On Reflection Principles ⋆ - Logic at Harvard - Harvard University

On Reflection Principles ⋆Peter KoellnerHarvard University, Massachusetts, USA, 02138, (617) 495-3970AbstractGödel initiated the program of finding and justifying axioms that effect a significantreduction in incompleteness and he drew a fundamental distinction between intrinsicand extrinsic justifications. Reflection principles are the most promising candidatesfor new axioms that are intrinsically justified. Taking as our starting point Tait’swork on general reflection principles, we prove a series of limitative results concerningthis approach. These results collectively show that general reflection principlesare either weak (in that they are consistent relative to the Erdös cardinal κ(ω)) orinconsistent. The philosophical significance of these results is discussed.Key words: Set Theory, Incompleteness, Large Cardinal Axioms, ReflectionPrinciples1991 MSC: 03E55The incompleteness phenomenon in set theory provides us with natural statementsof mathematics that cannot be settled on the basis of the standard axiomsof set theory, ZFC. Two classic examples of such statements are PU (thestatement that all projective sets admit of a projective uniformization) andCH (Cantor’s continuum hypothesis). This leads to the program of seeking andjustifying new axioms which settle the undecided statements. This programhas both a mathematical component and a philosophical component. On themathematical side, one must find axioms that are sufficiently strong to do thework. On the philosophical side, one must determine, first, what would countas a justification and, second, whether the axioms in question are justified. Inthis paper I will investigate these two aspects of one promising approach tojustifying new axioms—the approach based on reflection principles. 1⋆ I would like to thank Bill Tait for sparking my interest in the topic and for fruitfulcorrespondence.Email address: koellner@fas.harvard.edu (Peter Koellner).1 The result in Section 4 was proved in my dissertation (Koellner (2003)). Theremaining results were proved in the Spring and Summer of 2007.Preprint submitted to Elsevier March 1, 2008

1 IntroductionThe question motivating this work is: Can intrinsic justifications secure reflectionprinciples that are sufficiently strong to effect a significant reduction inincompleteness? To render this question more precise I will discuss the notionsof an “intrinsic justification” and a “significant reduction in incompleteness”(in the present section) and the notion of a “reflection principle” (in the nextsection).Intrinsic versus Extrinsic Justifications. In his classic paper on the continuumproblem (Gödel (1947) and Gödel (1964)) Gödel drew a fundamental distinctionbetween intrinsic and extrinsic justifications. The discussion pertains tothe iterative concept of set, that is, the concept of set “according to which aset is something obtainable from the integers (or some other well-defined objects)by iterated application of the operation “set of” ” (Gödel (1964), p. 259).Gödel maintains that the “axioms of set theory [ZFC] by no means form asystem closed in itself, but, quite on the contrary, the very concept of set onwhich they are based suggests their extension by new axioms which assert theexistence of still further iterations of the operation “set of” ” (260). He mentionsas examples the axioms asserting the existence of inaccessible and Mahlocardinals and maintains that “[t]hese axioms show clearly, not only that theaxiomatic system of set theory as used today is incomplete, but also that itcan be supplemented without arbitrariness by new axioms which only unfoldthe content of the concept of set as explained above” (260–261, my emphasis).Since Gödel later refers to such axioms as having “intrinsic necessary” I shallaccordingly speak of such axioms being intrinsically justified on the basis ofthe iterative concept of set. 2The notion of an intrinsic justification on the basis of the iterative conceptionof set begs for sharpening. It appears that Gödel took it to be a fundamentalform of justification, one that cannot be explained in more primitive terms.Nevertheless, one can explicate the idea of that Gödel appears to have inmind by comparing and contrasting it with other notions and by pointing toexamples. One such point of contrast is that of an extrinsic justification, whichGödel introduces as follows: “[E]ven disregarding the intrinsic necessity ofsome new axiom, and even in case it has no intrinsic necessity at all, a probabledecision about its truth is possible also in another way, namely, inductively bystudying its “success”. (261) Here by “success” Gödel means “fruitfulness inconsequences, in particular “verifiable” consequences”. In a famous passage he2 I shall also employ the more neutral notion of the iterative conception of setrather that the iterative concept of set since there is nothing in this discussion thatrests on a robust form of conceptual realism such as that of Gödel. For discussionsof Gödel’s conceptual realism see Parsons (1995) and Martin (2005).2

It is of interest then to determine whether such a view can be sharpened andupheld. We shall do this by taking Tait’s work on general reflection principles(Tait (1990), Tait (1998a), Tait (1998b), and Tait (2005a)) as our startingpoint. Tait (2005a) says that his bottom-up approach may have the resourcesto lead beyond the V = L barrier (p. 135). As we shall see, it follows from thelimitative results below that current reflection principles do not imply 0 # andhence cannot lead to a significant reduction in incompleteness (in the senseindicated above).2 Reflection PrinciplesReflection principles aim to articulate the informal idea that the height ofthe universe is “absolutely infinite” and hence cannot be “characterized frombelow”. These principles assert that any statement true in V is true in somesmaller V α . Thus, for any ϕ one cannot define V as the collection which satisfiesϕ since there will be a proper initial segment V α of V that satisfies ϕ. Moreformally, we shall write this asV |= ϕ(A) → ∃α V α |= ϕ α (A α )where ϕ α ( · ) is the result of relativizing the quantifiers of ϕ( · ) to V α andA α is the result of relativizing the A to V α . This schematic characterizationof a reflection principle will be filled in as we proceed by (1) specifying thelanguage and (2) specifying the nature of relativization.For the time being our language will be the language of set theory extendedwith variables of all finite orders. We shall use x, y, z, . . . as variables of thefirst order and, for m > 2, X (m) , Y (m) , Z (m) , . . . as variables of the m thorder. 8 Relative to V α the first-order variables are interpreted to range overthe elements of V α and, for m > 1, the m th -order variables are interpretedto range over the elements of an isomorphic version of V α+(m−1) . The reasonfor using an isomorphic copy of V α+(m−1) and not V α+(m−1) itself is that wewish to keep track of the set/class distinction in cases where a set and classhave the same extension. For definiteness, in the case of m = 2 we shall useα × V α+1 and we shall interpret class membership “y ∈ (α, x)” as y ∈ x. Thecases where m > 2 are handled similarly.We now turn to the nature of relativization. If A (2) is a second-order parameterover V α , then the relativization of A (2) to V β , written A (2),β , is A ∩ V β . 9section, have a different form.8 When the order of a variable or parameter is clear from context we shall oftendrop the superscript for notational simplicity.9 More precisely, given our coding apparatus, A (2),β is really {x ∈ V β | (α, x) ∈6

