is in fact my <strong>at</strong>titude, I by no means wish to dismiss <strong>the</strong>opposite viewpoint. Those who argue th<strong>at</strong> <strong>the</strong> concept of set is not sufficiently clear tofix <strong>the</strong> truth-value of CH have a position which is <strong>at</strong> present difficult to assail. As long asno new axiom is found which decides CH, <strong>the</strong>ir case will continue to grow stronger, andour assertion th<strong>at</strong> <strong>the</strong> meaning of CH is clear will sound more and more empty.Martin 1976, 90-91Abstract: The purpose of this article is to explain why I believe th<strong>at</strong> <strong>the</strong> <strong>Continuum</strong>Hypo<strong>the</strong>sis (CH) is not a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>ical problem. My reason for th<strong>at</strong> is th<strong>at</strong> <strong>the</strong>concept of arbitrary set essential to its formul<strong>at</strong>ion is vague or underdetermined and <strong>the</strong>reis no way to sharpen it without viol<strong>at</strong>ing wh<strong>at</strong> it is supposed to be about. In addition,<strong>the</strong>re is considerable circumstantial evidence to support <strong>the</strong> view th<strong>at</strong> CH is not <strong>definite</strong>.In more detail, <strong>the</strong> st<strong>at</strong>us of CH is examined from three directions, first a thoughtexperiment rel<strong>at</strong>ed to <strong>the</strong> Millennium Prize Problems, <strong>the</strong>n a view of <strong>the</strong> n<strong>at</strong>ure of1
ma<strong>the</strong>m<strong>at</strong>ics th<strong>at</strong> I call Conceptual Structuralism, and finally a proposed logicalframework for distinguishing <strong>definite</strong> from in<strong>definite</strong> concepts.My main purpose here is to explain why, in my view, <strong>the</strong> <strong>Continuum</strong> Hypo<strong>the</strong>sis (CH) isnot a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>ical problem. In <strong>the</strong> past, I have referred to CH as depending inan essential way on <strong>the</strong> inherently vague concept of arbitrary set, by which I mean th<strong>at</strong><strong>the</strong> concept of a set being arbitrary is vague or underdetermined and <strong>the</strong>re is no way tosharpen it without viol<strong>at</strong>ing wh<strong>at</strong> it is supposed to be about. I still believe th<strong>at</strong>, which is<strong>the</strong> main reason th<strong>at</strong> has led me to <strong>the</strong> view th<strong>at</strong> CH is not <strong>definite</strong>. O<strong>the</strong>rs, with <strong>the</strong>extensive set-<strong>the</strong>oretical independence results concerning CH in mind, have raised <strong>the</strong>question whe<strong>the</strong>r it is an absolutely undecidable proposition, th<strong>at</strong> is, in <strong>the</strong> words ofKoellner (2010), “undecidable rel<strong>at</strong>ive to any set of axioms th<strong>at</strong> are justified.” 1 I prefernot to pose <strong>the</strong> issue th<strong>at</strong> way, because it seems to me th<strong>at</strong> <strong>the</strong> idea of an absolutelyundecidable proposition presumes th<strong>at</strong> <strong>the</strong> st<strong>at</strong>ement in question has a <strong>definite</strong>ma<strong>the</strong>m<strong>at</strong>ical meaning. There is no disputing th<strong>at</strong> CH is a <strong>definite</strong> st<strong>at</strong>ement in <strong>the</strong>language of set <strong>the</strong>ory, whe<strong>the</strong>r considered formally or informally. And <strong>the</strong>re is no doubtth<strong>at</strong> th<strong>at</strong> language involves concepts th<strong>at</strong> have become an established, robust part ofma<strong>the</strong>m<strong>at</strong>ical practice. Moreover, many ma<strong>the</strong>m<strong>at</strong>icians have grappled with <strong>the</strong> problemand tried to solve it as a ma<strong>the</strong>m<strong>at</strong>ical problem like any o<strong>the</strong>r. Given all th<strong>at</strong>, how can wesay th<strong>at</strong> CH is not a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>ical problem?I shall examine this from three directions, first a thought experiment rel<strong>at</strong>ed to <strong>the</strong>Millennium Prize Problems, <strong>the</strong>n a view of <strong>the</strong> n<strong>at</strong>ure of ma<strong>the</strong>m<strong>at</strong>ics th<strong>at</strong> I callConceptual Structuralism, and finally a proposed logical framework for distinguishing<strong>definite</strong> from in<strong>definite</strong> concepts.1. The Millennium Prize Problem TestEarly in <strong>the</strong> year 2000, <strong>the</strong> Scientific Advisory Board (SAB) of <strong>the</strong> Clay Ma<strong>the</strong>m<strong>at</strong>icsInstitute (CMI) announced a list of seven ma<strong>the</strong>m<strong>at</strong>ical problems, each of which if solved1 Th<strong>at</strong> is not Koellner’s view; in fact his main aim in Koellner (2010) is to exposit workof Woodin th<strong>at</strong> it is hoped will lead to a decision as to CH.2
would lead to a prize of one million dollars. The list includes some of <strong>the</strong> most famousopen problems in ma<strong>the</strong>m<strong>at</strong>ics, some old, some new: <strong>the</strong> Riemann Hypo<strong>the</strong>sis, <strong>the</strong>Poincaré conjecture, <strong>the</strong> Hodge conjecture, <strong>the</strong> P vs NP question, and so on, but it doesnot include <strong>the</strong> <strong>Continuum</strong> Hypo<strong>the</strong>sis. The ground rules for <strong>the</strong> prize are th<strong>at</strong> anyproposed solution should be initi<strong>at</strong>ed by a refereed public<strong>at</strong>ion, followed by a two-yearwaiting period “to ensure acceptance of <strong>the</strong> work by <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ics community, before<strong>the</strong> CMI will even solicit expert opinions about <strong>the</strong> validity or <strong>at</strong>tribution of a presumedsolution.“ (Jaffe 2006, 655) One criterion for <strong>the</strong> selection of problems by <strong>the</strong> SAB wasth<strong>at</strong> “each of <strong>the</strong>se questions should be difficult and important.” (Jaffe 2006, 653). And itwas decided th<strong>at</strong> <strong>the</strong> simplest form of a question was to be preferred “<strong>at</strong> least wheneverth<strong>at</strong> choice seemed sensible on ma<strong>the</strong>m<strong>at</strong>ical and general scientific grounds.” Finally,“while each problem on <strong>the</strong> list was central and important…<strong>the</strong> SAB did not envisagemaking a definitive list, nor even a represent<strong>at</strong>ive set of famous unsolved problems.Ra<strong>the</strong>r, personal taste entered our choices; a different scientific advisory boardundoubtedly would have come up with a different list. … We do not wish to address <strong>the</strong>question, ‘Why is Problem A not on your list?’ Ra<strong>the</strong>r we say th<strong>at</strong> <strong>the</strong> list highlightsseven historic, important, and difficult open questions in ma<strong>the</strong>m<strong>at</strong>ics.” (Ibid., 654)We don’t know if CH was considered for inclusion in <strong>the</strong> list, though I would besurprised if it did not come up initially as one of <strong>the</strong> prime candid<strong>at</strong>es. In any case, a newopportunity for it to be considered may have emerged since Grigory Pereleman solved<strong>the</strong> Poincaré conjecture, was awarded <strong>the</strong> prize and declined to accept it. Let’s imagineth<strong>at</strong> <strong>the</strong> CMI sees this as an opportunity to set one new prize problem ra<strong>the</strong>r than let <strong>the</strong>million dollars go begging th<strong>at</strong> would have gone to Perelman. Here’s a scenario: <strong>the</strong>SAB solicits advice from <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ics community again for <strong>the</strong> selection of one newproblem and in particular asks an expert or experts in set <strong>the</strong>ory (EST) whe<strong>the</strong>r CHshould be chosen for th<strong>at</strong>. 2 Questions and discussion ensue.*********2 There was apparently no animus to <strong>the</strong> logic community on <strong>the</strong> part of <strong>the</strong> SABconcerning <strong>the</strong> problems it selected; logicians (along with computer scientists) wereconsulted as to <strong>the</strong> importance of <strong>the</strong> P = (?) NP problem.3
SAB: Thank you for joining us today. CH is a prima-facie candid<strong>at</strong>e to be chosen for <strong>the</strong>new problem and your inform<strong>at</strong>ion and advice will be very helpful in deciding whe<strong>the</strong>rwe should do so. Please tell me more about why it is important and wh<strong>at</strong> efforts havebeen made to solve it.EST: Set <strong>the</strong>ory is generally accepted to be <strong>the</strong> found<strong>at</strong>ion of all ma<strong>the</strong>m<strong>at</strong>ics, and this isone of <strong>the</strong> most basic problems in Cantor’s <strong>the</strong>ory of transfinite cardinals which led to hisdevelopment of set <strong>the</strong>ory starting in <strong>the</strong> 1880s. Hilbert recognized its importance veryquickly and in 1900 placed it in first position in his famous list of ma<strong>the</strong>m<strong>at</strong>icalproblems. As to <strong>the</strong> problem itself, soon after Cantor had shown th<strong>at</strong> <strong>the</strong> continuum isuncountable in <strong>the</strong> early 1880s, he tried but failed to show th<strong>at</strong> every uncountable subsetof <strong>the</strong> continuum has <strong>the</strong> same power as <strong>the</strong> continuum. Th<strong>at</strong>’s one form of CH,sometimes called <strong>the</strong> Weak <strong>Continuum</strong> Hypo<strong>the</strong>sis, <strong>the</strong> more usual form being <strong>the</strong>א = 0א st<strong>at</strong>ement 2 1 ; <strong>the</strong> two are equivalent assuming <strong>the</strong> Axiom of Choice (AC). [ESTcontinues with an account of <strong>the</strong> early history of work on CH beginning with Cantorthrough Sierpinski and Luzin in <strong>the</strong> mid 1930s; cf. Moore (1988, 2011).]SAB: Despite all th<strong>at</strong> work, <strong>the</strong> quote you give from Gödel [above] says th<strong>at</strong> nothing waslearned about <strong>the</strong> cardinality of <strong>the</strong> continuum beyond its uncountability and König’s<strong>the</strong>orem. <strong>Is</strong> th<strong>at</strong> fair? And wh<strong>at</strong>’s happened since he wrote th<strong>at</strong> in 1947?EST: Well, in th<strong>at</strong> quote Gödel did not mention a rel<strong>at</strong>ed problem th<strong>at</strong> Cantor had alsoiniti<strong>at</strong>ed, when he showed th<strong>at</strong> every uncountable closed subset X of <strong>the</strong> continuum has<strong>the</strong> perfect set property, i.e. contains a perfect subset and hence is of <strong>the</strong> power of <strong>the</strong>continuum; we also say th<strong>at</strong> CH holds of X. The question <strong>the</strong>n became, which sets haveth<strong>at</strong> property? Th<strong>at</strong> was actively pursued in <strong>the</strong> 1930s in <strong>the</strong> Russian school ofDescriptive Set Theory (DST, <strong>the</strong> study of definable sets of reals and of o<strong>the</strong>r topologicalspaces) led by Luzin. The strongest result he obtained, toge<strong>the</strong>r with his student Suslin,is th<strong>at</strong> all ∑ 1 1 (or “analytic” sets) have <strong>the</strong> perfect subset property. The ∑ 1 1 sets of realsare <strong>the</strong> projections of 2-dimensional Borel sets, and th<strong>at</strong> is <strong>the</strong> first level of sets in <strong>the</strong>projective hierarchy, which are gener<strong>at</strong>ed by altern<strong>at</strong>ing projection (giving <strong>the</strong> ∑ 1 n sets)and complement<strong>at</strong>ion (giving <strong>the</strong> ∏ 1 n sets). The workers in DST were unable to showth<strong>at</strong> <strong>the</strong> ∏ 1 1 sets have <strong>the</strong> perfect set property. The reason for th<strong>at</strong> was explained by4
Gödel (1938, 1940) who showed th<strong>at</strong> <strong>the</strong>re are uncountable ∏ 1 1 sets th<strong>at</strong> do not have <strong>the</strong>perfect set property in his constructible sets model L of ZFC + GCH. 3SAB: So <strong>the</strong> analytic sets would seem to be as far as one could go with <strong>the</strong> perfect setproperty.EST: Actually not, since it could equally well be consistent with ZFC and o<strong>the</strong>r axiomsth<strong>at</strong> not only ∏ 1 1 sets but all uncountable sets in <strong>the</strong> projective hierarchy have <strong>the</strong> perfectset property. And th<strong>at</strong> turned out to be <strong>the</strong> case through a surprising route, namely <strong>the</strong>use of a st<strong>at</strong>ement called <strong>the</strong> Axiom of Determinacy (AD) introduced in <strong>the</strong> early 1980sby Mycielski and Steinhaus. Th<strong>at</strong> is very powerful because it proves such things as th<strong>at</strong>all sets of reals are Lebesgue measurable, contradicting AC. So set <strong>the</strong>orists don’t acceptAD because AC is absolutely basic to set <strong>the</strong>ory and is intuitively true of <strong>the</strong> domain ofarbitrary sets. But set <strong>the</strong>orists realized th<strong>at</strong> rel<strong>at</strong>ivized forms of AD could be true andhave important consequences.SAB: Tell me⎯but first explain wh<strong>at</strong> AD is.EST: Associ<strong>at</strong>ed with any subset X of 2 N is a two person infinite game G X . Beginningwith player I, each chooses in altern<strong>at</strong>ion a 0 or a 1. At <strong>the</strong> “end” of play, one has asequence σ in 2 N ; player I wins if σ belongs to X while player II wins if not. AD for Xsays th<strong>at</strong> one player or <strong>the</strong> o<strong>the</strong>r has a winning str<strong>at</strong>egy for G X , and AD holds for a classΓ of sets if it holds for each X in Γ. Even though AD is not true for <strong>the</strong> class of allsubsets of <strong>the</strong> continuuum, it could well be true for substantial subclasses Γ of th<strong>at</strong>. Infact Martin (1975) proved in ZFC th<strong>at</strong> AD holds for <strong>the</strong> Borel sets and <strong>the</strong>n movedbeyond th<strong>at</strong> to study AD for sets in <strong>the</strong> projective hierarchy. Projective Determinacy(PD) st<strong>at</strong>es th<strong>at</strong> AD holds for <strong>the</strong> class of all projective sets. PD turns out to have manysignificant consequences including th<strong>at</strong> (i) every projective set is Lebesgue measurable,(ii) has <strong>the</strong> Baire property, and (iii) if uncountable contains a perfect subset. And, in3 ZFC consists of <strong>the</strong> usual Zermelo-Fraenkel axioms of set <strong>the</strong>ory (ZF) expanded by <strong>the</strong>Axiom of Choice (AC), and GCH is <strong>the</strong> Generalized <strong>Continuum</strong> Hypo<strong>the</strong>sis in <strong>the</strong> formth<strong>at</strong> for all ordinals α, 2 αא = א α+1 .5
1989, Tony Martin and John Steel succeeded in obtaining <strong>the</strong> stunning result th<strong>at</strong> PDholds; <strong>the</strong>ir proof was published in <strong>the</strong> Journal of <strong>the</strong> American Ma<strong>the</strong>m<strong>at</strong>ical Societyunder <strong>the</strong> title “A proof of projective determinacy” (Martin and Steel 1989).SAB: Th<strong>at</strong> sounds pretty impressive and like real progress. So wh<strong>at</strong> you’re telling me isth<strong>at</strong> not only is PD consistent with ZFC but it’s true, though I guess it can’t be true in Lfrom wh<strong>at</strong> you told me before.EST: You’re right about <strong>the</strong> l<strong>at</strong>ter, but something more has to be said about Martin andSteel’s proof to assert th<strong>at</strong> PD is true. They don’t just use <strong>the</strong> ordinary axioms of set<strong>the</strong>ory; wh<strong>at</strong> <strong>the</strong>y actually show (and are quite explicit about) is th<strong>at</strong> PD follows fromZFC plus <strong>the</strong> assumption th<strong>at</strong> <strong>the</strong>re exist infinitely “Woodin cardinals” with a measurablecardinal above all of <strong>the</strong>m. In fact, Woodin streng<strong>the</strong>ned th<strong>at</strong> to showing th<strong>at</strong> under <strong>the</strong>same assumption, AD holds in L(R), <strong>the</strong> constructible sets rel<strong>at</strong>ive to <strong>the</strong> real numbers R(cf. Koellner 2010, 205). Incidentally, he also showed th<strong>at</strong> if AD holds in L(R), <strong>the</strong>n it isconsistent with ZFC th<strong>at</strong> <strong>the</strong>re are infinitely many Woodin cardinals (ibid.).SAB: I know wh<strong>at</strong> a measurable cardinal is supposed to be since it was introduced byUlam, and happen to know th<strong>at</strong> Scott showed <strong>the</strong>re are no measurable cardinals in L; on<strong>the</strong> o<strong>the</strong>r hand, lots of large cardinals like those due to Mahlo are consistent with V = L,so <strong>the</strong> existence of measurables is pretty strong. But wh<strong>at</strong> are Woodin cardinals? Andwhen are you going to tell me about <strong>the</strong> st<strong>at</strong>us of CH itself?EST: First of all, <strong>the</strong> large cardinals like those of Mahlo, etc., th<strong>at</strong> are consistent with V =L are sometimes called “small” large cardinals, while those like <strong>the</strong> measurables andbeyond th<strong>at</strong> are inconsistent with V = L are called “large” large cardinals. Woodincardinals are among <strong>the</strong> l<strong>at</strong>ter; <strong>the</strong>y are loc<strong>at</strong>ed in a hierarchy of large large cardinalsabove measurable cardinals and below supercompact cardinals and are given by a ra<strong>the</strong>rtechnical definition (cf. Kanamori 1994, 471).SAB: You say <strong>the</strong> assumption <strong>the</strong>y exist was made by Martin and Steel, so th<strong>at</strong> seems tosuggest th<strong>at</strong> <strong>the</strong>ir existence hasn’t been proved. Or is it intuitively clear th<strong>at</strong> <strong>the</strong>irexistence should be accepted?6
would imply th<strong>at</strong> <strong>the</strong> cardinal number of <strong>the</strong> continuum is א 2 But <strong>the</strong> explan<strong>at</strong>ion of th<strong>at</strong>would take a bit more time (cf. also Koellner 2010, 212 ff).SAB: Hmm. I won’t ask you to explain th<strong>at</strong> to us, or to ask how one would convinceoneself of its truth, if even LLCAs are not enough. Anyhow, our time is up, and thankyou for all this valuable inform<strong>at</strong>ion and advice.Next!*********Wh<strong>at</strong> would be difficult for <strong>the</strong> SAB as represent<strong>at</strong>ives of <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ical establishmentin this imagined interview about whe<strong>the</strong>r to add CH to its list is th<strong>at</strong> <strong>the</strong> usual idea ofma<strong>the</strong>m<strong>at</strong>ical truth in its ordinary sense is no longer oper<strong>at</strong>ive in <strong>the</strong> research programs ofMartin, Steel, Woodin, et al. which, ra<strong>the</strong>r, are proceeding on <strong>the</strong> basis of wh<strong>at</strong> seem tobe highly unusual (one might even say, metaphysical) assumptions. And even though <strong>the</strong>experts in set <strong>the</strong>ory may find such assumptions compelling from <strong>the</strong>ir experience ofworking with <strong>the</strong>m and through <strong>the</strong> kind of plausibility arguments indic<strong>at</strong>ed above, <strong>the</strong>likelihood of <strong>the</strong>ir being accepted by <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ical community <strong>at</strong> large is practicallynil. So if a resolution of CH were to come out of a pursuit of <strong>the</strong>se programs,ma<strong>the</strong>m<strong>at</strong>icians outside of set <strong>the</strong>ory would have no way of judging <strong>the</strong> truth or falsity ofCH on th<strong>at</strong> basis. Thus, given our present understanding of <strong>the</strong> st<strong>at</strong>us of CH, it would notbe a good bet for <strong>the</strong> SAB to add it to its Prize list.The situ<strong>at</strong>ion is not <strong>at</strong> all like th<strong>at</strong> of <strong>the</strong> case of AC, which was long resisted bysignificant parts of <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ics community, but came to be accepted by <strong>the</strong> vastmajority when Zermelo’s arguments (and those of o<strong>the</strong>rs) sank in: namely AC is both asimple intuitively true st<strong>at</strong>ement about <strong>the</strong> universe of arbitrary sets (granted <strong>the</strong> conceptof such) and its use underlies many common informal ma<strong>the</strong>m<strong>at</strong>ical argumentspreviously considered unobjectionable even by <strong>the</strong> critics of AC (cf. Moore 1982) .Nei<strong>the</strong>r of <strong>the</strong>se applies to <strong>the</strong> extraordinary set <strong>the</strong>oretical hypo<strong>the</strong>ses in question here.Of course, none of this by itself establishes th<strong>at</strong> CH is not a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>icalproblem, but it surely has to give one pause and ask if <strong>the</strong> concepts of arbitrary set and8