This is how A (2) looks from “the point of view of V β ”. The relativization ofhigher-order parameters is defined inductively in the natural way: For m > 1,A (m+1),β = {B (m),β | B (m) ∈ A (m+1) }. The relativization of a higher-order formulaϕ to the level V β is obtained by interpreting the first-order variables torange over V β and, for m > 1, m th -order variables to range over the elementsof V β+(m−1) .With these specifications we now have a hierarchy of reflection principles. Forreasons which will become apparent in the next section, we shall restrict ourselvesfor the time being to parameters of second order. Let us recall some basicfacts: Let T be the theory ZFC−Infinity−Replacement. If one supplements Twith the scheme for second-order reflection then the resulting theory impliesInfinity and Replacement. Moreover, in the second-order language, one canexpress the statement “Ω is (strongly) inaccessible” (where ‘Ω’ designates theclass of ordinals) and so, assuming second-order reflection, there exists κ suchthat κ is inaccessible (and thus V κ |= ZFC). We can then reflect the statement“Ω is an inaccessible greater than κ” to get an inaccessible above κ and,continuing in this manner, we obtain a proper class of inaccessibles. Thus, Ωis an inaccessible limit of inaccessibles and hence, by reflection, there existsκ which is an inaccessible limit of inaccessibles. Continuing in this mannerone obtains the various orders of inaccessibles and Mahlos. One then obtainsweakly compact cardinals and, moving up through the higher-order languages,one obtains the higher-order indescribable cardinals.Before proceeding further let us examine some philosophical difficulties withthe claim that higher-order reflection principles are intrinsically justified onthe basis of the iterative conception of set.The most basic difficulty involves the interpretation of higher-order quantificationand turns on how one conceives of the “totality of sets”. There aretwo conceptions of the “totality of sets”—the actualist conception and thepotentialist conception. The actualist maintains that the totality of sets is a“completed totality”, while the potentialist denies this. These two viewpointsface complementary difficulties in providing an intrinsic justification of higherorderreflection principles. On the actualist view one can refer to the totalityof sets and thus one can articulate the idea that this totality cannot be describedfrom below and hence satisfies the reflection principles. However, sincethere are no sets beyond this totality it is hard on this view to make sense offull higher-order quantification over the universe of sets. 10 On the potentialistA (2) }, but we shall suppress such fine points in the future.10 One can, of course, simulate the construction of L over the universe for anyordinal that one can make sense of “internally” with the help of bootstrapping, butthe resulting principles are quite weak. Now, there are some actualists who thinkthat there is no problem in having full second-order quantification over the universeof sets. I am thinking here of advocates of the plural interpretation of second-order7

view the closest one can come to speaking of the totality of sets is throughspeaking of some V α . One can certainly make sense of higher-order quantificationover V α but now the difficulty lies in motivating and justifying reflectionprinciples.I know of two attempts to get around this difficulty. The first is the theory oflegitimate candidates of Reinhardt. The second is the bottom up approach ofTait. I will say something about the former in the final section of this paper.Here I will concentrate on the latter.Tait’s approach takes its starting point in what he calls the Cantorian Principle,namely, the principle which asserts that if and initial segment A ⊆ Ω isa set then it has a strict upper bound S(A) ∈ Ω. This is the principle whichCantor used to introduce (in a highly impredicative fashion) the totality ofordinals Ω. It follows (from the well-foundedness of the ordinals) that Ω isnot a set. The problem with this principle is that it involves reference to thenotion of “set” (in contrast to the notion of “class” or “inconsistent multiplicity”)and this notion (and distinction) is far from clear. For this reasonTait replaces the principle with a hierarchy of Relativized Cantorian Principles.For a given condition C (called an existence condition) such a principleasserts that if an initial segment A ⊆ Ω C satisfies C then it has a strict upperbound S(A) ∈ Ω C . It follows (from the well-foundedness of the ordinals) thatΩ C does not satisfy the condition C.One can now obtain reflection principles by the appropriate choice of an existencecondition. For example, suppose we wish to construct an ordinal Ω Csuch that for a given second-order formula ϕ(X), V ΩC satisfies ϕ-reflection,that is,(∀X (2) ⊆ V ΩC VΩC |= ϕ(X (2) ) → ∃α ϕ α (X (2),α ) ) .We simply take the condition C on initial segments A ⊆ Ω C to be∃X (2) ⊆ V A(VA |= ϕ(X (2) ) ∧ ∀α ∈ A ¬ϕ α (X (2),α ) ) .Applying the Relativized Cantorian Principle to this condition and appealingagain to the well-foundedness of the ordinals, we have that Ω C does not satisfyC, that is, V ΩC satisfies ϕ-reflection. The trouble is that this methodis too general. For example, suppose we wish to introduce Ω C ′ such thatV ΩC ′ satisfies that there is a ϕ-cardinal, where ‘ϕ’ could be anything, suchas ‘supercompact’ or ‘Reinhardt’. Let C ′ be the following condition on A:V A ̸|= “There is a ϕ-cardinal”. Applying the Relativized Cantorian Principleto this condition and appealing to the well-foundedness of the ordinals, wehave that Ω C does not satisfy C ′ , that is, V ΩC |= “There is a ϕ-cardinal”.quantification. It would take us too far afield to consider this view in detail. In anycase, the resulting principles fall under the limitative results to be presented below.8