function th<strong>at</strong> are essential to its formul<strong>at</strong>ion are indeed as <strong>definite</strong> as one thought, despite<strong>the</strong>ir ubiquity in modern ma<strong>the</strong>m<strong>at</strong>ics. For th<strong>at</strong> we have to dig deeper into <strong>the</strong>philosophical presumptions of set <strong>the</strong>ory within a view of <strong>the</strong> n<strong>at</strong>ure of ma<strong>the</strong>m<strong>at</strong>icaltruth more generally. If one is not a formalist, finitist, constructivist, or predic<strong>at</strong>ivist, etc.,wh<strong>at</strong> are <strong>the</strong> options? The well-known difficulties of pl<strong>at</strong>onism have left it with few ifany adherents, though it has re-emerged in some forms of ma<strong>the</strong>m<strong>at</strong>ical structuralism.O<strong>the</strong>rs have retre<strong>at</strong>ed to a defl<strong>at</strong>ionary view of ma<strong>the</strong>m<strong>at</strong>ical truth (e.g., Burgess inMaddy 2005, 361-362, but th<strong>at</strong> leaves us just as much <strong>at</strong> sea when we ask whe<strong>the</strong>r or notCH is true. Still o<strong>the</strong>rs would put philosophical consider<strong>at</strong>ions aside in favor of <strong>the</strong>judgment of working ma<strong>the</strong>m<strong>at</strong>icians: “ma<strong>the</strong>m<strong>at</strong>ics is as ma<strong>the</strong>m<strong>at</strong>ics does.” Someamong those (e.g. Maddy 1997, 2005) combine th<strong>at</strong> with specifically methodologicalguidelines in <strong>the</strong> case of set <strong>the</strong>ory, with slogans like “maximize”, but th<strong>at</strong> does nothingto tell us why we should accept something as true if it is <strong>the</strong> result of such.2. Conceptual Structuralism 42(a) Ma<strong>the</strong>m<strong>at</strong>ical structuralism and structuralist philosophiesMa<strong>the</strong>m<strong>at</strong>ical practice has been increasingly domin<strong>at</strong>ed by structuralist views since <strong>the</strong>beginning of <strong>the</strong> 20th century. Their explicit inception in <strong>the</strong> 19th century is oftencredited to Dedekind. But I would argue th<strong>at</strong> ma<strong>the</strong>m<strong>at</strong>icians always regarded <strong>the</strong>irsubject m<strong>at</strong>ter in structuralist terms, if only implicitly, since <strong>the</strong>ir concern throughout wasnot with wh<strong>at</strong> ma<strong>the</strong>m<strong>at</strong>ical objects “really are”, but with how <strong>the</strong>y rel<strong>at</strong>e to each o<strong>the</strong>r,and how <strong>the</strong>y can be a subject of system<strong>at</strong>ic comput<strong>at</strong>ion and reasoning. True, one had<strong>the</strong> successive questions: Wh<strong>at</strong> is zero?, Wh<strong>at</strong> are neg<strong>at</strong>ive quantities?, Wh<strong>at</strong> areirr<strong>at</strong>ional magnitudes?, Wh<strong>at</strong> are imaginary numbers?, Wh<strong>at</strong> are infinitesimal quantities?,etc., etc. But <strong>the</strong> “wh<strong>at</strong>” had to do with <strong>the</strong> problem as to how one could coherently fitsystem<strong>at</strong>ic use of <strong>the</strong>se successively new ideas with wh<strong>at</strong> had previously been recognizedas ma<strong>the</strong>m<strong>at</strong>ically meaningful. Hilbert is of course famous for his early espousal of anexplicit structuralist point of view in his work on <strong>the</strong> found<strong>at</strong>ions of geometry. The gre<strong>at</strong>4 This section of <strong>the</strong> paper is largely taken from Feferman (2009a, 2010a).9
contemporary of Hilbert, Henri Poincaré, though known for his emphasis on quitedifferent, more intuitive aspects of ma<strong>the</strong>m<strong>at</strong>ics from Hilbert, also voiced a structuralistview of <strong>the</strong> subject:Ma<strong>the</strong>m<strong>at</strong>icians do not study objects, but <strong>the</strong> rel<strong>at</strong>ions between objects; to <strong>the</strong>m it isa m<strong>at</strong>ter of indifference if those objects are replaced by o<strong>the</strong>rs, provided th<strong>at</strong> <strong>the</strong>rel<strong>at</strong>ions do not change. M<strong>at</strong>ter does not engage <strong>the</strong>ir <strong>at</strong>tention, <strong>the</strong>y are interestedby form alone. (Poincaré 1952, 20)In <strong>the</strong> 20th century <strong>the</strong> Bourbaki group led <strong>the</strong> way in promoting a system<strong>at</strong>icstructuralist approach to ma<strong>the</strong>m<strong>at</strong>ics, and th<strong>at</strong> has been continued in a specific way into<strong>the</strong> 21st century by <strong>the</strong> c<strong>at</strong>egory <strong>the</strong>orists led by Saunders Mac Lane.Structuralist philosophies of ma<strong>the</strong>m<strong>at</strong>ics have emerged in <strong>the</strong> last thirty years as acompetitor to <strong>the</strong> traditional philosophies of ma<strong>the</strong>m<strong>at</strong>ics. An early expression is foundin <strong>the</strong> famous article by Benacerraf, “Wh<strong>at</strong> numbers could not be” (1965). System<strong>at</strong>icefforts <strong>at</strong> <strong>the</strong> development of a structuralist philosophy of ma<strong>the</strong>m<strong>at</strong>ics have been givenin Hellman (1989), Resnik (1997), Shapiro (1997) and Chihara (2004), among o<strong>the</strong>rs; fora survey of some of <strong>the</strong> leading ideas, see Hellman (2005). Closest to <strong>the</strong> views here butwith quite opposite conclusions re CH is <strong>Is</strong>aacson (2008).2(b) The <strong>the</strong>ses of Conceptual StructuralismMy version of such a philosophy, Conceptual Structuralism, is meant to emphasize <strong>the</strong>source of all ma<strong>the</strong>m<strong>at</strong>ical thought in human conceptions. It differs in various respectsfrom <strong>the</strong> ones mentioned and is summarized in <strong>the</strong> following ten <strong>the</strong>ses. 51. The basic objects of ma<strong>the</strong>m<strong>at</strong>ical thought exist only as mental conceptions, though <strong>the</strong>source of <strong>the</strong>se conceptions lies in everyday experience in manifold ways, in <strong>the</strong>processes of counting, ordering, m<strong>at</strong>ching, combining, separ<strong>at</strong>ing, and loc<strong>at</strong>ing in spaceand time.5 I began developing <strong>the</strong>se ideas in <strong>the</strong> l<strong>at</strong>e 1970s and first circul<strong>at</strong>ed <strong>the</strong>m in notes in1978; cf. Feferman (2009a) 170, fn. 2.10
2. Theoretical ma<strong>the</strong>m<strong>at</strong>ics has its source in <strong>the</strong> recognition th<strong>at</strong> <strong>the</strong>se processes areindependent of <strong>the</strong> m<strong>at</strong>erials or objects to which <strong>the</strong>y are applied and th<strong>at</strong> <strong>the</strong>y arepotentially endlessly repe<strong>at</strong>able.3. The basic conceptions of ma<strong>the</strong>m<strong>at</strong>ics are of certain kinds of rel<strong>at</strong>ively simple idealworld-pictures which are not of objects in isol<strong>at</strong>ion but of structures, i.e. coherentlyconceived groups of objects interconnected by a few simple rel<strong>at</strong>ions and oper<strong>at</strong>ions.They are communic<strong>at</strong>ed and understood prior to any axiom<strong>at</strong>ics, indeed prior to anysystem<strong>at</strong>ic logical development.4. Some significant fe<strong>at</strong>ures of <strong>the</strong>se structures are elicited directly from <strong>the</strong> worldpictureswhich describe <strong>the</strong>m, while o<strong>the</strong>r fe<strong>at</strong>ures may be less certain. Ma<strong>the</strong>m<strong>at</strong>icsneeds little to get started and, once started, a little bit goes a long way.5. Basic conceptions differ in <strong>the</strong>ir degree of clarity. One may speak of wh<strong>at</strong> is true in agiven conception, but th<strong>at</strong> notion of truth may only be partial. Truth in full is applicableonly to completely clear conceptions.6. Wh<strong>at</strong> is clear in a given conception is time dependent, both for <strong>the</strong> individual andhistorically.7. Pure (<strong>the</strong>oretical) ma<strong>the</strong>m<strong>at</strong>ics is a body of thought developed system<strong>at</strong>ically bysuccessive refinement and reflective expansion of basic structural conceptions.8. The general ideas of order, succession, collection, rel<strong>at</strong>ion, rule and oper<strong>at</strong>ion are prema<strong>the</strong>m<strong>at</strong>ical;some understanding of <strong>the</strong>m is necessary to <strong>the</strong> understanding ofma<strong>the</strong>m<strong>at</strong>ics.9. The general idea of property is pre-logical; some understanding of th<strong>at</strong> and of <strong>the</strong>logical particles is also a prerequisite to <strong>the</strong> understanding of ma<strong>the</strong>m<strong>at</strong>ics. Thereasoning of ma<strong>the</strong>m<strong>at</strong>ics is in principal logical, but in practice relies to a considerableextent on various forms of intuition in order to arrive <strong>at</strong> understanding and conviction.10. The objectivity of ma<strong>the</strong>m<strong>at</strong>ics lies in its stability and coherence under repe<strong>at</strong>edcommunic<strong>at</strong>ion, critical scrutiny and expansion by many individuals often working11
independently of each o<strong>the</strong>r. Incoherent concepts, or ones which fail to withstand criticalexamin<strong>at</strong>ion or lead to conflicting conclusions are eventually filtered out fromma<strong>the</strong>m<strong>at</strong>ics. The objectivity of ma<strong>the</strong>m<strong>at</strong>ics is a special case of intersubjectiveobjectivity th<strong>at</strong> is ubiquitous in social reality.There is not time to elabor<strong>at</strong>e <strong>the</strong>se points, but various aspects of <strong>the</strong>m will come up inour discussion of two constell<strong>at</strong>ions of structural notions, first of objects gener<strong>at</strong>ed by oneor more “successor” oper<strong>at</strong>ions, and second of <strong>the</strong> continuum.Before digging into <strong>the</strong>se structures, I want to address a common objection to loc<strong>at</strong>ing<strong>the</strong> n<strong>at</strong>ure of ma<strong>the</strong>m<strong>at</strong>ics in human conceptions, namely th<strong>at</strong> it does not account for <strong>the</strong>objectivity of ma<strong>the</strong>m<strong>at</strong>ics. I disagree strongly, and in support of th<strong>at</strong> appeal to <strong>the</strong>objectivity of much of social reality. Th<strong>at</strong> is so pervasive, we are not even aware th<strong>at</strong>much of wh<strong>at</strong> we must deal with in our daily lives is constrained by social institutionsand social facts. In this respect I agree fully with John Searle, in his book, TheConstruction of Social Reality, and can hardly do better than quote him in support of th<strong>at</strong>:[T]here are portions of <strong>the</strong> real world, objective facts in <strong>the</strong> world, th<strong>at</strong> are onlyfacts by human agreement. In a sense <strong>the</strong>re are things th<strong>at</strong> exist only because webelieve <strong>the</strong>m to exist. ... things like money, property, governments, and marriages.Yet many facts regarding <strong>the</strong>se things are ‘objective’ facts in <strong>the</strong> sense th<strong>at</strong> <strong>the</strong>y arenot a m<strong>at</strong>ter of [our] preferences, evalu<strong>at</strong>ions, or moral <strong>at</strong>titudes. (Searle 1995, p.1)Searle goes on to give examples of such facts (<strong>at</strong> <strong>the</strong> time of writing) as th<strong>at</strong> he is acitizen of <strong>the</strong> United St<strong>at</strong>es, th<strong>at</strong> he has a five dollar bill in his pocket, th<strong>at</strong> his youngersister got married on December 14, th<strong>at</strong> he owns a piece of property in Berkeley, and th<strong>at</strong><strong>the</strong> New York Giants won <strong>the</strong> 1991 Superbowl. 6 He might well have added board gamesto <strong>the</strong> list of things th<strong>at</strong> exist only because we believe <strong>the</strong>m to exist, and facts such as th<strong>at</strong>in <strong>the</strong> game of chess, it is not possible to force a checkm<strong>at</strong>e with a king and two knights6 Objective facts for o<strong>the</strong>rs might be th<strong>at</strong> <strong>the</strong>y are not citizens of <strong>the</strong> United St<strong>at</strong>es, or th<strong>at</strong><strong>the</strong>y are not married, or th<strong>at</strong> <strong>the</strong>y are married in some US St<strong>at</strong>es but not in o<strong>the</strong>rs, or th<strong>at</strong><strong>the</strong>y have no money in <strong>the</strong>ir pockets. Purported facts th<strong>at</strong> might be up for dispute are th<strong>at</strong>one is fishing beyond <strong>the</strong> 12 nautical mile territorial w<strong>at</strong>ers off <strong>the</strong> coast of Maine.12
against a lone king. Unlike facts about one’s government, citizenship, finances, property,marital rel<strong>at</strong>ions, and so on, th<strong>at</strong> are vitally important to our daily welfare, since <strong>the</strong>yconstrain one’s actions and determine one’s “rights”, “responsibilities” and “oblig<strong>at</strong>ions”,facts about <strong>the</strong> structure and execution of <strong>at</strong>hletic games and board games are notessential to our well-being even though <strong>the</strong>y may engage us passion<strong>at</strong>ely. In this respect,ma<strong>the</strong>m<strong>at</strong>ics is akin to games; <strong>the</strong> fact th<strong>at</strong> <strong>the</strong>re are infinitely many prime numbers is anexample of a fact th<strong>at</strong> is about our conception of <strong>the</strong> integers, a conception th<strong>at</strong> is as clearas wh<strong>at</strong> we mean, for example, by <strong>the</strong> game of chess or <strong>the</strong> game of go.Searle’s main concern is to answer <strong>the</strong> question, “How can <strong>the</strong>re be an objective realityth<strong>at</strong> exists in part by human agreement?”, and his book is devoted to giving a specificaccount of <strong>the</strong> n<strong>at</strong>ure of certain kinds of social facts, and of wh<strong>at</strong> makes <strong>the</strong>m true. Th<strong>at</strong>account is open to criticism in various respects, but wh<strong>at</strong> is not open to criticism is wh<strong>at</strong> itis supposed to be an account of. Th<strong>at</strong> is, we must take for granted <strong>the</strong> phenomenon ofintersubjective objectivity about many kinds of social constructions, be <strong>the</strong>ygovernments, money, property, marriages, games, and so on. (It should be stressed th<strong>at</strong>we are not dealing with <strong>the</strong> in many respects antiscientific viewpoint of socialconstructivism in post-modernist and deconstructionist thought.) My claim is th<strong>at</strong> <strong>the</strong>basic conceptions of ma<strong>the</strong>m<strong>at</strong>ics and <strong>the</strong>ir elabor<strong>at</strong>ion are also social constructions andth<strong>at</strong> <strong>the</strong> objective reality th<strong>at</strong> we ascribe to ma<strong>the</strong>m<strong>at</strong>ics is simply <strong>the</strong> result ofintersubjective objectivity about those conceptions and not about a supposed independentreality in any pl<strong>at</strong>onistic sense. Also, this view of ma<strong>the</strong>m<strong>at</strong>ics does not require totalrealism about truth values. Th<strong>at</strong> is, it may simply be undecided under a given conceptionwhe<strong>the</strong>r a given st<strong>at</strong>ement in <strong>the</strong> language of th<strong>at</strong> conception has a determin<strong>at</strong>e truthvalue, just as, for example, our conception of <strong>the</strong> government of <strong>the</strong> United St<strong>at</strong>es isunderdetermined as to <strong>the</strong> presidential line of succession past a certain point. 72(c) Conceptions of Sequential Gener<strong>at</strong>ionThe most primitive ma<strong>the</strong>m<strong>at</strong>ical conception is th<strong>at</strong> of <strong>the</strong> positive integer sequence7 According to Wikipedia, th<strong>at</strong> is now specified up to #17, <strong>the</strong> Secretary of HomelandSecurity.13
epresented by <strong>the</strong> tallies: |, ||, |||, ... . Ma<strong>the</strong>m<strong>at</strong>ics begins when we conceive of <strong>the</strong>se asbeing gener<strong>at</strong>ed from a fixed initial unit by repe<strong>at</strong>edly associ<strong>at</strong>ing with each term n aunique successor Sc(n), without bound; <strong>the</strong> objects thus gener<strong>at</strong>ed form <strong>the</strong> collection N +of positive integers. Implicit in this conception is <strong>the</strong> order rel<strong>at</strong>ion m < n, which holdswhen m precedes n in <strong>the</strong> gener<strong>at</strong>ion of N + . Our primitive conception is thus th<strong>at</strong> of astructure (N + , 1, Sc,
th<strong>at</strong> <strong>the</strong>y are true or false, independently of whe<strong>the</strong>r we can establish <strong>the</strong>m one way or <strong>the</strong>o<strong>the</strong>r. In o<strong>the</strong>r words, in more modern terms, we should accept realism in truth valuesfor st<strong>at</strong>ements about this structure; <strong>the</strong>n <strong>the</strong> applic<strong>at</strong>ion of classical logic in reasoningabout such st<strong>at</strong>ements is autom<strong>at</strong>ically legitimized. Thus, despite <strong>the</strong> subjective source of<strong>the</strong> positive integer structure in <strong>the</strong> collective human understanding, <strong>the</strong>re is no reason torestrict oneself to intuitionistic logic on subjectivist grounds.I have described in Feferman (2009a) how reflection on <strong>the</strong> structure of positive integersleads us to conceive of and reduce to it <strong>the</strong> concepts of <strong>the</strong> arithmetical structures on <strong>the</strong>n<strong>at</strong>ural numbers N, <strong>the</strong> integers Z and <strong>the</strong> r<strong>at</strong>ional numbers Q, for which truth in <strong>the</strong>sestructures is a <strong>definite</strong> notion. But it should be emphasized th<strong>at</strong> <strong>the</strong> general scheme ofinduction,P(0) ∧ ∀n[P(n) → P(Sc(n))] → ∀n P(n),needs to be understood in an open-ended sense, by which I mean th<strong>at</strong> it is accepted forany <strong>definite</strong> property P of n<strong>at</strong>ural numbers th<strong>at</strong> one meets in <strong>the</strong> process of doingma<strong>the</strong>m<strong>at</strong>ics, no m<strong>at</strong>ter wh<strong>at</strong> <strong>the</strong> subject m<strong>at</strong>ter and wh<strong>at</strong> <strong>the</strong> notions used in <strong>the</strong>formul<strong>at</strong>ion of P. 8 The question⎯Wh<strong>at</strong> is a <strong>definite</strong> property?⎯may require in eachinstance <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>ician’s judgment. For example, <strong>the</strong> property,”n is an odd perfectnumber”, is a <strong>definite</strong> property, while “n is a feasibly representable number” is not. Wh<strong>at</strong>about “GCH is true <strong>at</strong> n”?Returning to <strong>the</strong> starting point of finite gener<strong>at</strong>ion under a successor oper<strong>at</strong>ion, it is alson<strong>at</strong>ural to conceive of gener<strong>at</strong>ion under more than one such oper<strong>at</strong>ion, Sc a where a is anelement of an index collection A. We may conceive of <strong>the</strong> objects of <strong>the</strong> resultingstructure as “words on <strong>the</strong> alphabet A”, with Sc a(w) = wa in <strong>the</strong> sense of conc<strong>at</strong>en<strong>at</strong>ion.The ma<strong>the</strong>m<strong>at</strong>ics of this conception is simply <strong>the</strong> <strong>the</strong>ory of conc<strong>at</strong>en<strong>at</strong>ion, which is8 In Feferman (1996) I introduced <strong>the</strong> concept of <strong>the</strong> unfolding of an open-ended systemT based on axioms in <strong>the</strong> usual sense of <strong>the</strong> word and open-ended schem<strong>at</strong>a, tocharacterize wh<strong>at</strong> oper<strong>at</strong>ions and rel<strong>at</strong>ions, and wh<strong>at</strong> principles concerning, <strong>the</strong>m ought toaccepted if one has accepted those provided by T. In Feferman and Strahm (2000) weshowed th<strong>at</strong> <strong>the</strong> unfolding of schem<strong>at</strong>ic number <strong>the</strong>ory is of proof-<strong>the</strong>oretical strength <strong>the</strong>same as predic<strong>at</strong>ive analysis.15