In short, whether the Relativized Cantorian Principle is intrinsically justifiedon the basis of the iterative conception of set will turn on the particular choiceof the condition C that one considers. One would need to argue that in the caseof conditions C that give rise to reflection principles the resulting instances ofthe Relativized Cantorian Principle are intrinsically justified. This is far fromimmediate. Furthermore, as we shall see, such a case cannot be made for abroad class of the instances that Tait considers since the resulting principlesare inconsistent. Moreover, it is hard to see how one could draw the line in aprincipled fashion.These philosophical difficulties show that at the moment we do not have astrong intrinsic justification of higher-order reflection principles. One wouldlike, however, to say something more definitive. In the remainder of this paperI will prove a number of limitative results which collectively show that higherorderreflection principles are either weak or inconsistent.3 Strong Reflection PrinciplesThe next step is to allow parameters of third- and higher-order. Unfortunately,when one does this the resulting reflection principles are inconsistent, as notedby Reinhardt (1974). To see this letA (3) = { {ξ | ξ < α} (2) | α ∈ Ω } (3)and let ϕ(A (3) ) be the statement that each element of A (3) is bounded. Thisstatement is true over V but for each α ∈ Ω the reflected version of thestatement, ϕ α (A (3),α ), is false since {ξ | ξ < α} (2) ∈ A (3),α is unbounded.This counter-example and related counter-examples force one to forgo negativestatements of the form X (m) ∉ Y (m+1) and X (m) ≠ Y (m) when m ≥ 2. Thisleads to the following notions, due to Tait.Definition 1 A formula in the language of finite orders is positive iff it isbuild up by means of the operations ∨, ∧, ∀ and ∃ from atoms of the formx = y, x ≠ y, x ∈ y, x ∉ y, x ∈ Y (2) , x ∉ Y (2) and X (m) = X ′(m) andX (m) ∈ Y (m+1) , where m ≥ 2.Surprisingly, even when one restricts the language in this way, there are reflectionprinciples which have significant strength.Definition 2 For 0 < n < ω, Γ (2)nis the class of formulas of the form∀X (2)1 ∃Y (k 1)1 · · · ∀X (2)n∃Y (kn)n ϕ(X (2)1 , Y (k 1)1 , . . . , X (2)n, Y (kn)n , A (l 1) , . . . , A (l n ′) )9

where ϕ does not have quantifiers of second- or higher-order and k 1 , . . . , k n ,l 1 , . . . , l n ′ are natural numbers.Definition 3 For 0 < n < ω, Γ (2)n -reflection is the schema asserting that foreach sentence ϕ ∈ Γ (2)n , if V |= ϕ then there is a δ ∈ Ω such that V δ |= ϕ δDefinition 4 (Baumgartner) For 0 < n < ω, κ is n-ineffable iff for any〈K α1 ,...,α n| α 1 < · · · < α n < κ〉 with K α1 ,...,α n⊆ α 1 for α 1 < · · · < α n < κ,there is an X ⊆ κ and an S stationary in κ such that for β 1 < · · · < β n , allin S, X ∩ β 1 = K β1 ,...,β n.Theorem 5 (Tait) Suppose n < ω and V κ |= Γ (2)n -reflection. Then κ is n-ineffable.Theorem 6 (Tait) Suppose κ is a measurable cardinal. Then, for each n