equivalent to <strong>the</strong> <strong>the</strong>ory of n<strong>at</strong>ural numbers when A has a single element. Of specialinterest below is <strong>the</strong> case th<strong>at</strong> A = {0, 1}, for which we also conceive of <strong>the</strong> words on Aas <strong>the</strong> finite p<strong>at</strong>hs in <strong>the</strong> binary branching tree, altern<strong>at</strong>ively as <strong>the</strong> end nodes of suchp<strong>at</strong>hs.2 (d). Conceptions of <strong>the</strong> continuumThe question of <strong>the</strong> determin<strong>at</strong>eness of CH in its weak form depends on our conceptionof three things: (1) <strong>the</strong> continuum, (2) subsets of <strong>the</strong> continuum, and (3) mappingsbetween such subsets. As to (1), <strong>the</strong>re is in fact no unique concept, but ra<strong>the</strong>r severalrel<strong>at</strong>ed ones, and part of <strong>the</strong> problem whe<strong>the</strong>r CH is a genuine ma<strong>the</strong>m<strong>at</strong>ical problem isth<strong>at</strong> <strong>the</strong>re is a certain amount of confl<strong>at</strong>ion of <strong>the</strong>se concepts with a resulting muddying of<strong>the</strong> picture. I shall be rel<strong>at</strong>ively brief about each of <strong>the</strong>se since I have expanded on <strong>the</strong>min Feferman (2009a).(i) The Euclidean continuum. The conception is semi-structural. Straight lines areconceived of as lying in <strong>the</strong> plane or in space. They are perfectly straight and have nobreadth. Points are perfectly fine; <strong>the</strong>y are conceived of as <strong>the</strong> objectific<strong>at</strong>ion of pureloc<strong>at</strong>ions. A point may lie on a line, but <strong>the</strong> line does not consist of <strong>the</strong> set of its points.The idea of arbitrary subset of a line or of <strong>the</strong> removal of some points from a line isforeign to Euclidean geometry.(ii) The Hilbertian continuum. The conception is structural: one has points, lines, planes,incidence, betweenness and equidistance rel<strong>at</strong>ions. Continuity is expressed set<strong>the</strong>oreticallyfollowing Dedekind.(iii) The Dedekind real line. This is a hybrid (arithmetical/geometric) notion th<strong>at</strong> can beexpressed in structural terms rel<strong>at</strong>ive to <strong>the</strong> r<strong>at</strong>ionals. Continuity is expressed by <strong>the</strong>Dedekind cut condition.(iv) The Cauchy-Cantor real line. This is again a hybrid structural notion rel<strong>at</strong>ive to Q.Continuity is expressed by <strong>the</strong> Cauchy convergence condition, used by Cantor to“construct” <strong>the</strong> reals.16
S(A) is a set, and hence th<strong>at</strong> it is a <strong>definite</strong> totality, so th<strong>at</strong> <strong>the</strong> oper<strong>at</strong>ion of quantific<strong>at</strong>ionover S(A), i.e. (∀X⊆A) P(X), has a determin<strong>at</strong>e truth value for each property P(X) th<strong>at</strong>has a determin<strong>at</strong>e truth value for each X ⊆ A. This is <strong>the</strong> usual conception of S(A) in set<strong>the</strong>ory, whe<strong>the</strong>r informal or axiom<strong>at</strong>ic. However, one may altern<strong>at</strong>ively consider S(A) tobe an in<strong>definite</strong> collection whose members are subsets of A, but whose exact extent isindetermin<strong>at</strong>e. To distinguish <strong>the</strong> two views of S(A), I will refer to <strong>the</strong> former as <strong>the</strong> set<strong>the</strong>oreticalone and <strong>the</strong> l<strong>at</strong>ter as <strong>the</strong> open-ended one. The set-<strong>the</strong>oretical view of S(A) iscertainly justified when A is finite; if A has n elements <strong>the</strong> 2 n elements of S(A) can becompletely listed (<strong>at</strong> least in principle). So <strong>the</strong> only case where <strong>the</strong> distinction would besignificant is when A is infinite, and <strong>the</strong> first case in which th<strong>at</strong> is met is for A = N, <strong>the</strong>set of n<strong>at</strong>ural numbers.The set-<strong>the</strong>oretical view of S(A) is justified by a thorough-going pl<strong>at</strong>onism 9 , as acceptedby Gödel. According to this view, sets in general have an existence independent ofhuman thoughts and constructions, and in particular, for any set A, S(A) is <strong>the</strong> <strong>definite</strong>totality of arbitrary subsets of A. A major problem with this view is <strong>the</strong> classic one ofepistemic access raised most famously by Paul Benacerraf (1965). Wh<strong>at</strong> <strong>the</strong> pl<strong>at</strong>onisticset-<strong>the</strong>oretical view has going for it, among o<strong>the</strong>r things, is <strong>the</strong> supposed determin<strong>at</strong>enessof st<strong>at</strong>ements about S(A) involving quantific<strong>at</strong>ion over S(A). The open-ended view, on<strong>the</strong> o<strong>the</strong>r hand, rests on conceptual structuralism, for which epistemic access is noproblem, but <strong>the</strong> determin<strong>at</strong>eness of various st<strong>at</strong>ements <strong>the</strong>n comes into question.So, now, on <strong>the</strong> set <strong>the</strong>oretical point of view, <strong>the</strong> power set oper<strong>at</strong>ion can <strong>at</strong> least beiter<strong>at</strong>ed finitely often and in particular both S(N) and S(S(N)) are <strong>definite</strong> totalities. Theweak form of CH takes <strong>the</strong> form: for every subset X of S(N) ei<strong>the</strong>r X is in 1-1correspondence with N or it is in 1-1 correspondence with S(N). In order for this to be adetermin<strong>at</strong>e question, both S(N) and S(S(N)) have to be <strong>definite</strong> totalities, and for eachsubset X of S(N) <strong>the</strong>re has to be a <strong>definite</strong> totality of all possible 1-1 functions whosedomain is X and whose range is N or S(N). Prima facie, <strong>the</strong> assumption of all <strong>the</strong>se<strong>definite</strong> totalities is only justified on <strong>the</strong> grounds of pl<strong>at</strong>onistic realism. One can try to9 Or Robust Realism, as Maddy (2005) terms it.18
sidestep th<strong>at</strong> by posing <strong>the</strong> question of CH within an axiom<strong>at</strong>ic <strong>the</strong>ory T of sets, but <strong>the</strong>none can only speak of CH as being a determin<strong>at</strong>e question rel<strong>at</strong>ive to T.In <strong>the</strong> familiar <strong>the</strong>ories T of sets, not only is it accepted th<strong>at</strong> <strong>the</strong> power set oper<strong>at</strong>ion canbe iter<strong>at</strong>ed finitely often, it is posited th<strong>at</strong> it can be iter<strong>at</strong>ed transfinitely often in acumul<strong>at</strong>ive way. This is a new idea of sets, not implicit in wh<strong>at</strong> has been said so far,which expands <strong>the</strong> concept of arbitrary set to th<strong>at</strong> of a “universe” V of “all sets” with <strong>the</strong>usual picture of V as fanning out from <strong>the</strong> class On of “all ordinals”, sliced by <strong>the</strong> sets V α<strong>at</strong> each α in On, where V 0 is <strong>the</strong> empty set, each V α+1 = S(V α ) and for limit α, V α = <strong>the</strong>union of all V β for β < α. On this picture, N is represented as a subset of <strong>the</strong> hereditarilyfinite sets V ω , and <strong>the</strong>n S(N) ⊆V ω+1 , and S(S(N)) ⊆V ω+2 . There is no problem to putoneself in <strong>the</strong> mental frame of mind of “this is wh<strong>at</strong> <strong>the</strong> cumul<strong>at</strong>ive hierarchy looks like”,for which one can see th<strong>at</strong> such and such propositions including <strong>the</strong> axioms of ZFC are(more or less) obviously true. I have taught set <strong>the</strong>ory many times and have presented itin terms of this ideal-world picture with only <strong>the</strong> cave<strong>at</strong> th<strong>at</strong> this is wh<strong>at</strong> things aresupposed to be like in th<strong>at</strong> world, ra<strong>the</strong>r than to assert th<strong>at</strong> th<strong>at</strong>’s <strong>the</strong> way <strong>the</strong> worldactually is. I am no more uncomfortable doing th<strong>at</strong> than if I were teaching Newtonianmechanics and talking of a world in which space-time is Euclidean, objects are pointmasses, and we have action <strong>at</strong> a distance, e.g. under gravit<strong>at</strong>ional force, or altern<strong>at</strong>ively ifI were teaching rel<strong>at</strong>ivistic mechanics. And one can see why little more is neededconceptually to develop an extraordinary part of set <strong>the</strong>ory; as in <strong>the</strong> case of number<strong>the</strong>ory, a little bit goes a long way. The issue is not whe<strong>the</strong>r we can think in such termsbut, ra<strong>the</strong>r, wh<strong>at</strong> <strong>the</strong> philosophical st<strong>at</strong>us is of th<strong>at</strong> kind of thought.How does this look from <strong>the</strong> viewpoint of conceptual structuralism? According to <strong>the</strong>ideal-world story th<strong>at</strong> has been told us: whe<strong>the</strong>r looked <strong>at</strong> directly over N or as part ofV ω+2 , S(S(N)) is a <strong>definite</strong> totality, quantific<strong>at</strong>ion over it is well-determined, and CH is a<strong>definite</strong> problem. But saying th<strong>at</strong> <strong>the</strong>se things are conceived as being <strong>definite</strong> does notmake <strong>the</strong>m <strong>definite</strong>. Daniel <strong>Is</strong>aacson espouses a form of conceptual structuralism inwhich he tries to make <strong>the</strong> case for th<strong>at</strong>. He writes:The basis of ma<strong>the</strong>m<strong>at</strong>ics is conceptual and epistemological, not ontological, andunderstanding particular ma<strong>the</strong>m<strong>at</strong>ical structures is prior to axiom<strong>at</strong>ic19
characteriz<strong>at</strong>ion. When such a resulting axiom<strong>at</strong>iz<strong>at</strong>ion is c<strong>at</strong>egorical, a particularma<strong>the</strong>m<strong>at</strong>ical structure is established. Particular ma<strong>the</strong>m<strong>at</strong>ical structures are notma<strong>the</strong>m<strong>at</strong>ical objects. They are characteriz<strong>at</strong>ions. (<strong>Is</strong>aacson 2008, 31)So <strong>Is</strong>aacson makes a very strong demand on conceptions of certain particular structures,namely th<strong>at</strong> <strong>the</strong>y have c<strong>at</strong>egorical axiom<strong>at</strong>iz<strong>at</strong>ions, i.e. <strong>the</strong>y are exemplified in <strong>the</strong>conceived structure and any two structures th<strong>at</strong> exemplify <strong>the</strong>m are isomorphic. Now <strong>the</strong>only axiom<strong>at</strong>iz<strong>at</strong>ions in sight th<strong>at</strong> will do this for number <strong>the</strong>ory and set <strong>the</strong>ory areformul<strong>at</strong>ed in 2nd order logic where <strong>the</strong> quantifiers are interpreted as ranging overarbitrary subsets of <strong>the</strong> domains of <strong>the</strong> particular structures considered. Certainly nocharacteriz<strong>at</strong>ions in 1st order logic will do <strong>the</strong> trick, by <strong>the</strong> Löwenheim-Skolem results.And indeed, <strong>the</strong>re are 2nd order characteriz<strong>at</strong>ions not only of <strong>the</strong> n<strong>at</strong>ural numbers, but ofeach of <strong>the</strong> structures mentioned above in which CH is formul<strong>at</strong>ed. Moreover, <strong>the</strong>re issuch a characteriz<strong>at</strong>ion of (V α , ∈) for α = <strong>the</strong> first strongly inaccessible ordinal, and th<strong>at</strong>structure is a model of ZFC. All of this is established in ZFC as a background set <strong>the</strong>ory(in <strong>the</strong> last case under <strong>the</strong> assumption th<strong>at</strong> th<strong>at</strong> α exists). 10 Clear as <strong>the</strong>se results are as<strong>the</strong>orems within ZFC about 2nd order ZFC, <strong>the</strong>y are evidently begging <strong>the</strong> question about<strong>the</strong> <strong>definite</strong>ness of <strong>the</strong> above conceptions required for <strong>the</strong> formul<strong>at</strong>ion of CH. <strong>Is</strong>aacsonfollows Kreisel (1971) in <strong>the</strong> argument th<strong>at</strong> CH is a <strong>definite</strong> ma<strong>the</strong>m<strong>at</strong>ical problembecause of such c<strong>at</strong>egoricity results. But in relying on so-called standard 2nd order logicfor this purpose we are presuming <strong>the</strong> <strong>definite</strong>ness of <strong>the</strong> very kinds of notions whose<strong>definite</strong>ness is to be established. 11 The circularity is so p<strong>at</strong>ent, one does not see th<strong>at</strong> anymore can be claimed <strong>the</strong>reby than th<strong>at</strong> <strong>the</strong> <strong>definite</strong>ness of those notions may be acceptedas a m<strong>at</strong>ter of faith. Moreover, <strong>the</strong> argument via c<strong>at</strong>egoricity, which takes structures asobjects of a particular kind, viti<strong>at</strong>es <strong>the</strong> claim th<strong>at</strong> <strong>the</strong> issues are conceptual and10 <strong>Is</strong>aacson makes much of <strong>the</strong> work of Zermelo (1930) in which a general semic<strong>at</strong>egoricityresult is proved for ZFC 2 , i.e. ZFC formul<strong>at</strong>ed in 2nd order terms, and th<strong>at</strong>was reproved in gre<strong>at</strong>er detail by John Shepherdson in his three part paper on innermodels for set <strong>the</strong>ory in <strong>the</strong> J. Symbolic <strong>Logic</strong> 16-18. This takes <strong>the</strong> form th<strong>at</strong> if M 1 , M 2are any two structures s<strong>at</strong>isfying ZFC 2 , M 1 is isomorphically embeddable in M 2 as <strong>at</strong>ransitive submodel, or vice versa.11 Väänänen (2001) has fully deconstructed <strong>the</strong> facile appeal to “standard” 2 nd order logicin this and o<strong>the</strong>r found<strong>at</strong>ional arguments.20
epistemological, not ontological.My own conceptual structuralism forces me to be much more modest. First of all, eachof <strong>the</strong> conceptions (i)-(vi) of <strong>the</strong> continuum is tre<strong>at</strong>ed in its own right, not necessarilyequivalent to <strong>the</strong> o<strong>the</strong>rs in terms of cardinality. Secondly, <strong>the</strong> intuitive distinctionbrought up above between sets and sequences is maintained, with sequences having agre<strong>at</strong>er clarity than sets. For example, we have a much clearer conception of arbitrarysequences of points on <strong>the</strong> Hilbert (or Dedekind, or Cauchy-Cantor) line, or <strong>at</strong> least ofbounded strictly monotone sequence, than we do of arbitrary subsets of <strong>the</strong> line. And th<strong>at</strong>also makes a difference when we come to <strong>the</strong> conceptions (v) of 2 N and (vi) of S(N). Wehave a clearer conception of wh<strong>at</strong> it means to be an arbitrary infinite p<strong>at</strong>h through <strong>the</strong> fullbinary tree than of wh<strong>at</strong> it means to be an arbitrary subset of N, but in nei<strong>the</strong>r case do wehave a clear conception of <strong>the</strong> totality of such p<strong>at</strong>hs, resp. sets. In both cases, ourconception allows us to reason th<strong>at</strong> <strong>the</strong>re is no enumer<strong>at</strong>ion of all infinite p<strong>at</strong>hs through<strong>the</strong> binary tree, resp. of all subsets of N, simply by <strong>the</strong> appropri<strong>at</strong>e diagonal argument.But it is an idealiz<strong>at</strong>ion of our conceptions to speak of 2 N , resp. S(N), as being <strong>definite</strong>totalities. And when we step to S(S(N)) <strong>the</strong>re is a still fur<strong>the</strong>r loss of clarity, but it is just<strong>the</strong> <strong>definite</strong>ness of th<strong>at</strong> th<strong>at</strong> is needed to make <strong>definite</strong> sense of CH.If <strong>the</strong> concept in general of <strong>the</strong> totality of arbitrary subsets of a given infinite set A is not<strong>definite</strong> <strong>the</strong>n how could we sharpen it to make it <strong>definite</strong>? Well, we could say th<strong>at</strong> we aretalking only about those subsets of A th<strong>at</strong> are defined in a certain way⎯but <strong>the</strong>n we areno longer talking about arbitrary subsets. This holds for any proposed sharpening orconstraint on <strong>the</strong> concept of being arbitrary in <strong>the</strong> case of sets. In o<strong>the</strong>r words, <strong>the</strong>concept of <strong>the</strong> totality of arbitrary subsets of A is essentially underdetermined or vague,since any sharpening of it viol<strong>at</strong>es wh<strong>at</strong> <strong>the</strong> concept is supposed to be about. But this isalso why (almost all) set-<strong>the</strong>orists <strong>the</strong>mselves reject Gödel’s Axiom of Constructibility,V = L, th<strong>at</strong> says all sets in <strong>the</strong> cumul<strong>at</strong>ive hierarchy are successively definable in aspecific way.One argument th<strong>at</strong> is made for accepting <strong>the</strong> <strong>definite</strong>ness of <strong>the</strong> continuum according to<strong>the</strong> set-<strong>the</strong>oretical point of view is th<strong>at</strong>, as applied to it in its guise as <strong>the</strong>arithmetical/geometric structure of <strong>the</strong> real line, it is unquestionably needed to develop21
modern analysis, and modern analysis is used in an essential way in <strong>the</strong> formul<strong>at</strong>ion of<strong>the</strong> laws of physical science and deriv<strong>at</strong>ion of <strong>the</strong>ir consequences. It may even be arguedth<strong>at</strong> this shows th<strong>at</strong> <strong>the</strong> real line in <strong>the</strong> set-<strong>the</strong>oretical sense is somehow physicallyexemplified in <strong>the</strong> real world. However, a number of case studies support <strong>the</strong> <strong>the</strong>sis th<strong>at</strong>I advanced in Feferman (1993) th<strong>at</strong> all scientifically applicable ma<strong>the</strong>m<strong>at</strong>ics can becarried out in a completely predic<strong>at</strong>ive way⎯in fact in a certain formal system W th<strong>at</strong>has been shown in Feferman and Jäger (1993, 1996) to be a conserv<strong>at</strong>ive extension ofPeano Arithmetic (PA). The design<strong>at</strong>ion ‘W’ is in honor of Hermann Weyl, who gotpredic<strong>at</strong>ive analysis underway in his groundbreaking work Das Kontinuum (1918), using<strong>the</strong> open-ended sequential (Cauchy-Cantor) view of <strong>the</strong> real numbers ra<strong>the</strong>r than <strong>the</strong>Dedekind point of view. Whe<strong>the</strong>r or not this <strong>the</strong>sis will hold for all future applic<strong>at</strong>ions ofma<strong>the</strong>m<strong>at</strong>ics to physical science, wh<strong>at</strong> this shows is th<strong>at</strong> <strong>the</strong> argument from <strong>the</strong> success ofphysics to <strong>the</strong> reality of <strong>the</strong> set-<strong>the</strong>oretical conception of <strong>the</strong> real numbers is completelyviti<strong>at</strong>ed.Never<strong>the</strong>less, one might argue for an intermedi<strong>at</strong>e position between th<strong>at</strong> of conceptualstructuralism, which rejects <strong>the</strong> continuum as a <strong>definite</strong> totality, and <strong>the</strong> set-<strong>the</strong>oreticalaccount which not only accepts th<strong>at</strong> but also much, much more. Namely, one may grantas a working apparently robust idea <strong>the</strong> concept of S(N), but nothing higher in <strong>the</strong>cumul<strong>at</strong>ive hierarchy. This would justify <strong>the</strong> assumption of Dedekind or Cantorcompleteness of <strong>the</strong> real line with respect to all sets definable by quantific<strong>at</strong>ion over <strong>the</strong>continuum, thus going far beyond predic<strong>at</strong>ive ma<strong>the</strong>m<strong>at</strong>ics into <strong>the</strong> domain of descriptiveset <strong>the</strong>ory. In logical terms, th<strong>at</strong> would justify working in a system of strength full 2 ndorder number <strong>the</strong>ory, or “analysis” as it is justly called. There is no question th<strong>at</strong> all ofextant scientifically applicable ma<strong>the</strong>m<strong>at</strong>ics can be reduced to working in th<strong>at</strong> and in factin its rel<strong>at</strong>ively weak subsystem denoted ∏ 1 1-CA (cf. Simpson 1999, 2010).3. A proposed logical framework for wh<strong>at</strong>’s <strong>definite</strong> (and wh<strong>at</strong>’s not)According to <strong>the</strong> finitists, <strong>the</strong> n<strong>at</strong>ural numbers form an incompleted or in<strong>definite</strong> totalityand <strong>the</strong> general oper<strong>at</strong>ions of universal and existential quantific<strong>at</strong>ion over <strong>the</strong> n<strong>at</strong>uralnumbers are not to be admitted in reasoning about <strong>the</strong>m except in bounded form. Theintuitionists share <strong>the</strong> finitists’ general point of view about <strong>the</strong> n<strong>at</strong>ural numbers but regard22