Step 1: Consider the structure N = 〈V κ , ∈,

Lemma 10 Suppose ϕ(A 1 , . . . , A m ) ∈ Γ (2)n . If Vι M0Vι M0|= ϕ(j(A 1 ) ι 0, . . . , j(A m ) ι 0).|= ϕ(A 1 , . . . , A m ) thenProof. Base Case: It will be convenient to separate the second-order variablesfrom the higher order variables, so let us write ϕ(A 1 , . . . , A m ) as, B (n )j+1)j+1 , . . . , B m(nm) . Since ι0 is inacces-Suppose that Vι M0|= ϕ ( A (2)sible in M,C = { α < ι 0∣ ∣∣〈VMαϕ ( A (2)1 , . . . , A (2)j , B (n )j+1)j+1 , . . . , B m(nm) .1 , . . . , A (2)is club in ι 0 . We claim that for α ∈ C,j, ∈, A (2),α1 , . . . , A (2),α 〉 〈j ≺ VMι0, ∈, A (2)1 , . . . , A (2) 〉}jVα M |= ϕ ( A (2)1 , . . . , A (2)j , B (n )j+1)j+1 , . . . , B m(nm) .The key point is this: Suppose that A (2) ∈ B (3) or B (k) ∈ B (k+1) are constituentsof ϕ. If such a constituent is false (evaluated at Vι M0) then, since itoccurs positively in ϕ, it does not contribute to the truth of ϕ (evaluated atVι M0). If such a constituent is true (evaluated at Vι M0) then its reflected versionto the α th -level will be true (evaluated at V α ) for every α < ι 0 .Now the statement that ϕ reflects to each α in C is a first-order statementof M about the parameters ι 0 , C, A 1 , . . . , A j , B j+1 , . . . , B m . Thus, by the elementarityof j, the corresponding statement holds with respect to the imageof these parameters, that is, the statementϕ(j(A 1 ), . . . , j(A j ), j(B j+1 ), . . . , j(B m ))reflects to the club of points j(C) below j(ι 0 ). Since j(C) ∩ C = C and sinceC is unbounded in ι 0 and j(C) is club, it follows that ι 0 ∈ j(C), that is, thestatement ϕ(j(A 1 ), . . . , j(A j ), . . . , j(B j+1 ), . . . , j(B m )) reflects to ι 0 .Induction Step: Assume the lemma is true for ψ ∈ Γ (2)n . Our aim is to showthat it is true for ∀X (2) ∃Y (k) ψ(X (2) , Y (k) , A): ⃗V Mι 0|= ∀X (2) ∃Y (k) ψ(X (2) , Y (k) , A) ⃗ [↔ ∀B ⊆ Vι M0 VMι0|= ψ(B (2) , f(B) (k) , A) ⃗ ][→ ∀B ⊆ Vι M0 VMι0|= ψ(j(B) (2),ι 0, j(f(B)) (k),ι 0, j( A) ⃗ ι 0) ][→ ∀B ⊆ Vι M0 VMι0|= ψ(B (2) , f ′ (B) (k) , j( A) ⃗ ι 0) ]↔ V Mι 0|= ∀X (2) ∃Y (k) ψ(X (2) , Y (k) , j( ⃗ A) ι 0).12

In the first equivalence f : P(V Mι 0) → P k (Vι M0) is a Skolem function. Thesecond implication holds by the induction hypothesis. The final equivalence isimmediate. The third implication requires further comment: The first line ofthe implication provides us with a mapj(B) (2),ι 0↦→ j(f(B)) (k),ι 0that is defined for each B ⊆ Vι M0. We would like to extract from this map aSkolem function for the quantifier alternation ∀X (2) ∃Y (k) . Fortunately, (andthis is the key point), for each B ⊆ Vι M0, j(B) (2),ι 0= B. Thus,f ′ : P(Vι M0) → P k (Vι M0)B ↦→ j(f(B)) (k),ι 0is the desired Skolem function.✷Step 3: We can now show that Vι M0|= Γ (2)n -reflection, for all n < ω. Assume|= ϕ(A 1 , . . . , A n ). ThenV Mι 0by the lemma. SoV Mι 0|= ϕ(j(A 1 ) ι 0, . . . , j(A n ) ι 0)V Mj(ι 0 ) |= ∃α < j(ι 0 ) ( ϕ α (j(A 1 ) α , . . . , j(A n ) α ) )as witnessed by α = ι 0 . Finally, Vι M0elementarity of j.|= ∃α < ι 0(ϕ α (A α 1 , . . . , A α n) ) , by theFinally, let δ = π(ι 0 ). Applying π we have that V δ |= Γ (2)n -reflection for alln < ω, which completes the proof. ✷It is important to note that in the above proof we make key use of the factthat the higher-order universal quantifiers in a Γ (2)n -formula are second-order.The proof does not generalize to establish the consistency of Γ (m)n -reflectionfor m > 2 (contrary to what is suggested at the end of Tait (2005a)). The keystep in which m = 2 is used is the step where we derive the choice functionf ′ from f using j. Here it is crucial that the domain of the choice functionbe second-order since it is only for second-order parameters B that we canbe guaranteed that j(B) ι 0= B and hence that the derived choice function istotal. To be more specific, the third implication in the induction step of thekey lemma fails for m = 3 since in this case the domain of the derived choicefunction is {j(B) (3),ι 0| B ⊆ P(Vι M0)} which is a proper subset of P 2 (Vι M0).The question remains whether Γ (m)n -reflection for m > 2 is consistent relativeto large cardinal axioms. There is a high-level reason for thinking that any13

eflection principle which is consistent relative to large cardinals is consistentrelative to κ(ω). To see this recall that a canonical class of large cardinalaxioms assert the existence of a nontrivial elementary embedding j : V → M,where M is a transitive proper class and that as one increases the agreementbetween M and V the reflection properties of the critical point increase. Thelimiting case in which M = V was shown to be inconsistent (with AC) byKunen. If one drops AC and takes the embedding j : V → V one is in asituation that closely resembles our situation with j : M → M. The difference,of course, is that we are dealing with a countable model M and not the entireuniverse. However, from the point of view of a consistency proof it wouldappear that whatever reflection is provable from j : V → V should also beprovable from j : M → M. Since reflection would appear to be an entirelyinternal matter, this is a reason for thinking that any conceivable reflectionprinciple must have consistency strength below that of κ(ω).5 InconsistencyIt turns out that the above consistency proof is optimal in that Γ (3)1 -reflectionis inconsistent (using a fourth-order parameter). The counterexample is bestthought of in terms of a combinatorial consequence of Γ (3)1 -reflection.Definition 11 Suppose that κ is an uncountable regular cardinal. Let m ≥ 2.A 1-sequence (m) is a function K : κ → V κ such that for all α < κ, K(α) ⊆V α+(m−2) .Definition 12 Suppose that κ is an uncountable regular cardinal, K is a 1-sequence (m) , and X (m) is an m th order class over V κ . Then[K, X] = {α < κ | K(α) = X α }.This is the set of points at which X “correctly guesses” K.Definition 13 Suppose κ is an uncountable regular cardinal and m ≥ 2. LetD ⊆ κ.(1) D is 0-stationary (m) iff D is stationary.(2) D is (n + 1)-stationary (m) iff for all 1-sequences (m) K there exists X (m)such that [K, X] ∩ D is n-stationary (m) .We verify that (the relevant case of) one of Tait’s results generalizes from thesecond-order to the third-order context.Theorem 14 (Tait) Suppose that V κ satisfies Γ (3)1 -reflection. Then κ is 1-stationary (3) .14