unbounded quantific<strong>at</strong>ion over <strong>the</strong>m as meaningful though not when combined withclassical logic. According to <strong>the</strong> predic<strong>at</strong>ivists, <strong>the</strong> n<strong>at</strong>ural numbers form a <strong>definite</strong>totality, but not <strong>the</strong> collection of “all” sets of n<strong>at</strong>ural numbers. Never<strong>the</strong>less systems ofpredic<strong>at</strong>ive analysis have been formul<strong>at</strong>ed using quantific<strong>at</strong>ion with respect to variablesfor such sets within classical logic. Finally, we have <strong>the</strong> case of set <strong>the</strong>ory, in whichaccording to practically all points of view⎯including th<strong>at</strong> of <strong>the</strong> ma<strong>the</strong>m<strong>at</strong>icalpractitioners from <strong>the</strong> very beginning to <strong>the</strong> present as well as both <strong>the</strong> philosophicaldefenders and critics⎯sets in general form an in<strong>definite</strong> totality, o<strong>the</strong>rwise <strong>the</strong> universeof all sets would be a set and thus give rise to Russell’s Paradox. All of this posesquestions about <strong>the</strong> proper formul<strong>at</strong>ion of axiom<strong>at</strong>ic systems in which some of <strong>the</strong>domains involved are conceived to form in<strong>definite</strong> totalities, while o<strong>the</strong>rs, or parts of<strong>the</strong>m, are considered to be <strong>definite</strong> totalities. There has been much interestingphilosophical discussion in recent years about <strong>the</strong> problems raised by absolute generality(cf. Rayo and Uzquiano 2006) both in its unrestricted sense and more specifically asapplied to set <strong>the</strong>ory.My main concern here is ra<strong>the</strong>r to see wh<strong>at</strong> precisely can be said about certain ways ofconstruing in formal terms <strong>the</strong> distinction between <strong>definite</strong> and in<strong>definite</strong> concepts. Oneway of saying of a st<strong>at</strong>ement φ th<strong>at</strong> it is <strong>definite</strong> is th<strong>at</strong> it is true or false; on a defl<strong>at</strong>ionaryaccount of truth th<strong>at</strong>’s <strong>the</strong> same as saying th<strong>at</strong> <strong>the</strong> Law of Excluded Middle (LEM) holdsof φ, i.e. one has φ∨¬φ. Since LEM is rejected in intuitionistic logic as a basic principle,th<strong>at</strong> suggests <strong>the</strong> slogan, “Wh<strong>at</strong>’s <strong>definite</strong> is <strong>the</strong> domain of classical logic, wh<strong>at</strong>’s not isth<strong>at</strong> of intuitionistic logic.” In <strong>the</strong> case of predic<strong>at</strong>ivity, this would lead us to considersystems in which quantific<strong>at</strong>ion over <strong>the</strong> n<strong>at</strong>ural numbers is governed by classical logicwhile only intuitionistic logic may be used to tre<strong>at</strong> quantific<strong>at</strong>ion over sets of n<strong>at</strong>uralnumbers or sets more generally. And in <strong>the</strong> case of set <strong>the</strong>ory, where every set isconceived to be a <strong>definite</strong> totality, we would have classical logic for boundedquantific<strong>at</strong>ion while intuitionistic logic is to be used for unbounded quantific<strong>at</strong>ion. Theformal study of systems of <strong>the</strong> l<strong>at</strong>ter kind goes back <strong>at</strong> least to a Wolf(1974) and23
Friedman (1980) and has most recently been carried on in <strong>the</strong> paper, Feferman (2010b). 12The general p<strong>at</strong>tern of <strong>the</strong> studies <strong>the</strong>re is th<strong>at</strong> we start with a system T formul<strong>at</strong>ed infully classical logic, <strong>the</strong>n consider an associ<strong>at</strong>ed system SI-T formul<strong>at</strong>ed in a mixed,semi-intuitionistic logic, and ask whe<strong>the</strong>r <strong>the</strong>re is any essential loss in proof-<strong>the</strong>oreticalstrength when passing from T to SI-T. In <strong>the</strong> cases th<strong>at</strong> are studied, it turns out th<strong>at</strong> <strong>the</strong>reis no such loss, and moreover, <strong>the</strong>re can be an advantage in going to such an SI-T;namely, we can beef it up to a semi-constructive system SC-T without changing <strong>the</strong>proof-<strong>the</strong>oretical strength, by <strong>the</strong> adjunction of certain principles th<strong>at</strong> go beyond wh<strong>at</strong> isadmitted in SI-T, because <strong>the</strong>y happened to be verified in a certain kind of functionalinterpret<strong>at</strong>ion of <strong>the</strong> intuitionistic logic.For example, if one starts with S equal to <strong>the</strong> classical system KPω of admissible set<strong>the</strong>ory (including <strong>the</strong> Axiom of Infinity), <strong>the</strong> associ<strong>at</strong>ed system SI-KPω has <strong>the</strong> sameaxioms as KPω, and is based on intuitionistic logic plus <strong>the</strong> law of excluded middle forall bounded (i.e. Δ 0 ) formulas. But <strong>the</strong> beefed up system SC-KPω proves <strong>the</strong> Full Axiomof Choice Scheme for sets,∀x ∈ a ∃y φ(x,y) → ∃r [Fun(r) ∧∀x ∈a φ(x, r(x))],where φ(x, y,…) is an arbitrary formula of <strong>the</strong> language of set <strong>the</strong>ory. Th<strong>at</strong> in turnimplies <strong>the</strong> Full Collection Axiom Scheme,∀x ∈ a ∃y φ(x,y) → ∃b ∀x ∈a ∃y ∈b φ(x,y)for φ an arbitrary formula, while this holds only for ∑ 1 formulas in SI-KPω.In addition, SC-KPω contains some o<strong>the</strong>r non-intuitionistic principles th<strong>at</strong> can,never<strong>the</strong>less, be given a semi-constructive interpret<strong>at</strong>ion. The main result in Feferman(2010b) is th<strong>at</strong> all of <strong>the</strong>se systems are of <strong>the</strong> same proof-<strong>the</strong>oretical strength, KPω ≡ SI-KPω ≡ SC- KPω. Moreover, <strong>the</strong> same result holds when we add <strong>the</strong> power set axiom12 See also <strong>the</strong> slides Feferman (2009b) for <strong>the</strong> conference in honor of Harvey Friedman’s60 th birthday.24
Pow as providing a <strong>definite</strong> oper<strong>at</strong>ion on sets, or just its restricted form Pow(ω) asserting<strong>the</strong> existence of <strong>the</strong> power set of ω.Now <strong>the</strong> choice of KPω ± Pow(resp. Pow(ω)) and <strong>the</strong> associ<strong>at</strong>ed semi-constructivesystems for initial consider<strong>at</strong>ion ra<strong>the</strong>r than ZFC ± Pow(resp. Pow(ω)) is dict<strong>at</strong>ed by <strong>the</strong>idea th<strong>at</strong> we should only accept <strong>the</strong> Separ<strong>at</strong>ion and Replacement schemes for <strong>definite</strong>properties. And since our idea of quantific<strong>at</strong>ion as a <strong>definite</strong> logical oper<strong>at</strong>ion is th<strong>at</strong> it isonly to range over <strong>definite</strong> totalities, <strong>at</strong> first sight only those formulas in whichquantific<strong>at</strong>ion is restricted to sets are to be admitted to those schemes. In particular, inview of <strong>the</strong> discussion <strong>at</strong> <strong>the</strong> end of <strong>the</strong> preceding section it should be of particularinterest to investig<strong>at</strong>e wh<strong>at</strong> properties and st<strong>at</strong>ements are <strong>definite</strong> in <strong>the</strong> specific case of<strong>the</strong> semi-constructive version of KPω + Pow(ω).In general, given one of <strong>the</strong> semi-constructive systems SC-T under consider<strong>at</strong>ion, onewould like to establish th<strong>at</strong> if a formula φ(x) is <strong>definite</strong> in <strong>the</strong> sense th<strong>at</strong> ∀x[φ(x)∨¬φ(x)]is provable <strong>the</strong>re, <strong>the</strong>n φ is <strong>at</strong> least equivalent to a Δ 1 formula and hence is absolute forend-extensions. I conjecture th<strong>at</strong> th<strong>at</strong> is <strong>the</strong> case for <strong>the</strong> systems of SC-KPω and itsextensions by Pow(ω) and even full Pow, but have not yet found a proof. My guess isth<strong>at</strong> th<strong>at</strong> can be done using a detailed analysis of <strong>the</strong> functional interpret<strong>at</strong>ions used inestablishing <strong>the</strong> results about <strong>the</strong> proof-<strong>the</strong>oretical strength of <strong>the</strong>se systems in my(2010b) paper.Similarly, a st<strong>at</strong>ement φ should be considered <strong>definite</strong> in this framework for a given semiconstructivesystem SC-T if LEM holds for φ, i.e. φ ∨¬φ, can be proved <strong>the</strong>re. In pureintuitionistic systems we have <strong>the</strong> disjunction property, th<strong>at</strong> if φ ∨ψ is proved <strong>the</strong>n ei<strong>the</strong>rφ or ψ is provable; but we don’t have such a result for <strong>the</strong> semi-constructive systemsconsidered. For, every arithmetical st<strong>at</strong>ement s<strong>at</strong>isfies <strong>the</strong> LEM in our system but wecertainly can’t prove or disprove every such st<strong>at</strong>ement in <strong>the</strong> system. But we could hopeto show th<strong>at</strong> if LEM is provable for φ <strong>the</strong>n it must <strong>at</strong> least be equivalent to a Δ 1 sentence.Turning to st<strong>at</strong>ements of classical set <strong>the</strong>ory th<strong>at</strong> are of ma<strong>the</strong>m<strong>at</strong>ical interest, one wouldlike to know which of <strong>the</strong>m make essential use of full non-constructive reasoning and25
which can already be established on semi-constructive grounds. My guess is th<strong>at</strong> a gre<strong>at</strong>deal⎯including all of classical DST⎯can already be done in SC-KPω + Pow(ω).Turning finally to <strong>the</strong> <strong>Continuum</strong> Hypo<strong>the</strong>sis (CH), th<strong>at</strong>, in particular, is certainlymeaningful in SCS + (Pow(ω)) and formally it is a <strong>definite</strong> st<strong>at</strong>ement in SCS + (Pow), i.e.one can prove CH ∨ ¬CH <strong>the</strong>re. As with <strong>the</strong> semi-intuitionists, one can regard <strong>the</strong>conception of <strong>the</strong> set of all subsets of ω as clear enough, but not th<strong>at</strong> of arbitrary subset ofth<strong>at</strong>, and <strong>the</strong> meaning of CH depends essentially on quantifying over all such subsets.Presumably, CH is not <strong>definite</strong> in SCS + (Pow(ω)), and it would be interesting to see whyth<strong>at</strong> is so. 13 My own view, voiced elsewhere, is th<strong>at</strong> CH is wh<strong>at</strong> I have called anessentially vague st<strong>at</strong>ement, which says something like: <strong>the</strong>re is no way to sharpen it to a<strong>definite</strong> st<strong>at</strong>ement without essentially changing <strong>the</strong> meaning of <strong>the</strong> concepts involved init. But to formul<strong>at</strong>e th<strong>at</strong> idea more precisely within <strong>the</strong> semi-constructive framework,some stronger notion than formal <strong>definite</strong>ness may be required.ReferencesBenacerraf, P. (1965), Wh<strong>at</strong> numbers could not be, Phil. Review 74, 47-73, reprinted inBenacerraf and Putnam (1983), 272-294.Benacerraf, P. and H. Putnam (eds.) (1983) Philosophy of Ma<strong>the</strong>m<strong>at</strong>ics. SelectedReadings, 2 nd edn., Cambridge Univ. Press, Cambridge.Browder, F. E. (1976) (ed.), Ma<strong>the</strong>m<strong>at</strong>ical Developments arising from Hilbert Problems,Proc. Symposia in Pure Ma<strong>the</strong>m<strong>at</strong>ics 28, Amer. M<strong>at</strong>h. Soc., Providence, RI.Chihara, C. S. (2004), A Structural Account of Ma<strong>the</strong>m<strong>at</strong>ics, Clarendon Press, Oxford.Feferman, S. (1993), Why a little bit goes a long way: <strong>Logic</strong>al found<strong>at</strong>ions ofscientifically applicable ma<strong>the</strong>m<strong>at</strong>ics, in PSA 1992, Vol. 2, Philosophy of ScienceAssoci<strong>at</strong>ion, East Lansing; reprinted as Ch. 14 in Feferman (1998).Feferman, S. (1996), Gödel’s program for new axioms. Why, where, how and wh<strong>at</strong>?, in(P. Hajek, ed.) Gödel '96, Lecture Notes in <strong>Logic</strong> 6 (1996), 3-22.13 One would not expect CH would to be unique in this respect; any st<strong>at</strong>ement th<strong>at</strong> makesessential use of <strong>the</strong> assumed totalities S(S(N)) would be a candid<strong>at</strong>e for such a result.26
Feferman, S. (1998), In <strong>the</strong> Light of <strong>Logic</strong>, Oxford University Press, New York.Feferman, S. (2009a), Conceptions of <strong>the</strong> continuum, Intellectica 51, 169-189.Feferman, S. (2009b) Wh<strong>at</strong>’s <strong>definite</strong>? Wh<strong>at</strong>’s not?, slides for talk <strong>at</strong> Harvey Friedman60th birthday conference, Ohio St<strong>at</strong>e Univ., May 16, 2009,http://m<strong>at</strong>h.stanford.edu/~feferman/papers/wh<strong>at</strong>sdef.pdf .Feferman, S. (2010a), Conceptual structuralism and <strong>the</strong> continuum, slides for talk <strong>at</strong> PhilM<strong>at</strong>h Intersem 2010 Université Paris 7-Diderot, 6/08/10 (revision of slides for talk <strong>at</strong>VIIIth Intern<strong>at</strong>ional Ontology Congress, San Sebastián, 10/01/08).http://m<strong>at</strong>h.stanford.edu/~feferman/papers/Conceptual_Structuralism.pdf .Feferman, S. (2010b), On <strong>the</strong> strength of some semi-constructive <strong>the</strong>ories, in Fefermanand Sieg (2010), 109-129.Feferman, S. and G. Jäger (1993), Systems of explicit ma<strong>the</strong>m<strong>at</strong>ics with non-constructiveµ-oper<strong>at</strong>or, I, Annals of Pure and Applied <strong>Logic</strong> 65, 243-263.Feferman, S. and G. Jäger (1993), Systems of explicit ma<strong>the</strong>m<strong>at</strong>ics with non-constructiveµ-oper<strong>at</strong>or, II, Annals of Pure and Applied <strong>Logic</strong> 79, 37-52.Feferman, S., Parsons, C., and Simpson, S. G. (2010) (eds.), Kurt Gödel. Essays for hiscentennial, Lecture Notes in <strong>Logic</strong> 33, Assoc. for Symbolic <strong>Logic</strong>, Cambridge Univ.Press, Cambridge.Feferman, S. and W. Sieg (eds.) (2010), Proofs, C<strong>at</strong>egories and Comput<strong>at</strong>ions. Essays inhonor of Grigori Mints, College Public<strong>at</strong>ions, London.Feferman, S. and T. Strahm (2000), The unfolding of non-finitist arithmetic, Annals ofPure and Applied <strong>Logic</strong> 104, 75-96.Friedman, H. (1980), A strong conserv<strong>at</strong>ive extension of Peano Arithmetic, in (J.Barwise, et al., eds.) The Kleene Symposium, North-Holland, Amsterdam, 113-122.Gödel, K. (1938), The consistency of <strong>the</strong> axiom of choice and of <strong>the</strong> generalizedcontinuum hypo<strong>the</strong>sis, Proc. N<strong>at</strong>ional Academy of Sciences, USA 24, 556-557; reprintedin Gödel (1990), 26-27.Gödel, K. (1940), The Consistency of <strong>the</strong> Axiom of Choice and of <strong>the</strong> Generalized<strong>Continuum</strong> Hypo<strong>the</strong>sis with <strong>the</strong> Axioms of Set Theory, Annals of Ma<strong>the</strong>m<strong>at</strong>ics Studies 3,Princeton Univ. Press, Princeton; reprinted in Gödel (1990), 33-101.Gödel, K. (1947), Wh<strong>at</strong> is Cantor’s continuum problem?, American Ma<strong>the</strong>m<strong>at</strong>icalMonthly 54, 515-525; err<strong>at</strong>a 55, 151; reprinted in Gödel (1990), 240-251Gödel, K. (1964). Wh<strong>at</strong> is Cantor’s continuum problem?, revised and expanded versionof Gödel (1947), in Benacerraf and Putnam (1964), 258-273, reprinted in Gödel (1990),254-270.27
Gödel, K. (1990) Collected Works, Vol. II: Public<strong>at</strong>ions 1938-1974 (S. Feferman, et al.,eds.), Oxford University Press, New York.Hellman, G. (1989), Ma<strong>the</strong>m<strong>at</strong>ics Without Numbers, Oxford University Press, Oxford.Hellman, G. (2005), Structuralism, in Shapiro (2005), 536-562.<strong>Is</strong>aacson, D. (2008), The reality of ma<strong>the</strong>m<strong>at</strong>ics and <strong>the</strong> case of set <strong>the</strong>ory, in (Z. Novakand A. Simonyi, eds.), Truth, Reference, and Realism, Central European UniversityPress, Budapest, 1-64.Jaffe, A. M. (2006), The millennium grand challenge in ma<strong>the</strong>m<strong>at</strong>ics, Notices Amer.M<strong>at</strong>h. Soc. 53, no. 6, 652-660.Kanamori, A. (1997), The Higher Infinite, Perspectives in Ma<strong>the</strong>m<strong>at</strong>ical <strong>Logic</strong>, Springer,Berlin.Koellner, P. (2010), On <strong>the</strong> question of absolute undecidability, in Feferman, Parsons andSimpson (2010), 189-225.Kreisel, G. (1971), Observ<strong>at</strong>ions on popular discussions of found<strong>at</strong>ions, in (D. S. Scott,ed.), Axiom<strong>at</strong>ic Set Theory, Proc. Symposia in Pure Ma<strong>the</strong>m<strong>at</strong>ics 13, Part I, Amer. M<strong>at</strong>h.Soc., Providence, RI, 189-198.Levy, A. and R. M. Solovay (1967), Measurable cardinals and <strong>the</strong> continuum hypo<strong>the</strong>sis,<strong>Is</strong>rael J. of Ma<strong>the</strong>m<strong>at</strong>ics 5, 234-248.Maddy, P. (1997), N<strong>at</strong>uralism in Ma<strong>the</strong>m<strong>at</strong>ics, Oxford University Press, Oxford.Maddy, P. (2005), Ma<strong>the</strong>m<strong>at</strong>ical existence, Bull. Symbolic <strong>Logic</strong> 11, 351-376.Martin, D. A. (1975), Borel determinacy, Annals of Ma<strong>the</strong>m<strong>at</strong>ics 102, 363-371.Martin, Donald A. (1976), Hilbert’s first problem: The continuum hypo<strong>the</strong>sis, in Browder(1976), 81-92.Martin, D. A. and J. Steel (1989), A proof of projective determinacy, J. Amer. M<strong>at</strong>h. Soc.2, 71-125.Moore, G. H. (1982), Zermelo’s Axiom of Choice: Its origins, development and influence,Springer, New YorkMoore, G. H. (1989) Towards a history of Cantor’s continuum problem, The historyof modern ma<strong>the</strong>m<strong>at</strong>ics, vol. I: Ideas and <strong>the</strong>ir reception (David E. Rowe and JohnMcCleary, eds.), Academic Press, Boston, pp. 78–121.Moore, G. H. (2011), Early history of <strong>the</strong> generalized continuum hypo<strong>the</strong>sis: 1878-1938,Bull. Symbolic <strong>Logic</strong> (to appear).28
Poincaré, H. (1952), Science and Hypo<strong>the</strong>sis, Dover Public<strong>at</strong>ions, New York. (Englishtransl<strong>at</strong>ion of La Science et l’hypothèse, Flammarion, Paris,1902.)Rayo, A. and G. Uzquiano (eds.) (2006), Absolute Generality, Clarendon Press, Oxford.Resnik, M. (1997), Ma<strong>the</strong>m<strong>at</strong>ics as a Science of P<strong>at</strong>terns, Oxford University Press,Oxford.Searle, J. R. (1995), The Construction of Social Reality, Free Press, New York.Shapiro, S. (1997), Philosophy of Ma<strong>the</strong>m<strong>at</strong>ics: Structure and Ontology, OxfordUniversity Press, New York.Shapiro, S., ed. (2005), The Oxford Handbook of Philosophy of Ma<strong>the</strong>m<strong>at</strong>ics and <strong>Logic</strong>,Oxford University Press, Oxford.Simpson, S. G. (1999), Subsystems of Second Order Arithmetic, Perspectives inMa<strong>the</strong>m<strong>at</strong>ical <strong>Logic</strong>, Assoc. for Symbolic <strong>Logic</strong>, Springer, Berlin; 2 nd edn. (2009),Perspectives in Ma<strong>the</strong>m<strong>at</strong>ical <strong>Logic</strong>, Assoc. for Symbolic <strong>Logic</strong>, Cambridge Univ. Press,Cambridge.Simpson, S. G. (2010), The Gödel hierarchy and reverse ma<strong>the</strong>m<strong>at</strong>ics, in Feferman et al.(2010), 109-127.Steel, J. (2000), Ma<strong>the</strong>m<strong>at</strong>ics needs new axioms, Bull. Symbolic <strong>Logic</strong> 6, 422-433.Väänänen, J. (2001), Second-order logic and found<strong>at</strong>ions of ma<strong>the</strong>m<strong>at</strong>ics, Bull. Symbolic<strong>Logic</strong> 7, 504-520.Weyl, H. (1918), Das Kontinuum. Kritische Untersuchungen über die Grundlagen derAnalysis, Veit, Leipzig. (English transl<strong>at</strong>ion by S. Pollard and T. Bole, The <strong>Continuum</strong>. ACritical Examin<strong>at</strong>ion of <strong>the</strong> Found<strong>at</strong>ions of Analysis, Univ. Press of America, L<strong>at</strong>ham(1987).Wolf, Robert S. (1974), Formally Intuitionistic Set Theories with Bounded Predic<strong>at</strong>esDecidable, PhD Dissert<strong>at</strong>ion, Dept. of Ma<strong>the</strong>m<strong>at</strong>ics, Stanford University.Woodin, H. (2005a), The continuum hypo<strong>the</strong>sis, in (R. Cori, et al., eds.), <strong>Logic</strong>Colloquium 2000, Lecture Notes in <strong>Logic</strong> 19, Assoc. for Symbolic <strong>Logic</strong>, 143-197.Woodin, H. (2005b), Set <strong>the</strong>ory after Russell: The journey back to Eden, in (G. Link,ed.), One Hundred Years of Russell’s Paradox: Ma<strong>the</strong>m<strong>at</strong>ics, <strong>Logic</strong>, Philosophy, deGruyter Series in <strong>Logic</strong> and its Applic<strong>at</strong>ions, de Gruyter, Berlin.Zermelo, E. (1930), Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen überdie Grundlagen der Mengenlehre, Fundamenta Ma<strong>the</strong>m<strong>at</strong>icae 16, 29-47.29