Proof. Suppose, for contradiction, that κ is not 1-stationary (3) . We claim thatthere exists ϕ ∈ Γ (3)1 and a fourth-order parameter T (4) such that ϕ(T (4) ) doesnot reflect, that is,(1) V κ |= ϕ(T (4) ) and(2) for all β < κ, V β ̸|= ϕ(T (4),β ).Let K : κ → V κ be a 1-sequence (3) which is a counterexample to the 1-stationarity (3) of κ. For each X (3) ⊆ V κ+1 let C X be a club such that[K, X] ∩ C X = ∅.LetLetT (4) = {(K (2) , X (3) , C (2)X ) | X (3) ⊆ V κ+1 }.ϕ(T (4) ) = ∀X (3) ∃K (2) ∃C (2)( (K, X, C) ∈ T (4) ∧ C is unbounded ) .Notice that this is a Γ (3)1 -statement about a fourth-order parameter. (We areimplicitly using coding devices to collapse the existential quantifiers and codethe heterogeneous relation T (4) as a fourth-order class T ∗,(4) in such a way thatfor all α ∈ Lim, T (4),α is coded (in the same way) by T ∗,(4),α . See Tait (2005a)for details.)Claim 15 For each X (3) ⊆ V κ+1 ,(i) V κ |= “C X is unbounded” and(ii) for all β ∈ [K, X], V β ̸|= “C β X is unbounded”.Proof. (i) This is immediate since C X is club in κ. (ii) Suppose, for contradiction,that β ∈ [K, X] is such that V β |= “C β X is unbounded”. Since C Xis club in κ this implies β ∈ C X ∩ [K, X], which contradicts the fact that C Xwas chosen to be such that C X ∩ [K, X] = ∅. ✷It follows that V κ |= ϕ(T (4) ), since for each X (3) ⊆ V κ+1 our fixed K andchosen C X are witnesses. So we have proved (1).It remains to prove (2), namely, V β ̸|= ϕ β (T (4),β ), for all β < κ. Suppose, forcontradiction, that β < κ is such that V β |= ϕ β (T (4),β ), that is,V β |= ∀X (3) ∃K (2) ∃C (2) ( (K, X, C) ∈ T (4),β ∧ C is unbounded ) .For the particular choice X = K(β), let K (2)0 and C (2)0 be such that15

(*) V β |= (K (2)0 , X, C (2)0 ) ∈ T (4),β ∧ C (2)0 is unbounded.Since (K 0 , K(β), C 0 ) ∈ T (4),β we have(a) K 0 = K β ,(b) K(β) = X ′β for some X ′ ⊆ V κ+1 , and(c) C 0 = C β X ′ (where this is our canonical choice for X′ ),where (K, X ′ , C X ′) ∈ T (4) .Now, in defining T (4) we chose C X to be such that when (K, X, C X ) ∈ T (4) wehave[K, X] ∩ C X = ∅.Thus, in particular, [K, X ′ ] ∩ C X ′ = ∅, that is, for all α ∈ C X ′, K(α) ≠ X ′α .Now, by the Claim,(d) V κ |= “C X ′ is unbounded” and(e) ∀β ∈ [K, X ′ ] V β ̸|= “C β X is unbounded.”′However, by (b), β ∈ [K, X ′ ] and so, by (e),V β ̸|= “C β X ′is unbounded.”But by (c), C 0 = C β X ′, so V β ̸|= “C 0 is unbounded,”which contradicts (*).✷Theorem 16 Γ (3)1 -reflection is inconsistent.Proof. Suppose, for contradiction, that V κ satisfies Γ (3)1 -reflection. Then, bythe above theorem, κ is 1-stationary (3) . We arrive at a counterexample byconstructing a 1-sequence (3) which cannot be stationarily guessed. For α ∈Lim, letA α = { {ξ | ξ < γ} | γ < α } .Let K A : κ → V κ be such that K(α) = A α if α ∈ Lim and K(α) = ∅ otherwise.We claim that for each X (3) ⊆ V κ+1 , [K A , X]∩Lim contains at most one point.Suppose X α = K A (α), where α ∈ Lim. If α ′ ∈ Lim is such that α ′ > α andX α′ = K A (α ′ ) then there exists Y (2) ∈ X such that Y (2),α′ = {ξ | ξ < α}, inwhich case Y (2),α = {ξ | ξ < α} ∈ K A (α), which is a contradiction. Similarly,since X α = K A (α), for each ᾱ ∈ Lim ∩ α there exists Y (2) ∈ X such thatY (2),a = {ξ | ξ < ᾱ}, in which case Y (2),ᾱ = {ξ | ξ < ᾱ} ∈ K A (ᾱ) and henceXᾱ ≠ K A (ᾱ). Hence [K A , X] ∩ Lim contains at most one point.16

It follows that X (3) cannot stationarily guess K A since if [K A , X] is stationarythen [K, X] ∩ Lim is stationary, which is clearly impossible since it containsat most one point. ✷6 DichotomyThe results of the previous two sections show that the reflection principles wehave considered can be divided into two classes:(1) Weak: Γ (2)n -reflection, for n < ω.(2) Inconsistent: Γ (m)n -reflection, for m > 2 and n ≥ 1.Since Γ (3)1 comes directly after ⋃ n ξ ≤ α} | γ < α } .Let K B : κ → V κ be such that K(α) = B α if α ∈ Lim and K(α) = ∅otherwise. For X (2) ⊆ V κ , [K B , X] ∩ Lim = ∅. For X (3) ⊆ V κ+1 , [K B , X] ∩ Limcan contain at most one point.17

Second Modification. One response to the counterexample K B is to restrictour attention to “full” 1-sequences (3) , that is, 1-sequences (3) such thatfor each α ∈ Lim, min{rank(Y ) | Y ∈ K(α)} < α. However, this also has acounterexample. For α ∈ Lim, letC α = { α − {ξ} | ξ < α } .Let K B : κ → V κ be such that K(α) = C α if α ∈ Lim and K(α) = ∅ otherwise.This is a full 1-sequence (3) such that for each X (2) ⊆ V κ , [K C , X] ∩ Lim = ∅and for each X (3) ⊆ V κ+1 , [K C , X] ∩ Lim contains at most one point.Third Modification. One response to the counterexample K C is that althoughit is full and cannot be stationarily guessed by any X (2) ⊆ V κ or anyX (3) ⊆ V κ , it can be recast as a 1-sequence (2) which can be guessed by someX (2) ⊆ V κ . More generally, suppose K : κ → V κ is a full 1-sequence (3) which is“narrow” in the sense that for each α ∈ Lim, |K(α)| ≤ |α|. Relative to a fixedwell-ordering let 〈K(α) ξ | ξ < α〉 enumerate K(α). Now define the derivedsequence K ′ : κ → V κ to be such that K ′ (α) = {〈ξ, x〉 | ξ < α ∧ x ∈ K(α) ξ }if α ∈ Lim and K ′ (α) = ∅ otherwise. The sequence K ′ codes K and canbe stationarily guessed by some X (2) ⊆ V κ . So, to obtain strength, the abovecounter-examples suggest restricting attention to 1-sequences K which are“full” and “wide”, that is, such that (1) for all Y ∈ K(α), Y ⊆ α × V α anddom(Y ) = α and (2) |K(α)| > |α|. However, although the restriction to fulland wide 1-sequences (3) rules out counterexamples like K A , K B , and K C itdoes not rule out a simple “wide” version of K C : For α ∈ Lim, ξ < α, andY ⊆ α, letY ξ = { 〈γ, i, j〉 | (γ < α) ∧ (i = 0 ↔ γ ≠ ξ) ∧ (j = 0 ↔ γ ∈ Y ) } ,where i and j range over {0, 1}. (The role of the second coordinate is to codeα−{ξ} and the role of the third coordinate is to code Y . Thus we are “tagging”α − {ξ} with Y .) For α ∈ Lim, letC ∗ α = {Y ξ | ξ < α ∧ Y ⊆ α}.Finally, let K C ∗ : κ → V κ be such that K C ∗ = C ∗ α if α ∈ Lim and K C ∗ = ∅otherwise. The idea is that K C ∗(α) has 2 α -many subcollections, each correspondingto a given Y ⊆ α, each of size |α|, and each such that somethingspecific happens unboundedly often. This is a 1-full-wide sequence which cannotbe stationarily guessed.What all of the above counter-examples have in common is that they involvecollections which lack a certain “closure”. To make this precise we concentrateon third-order classes consisting of second-order class of ordinals. Over V α sucha class B is canonically coded by a collection of branches through 2 α , eachbranch being the characteristic function of a second-order class of ordinals. To18

say that a third-order class of second-order classes of ordinals is closed is simplyto say that the associated collection of branches is closed in the standardtopology (where basic open intervals have the form O s = {t ∈ 2 α | s ⊆ t},where s ∈ 2 α.The above counter-examples all involve restrictions of the combinatorial notionof 1-stationarity (3) but our main interest is in restrictions of the notion of Γ (3)1 -reflection. It will therefore be of importance to investigate the connectionbetween the two.Since we are interested in reflection principles extending second-order reflectionprinciples we may assume that the height of the universe V κ is Mahlo (ormore) and that in any reflection argument we reflect to a level V α where α isalso Mahlo (or more). Since in this case |V α | = |α|, we may, without loss ofgenerality, concentrate on third-order quantifiers which range over P(P(α)).Let X c(3) range over the closed third-order classes of second-order classes ofordinals. This range of quantification is uniformly defined with respect to V αfor each α ∈ Lim. More generally, let X Γ(3) range over the Γ(3)-third-orderclasses of second-order classes of ordinals, where Γ(3) is a pointclass which isuniformly defined with respect to V α for each α ∈ Lim. The pointclasses Γ(3)that will be of particular importance for us are those in the generalized Borelhierarchy which starts with the closed third-order classes over 2 α and proceedsby iterating the operations of α-union and complementation.Definition 17 Suppose κ is regular and uncountable. Suppose Γ(3) is a thirdorderpointclass as above. A 1-sequence Γ(3) is a function K : κ → V κ such thatthere is a club C K in κ such that for all α ∈ C K , K(α) ⊆ 2 α is in Γ(3).A cardinal κ is 1-stationary Γ(3) iff for all 1-sequences Γ(3) K there exists X (3)such that [K, X] is stationary.Definition 18 Suppose Γ(3) is a third-order pointclass as above. The collectionof Γ Γ(3)n -formulas is defined exactly as before except that now all third-orderuniversal quantifiers are replaced with ∀X Γ(3) and interpreted to range over thepointclass Γ(3).Theorem 19 Suppose V κ satisfies Γ Γ(3)1 -reflection. Then κ is 1-stationary Γ(3) .19

Proof. The proof is a modification of the proof of Theorem 14. Suppose,for contradiction, that κ is not 1-stationary Γ(3) . Let K : κ → V κ be a 1-sequence Γ(3) which is a counter-example to the 1-stationarity Γ(3) of κ. Foreach X Γ(3) ∈ P(P(κ)) let C X be a club such that [K, X] ∩ C X = ∅. Let C Kbe the club from the definition of a 1-sequence Γ(3) . LetLetT (4) = {(K (2) , X Γ(3) , C (2)X , C (2)K ) | X Γ(3) ∈ P(P(κ))}.ϕ(T (4) ) = ∀X Γ(3) ∃K (2) ∃C (2) ∃C ′(2) ( (K, X, C, C ′ ) ∈ T (4) ∧C 1 and C 2 are unbounded ) .This is a Γ Γ(3)1 -formula. As before, for each X Γ(3) ∈ P(P(κ)),(1) V κ |= “C X is unbounded” and(2) for all β ∈ [K, X], V β ̸|= “C β X is unbounded”.It follows thatV κ |= ϕ(T (4) ),since for each X Γ(3) ∈ P(P(κ)) our fixed K, C K and chosen C X are witnesses.It remains to prove thatV β ̸|= ϕ β (T (4),β ),for all β < κ. Suppose, for contradiction, that β < κ is such thatthat is,V β |= ϕ β (T (4),β ),V β |= ∀X Γ(3) ∃K (2) ∃C (2) ∃C ′(2) ( (K, X, C, C ′ ) ∈ T (4),β ∧C and C ′ are unbounded ) .Notice that C 1 = CK. β Moreover, since C β K is unbounded in β and since C Kis club, it follows that β ∈ C K . Thus, X = K(β) is in Γ(3) and hence isa legitimate substituent for the universal quantifier ∀X Γ(3) in the formuladisplayed above. The rest of the proof is as before. ✷Theorem 20 Suppose Γ(3) is a pointclass in the generalized Borel hierarchythat properly extends the closed classes. Then Γ Γ(3)1 -reflection is inconsistent.Proof. This follows from the previous theorem in conjunction with the earliercounter-examples. ✷20

Because of this one is essentially forced to pare the third-order quantifiers downto the closed sets. The question remains whether doing so leads to consistentreflection principles.Theorem 21 Assume κ = κ(ω) exists. Then there is a δ < κ such that V δsatisfies Γ c(3)n -reflection for all n < ω.Proof. The proof is as before. The key point is that for B c(3) ,j(B) c(3),ι 0= B c(3) .Thus, we can extract the derived Skolem function in the third implication ofthe induction step as before. ✷It remains to consider quantifiers of order beyond third-order. For the reasonsnoted earlier we may, without loss of generality, concentrate on n th -orderquantifiers which range over P n−1 (α) when interpreted over V α .The earlier counter-examples easily generalize to higher-orders and enableus to isolate the appropriate notion of closure needed to avoid them. Forillustrative purposes we concentrate on the fourth-order.For α ∈ Lim and for γ < α let Tγα be the tree consisting of the single branchb ∈ 2 α such that for all ξ < γ + 1, b(ξ) = 1 and for all ξ ≥ γ + 1, b(ξ) = 0. Forα ∈ Lim, letD α = {Tγα | γ < α}.Let K D : κ → V κ be such that K(α) = D α if α ∈ Lim and K D (α) = ∅otherwise. This is a 1-sequence (4) such that for each X (4) ⊆ V κ+2 , [K D , X]∩Limcontains at most one point.To rule out such counter-examples we must restrict to fourth-order classesthat are “closed” in the following sense: Suppose 〈T γ | γ < α〉 is a sequenceof trees such that each T γ ⊆ 2

level Γ(m) of this hierarchy beyond c(m), Γ Γ(m)1 -reflection is inconsistent forall m > 2.It remains to see that for m > 2 and n < ω, Γ c(m)n -reflection is weak. Workover V κ . The key point is that we can code trees T ⊆ 2 2 and n < ω.(2) Inconsistent: Γ Γ(m)n -reflection, for all m > 2 and n ≥ 1 and any pointclassΓ(m) containing the first level of the generalized Borel hierarchybeyond the closed sets.22

7 DiscussionIn the above results we have concentrated on quantifiers and parameters offinite order. However, one can make sense of quantifiers and parameters oftransfinite orders and, for each ordinal α, one can define the notion of Γ (α)n -reflection in the natural way. The results generalize to show (i) that to avoidinconsistency one must impose a closure constraint on the universal quantifiersof third- and higher-order and (ii) that resulting reflection principles arebounded below κ(ω).One would like to conclude from this that “reflection principles” in generalare weak. But absent a precise characterization of the notion of a reflectionprinciple one cannot state, let alone prove, a limitative result to this effect.Moreover, it is hard to see how one could give an adequate precise characterizationof the informal notion of a reflection principle since the notion appearsto be inherently schematic and “indefinitely extendible” in the sense that anyattempted precisification can be transcended by reflecting on reflection. However,our main limitative result is also schematic and the proof would appearto be able to track any degree of reflecting on reflection—the Erdös cardinalκ(ω) appears to be an impassable barrier as far as reflection is concerned.This is not a precise statement. But it leads to the following challenge: Formulatea strong reflection principle which is intrinsically justified on the iterativeconception of set and which breaks the κ(ω) barrier.It is natural at this point to think of the classic discussion of the justificationof new axioms in Reinhardt (1974). It is important to note, however,that this discussion involves a very difference conception of set which has itsroots in Reinhardt’s dissertation (Reinhardt (1967)). This conception involvessupplementing the iterative conception with what one might call the theory oflegitimate candidates. On this view there are a number of “possible alternativeinterpretations of V ”, each of which has the form V α . Let V α0 , V α1 be a pairof such candidates, where V α0 V α1 . Reinhardt’s basic method for obtainingstrong principles is “to exploit the principle which says that mathematicaltruths should be necessary truths” and “[a]ccording to this principle, if thenotion of possibility we have introduced is a good one, something true in oneinterpretation of V should be necessarily true, that is, true in all possible alternativeinterpretations of V ” (Reinhardt (1967), p. 76). In particular, takingthe language to be first-order with parameters from V α0 one should have thatfor each ϕ and for each parameter a ∈ V α0 , V α0 |= ϕ[a] iff V α1 |= ϕ[a], thatis, V α0 ≺ V α1 . The next step is to enrich the language to second-order and allowsecond-order parameters. Reinhardt assumes that for each class X ⊂ V α0one can “reinterpret” the class over V α1 as j(X) ⊂ V α1 in such a way that(V α0 , X) ≺ (V α1 , j(X)). Letting the interpretation function j be constant onelements of V α1 this is equivalent to asserting that there is an elementary em-23

edding j : V α0 +1 → V α1 +1, with critical point α 0 , that is, it is equivalent toasserting that α 0 is a 1-extendible.There are a number of difficulties with this approach—there are problemswith the underlying conception and problems with the derivation of strongprinciples. One problem with the underlying conception is that the theory ofpotential candidates is difficult to defend. For example, since candidates whichcome later in the sequence will have greater closure properties than candidateswhich come earlier it is hard to defend the idea that they are both equallylegitimate interpretations of V . Another difficulty is that underlying notionof mathematical modality would require considerable clarification and defense(especially in light of the fact that mathematics is traditionally thought toconcern objects that necessarily exist).But even if the underlying conception can be clarified in a satisfactory way,there are two problems with the derivation of strong principles. The first problemis what one might call the problem of tracking: In reinterpreting the classX ⊂ V α0 as j(X) ⊂ V α1 there must be some intensional notion at play. Now,one can certainly track definable classes by using their definitions. But Reinhardtwishes to shift every subset of V α0 and for this he requires an exceedinglyrich collection of intensional notions. Moreover, these intensional notions mustbe of a very special sort. For example, it would not do to associate to each setthe concept of being that set since Reinhardt needs to “stretch” the classes.It is unclear that such a collection of concepts exists. Moreover, even if itdid it would be a further step to assume that it gave rise to an elementaryembedding of the required sort.The second problem is what one might call the problem of extendibility toinconsistency: Even if one could provide and defend a theory of the requiredintensional objects it would appear that the theory would generalize and leadto inconsistency. In his dissertation Reinhardt did indeed think that the theorygeneralized: “[I]n order to extend [the above schema] to allow parametersof arbitrary (in the sense of V 2 ) order over V 0 we simply remove the restrictionX ⊆ V 0 .” (Reinhardt (1967), p. 79). Here V 0 is our V α1 and V 2 is some legitimatecandidate V α2 beyond V α0 and V α1 . The trouble is that when one generalizesin this way the result is a non-trivial elementary embedding j : V → Vwhich Kunen showed to be inconsistent (with AC). (In fact, Kunen showedthat even the existence of a non-trivial embedding j : V λ+2 → V λ+2 is inconsistent.)By the time he wrote his 1974 paper Reinhardt knew of this result.The point, however, is that the case Reinhardt makes for 1-extendibles appearsto extend to a case for the inconsistent axiom. Hence, unless one cangive principled reasons for blocking the extension, the case falters.One can overcome both the problem of tracking and the problem of extendibilityto consistency by restricting to classes which are definable with parameters.24

Iterating this through the constructible universe built over a given legitimatecandidate V α0 one can make a case for the following axiom, which is moreappropriately called an extension principle: For all γ there exist α 0 and α 1and an elementary embedding j : L(V α0 ) → L(V α1 ) where γ < α 0 < α 1 . Thisaxiom is consistent (relative to mild large cardinal assumptions). So by overcomingthe problem of tracking in this way one also overcomes the problem ofextendibility to inconsistency. Furthermore, the resulting axiom implies thatX # 1exists for all X and so freezes Σ ∼3 . This still leaves us with the problemof defending the underlying conception. In Koellner (2003) I examined thisconception and concluded on a skeptical note. In any case, although such anaxiom might be intrinsically plausible on such an alternative conception, itis hard to see how it could be intrinsically justified solely on the basis of theiterative conception of set that we have been discussing.There is another way in which one might try to justify strong principles resemblingreflection principles. Up until now we have concentrated on principleswhich say that the height of the universe cannot be approximated from below.One might consider related principles which articulate the idea that the widthof the universe cannot be approximated from within. On this approach to saythat the universe cannot be approximated from within is to say that there is no“L-like model” which “approximates” or “covers” the universe. This generalprinciple—the principle of width reflection—would then be rendered precise interms of the various models occurring in inner model theory and their correspondingcovering properties. For example, at the first stage one would simplytake Gödel’s constructible universe L as the approximating universe and asthe notion of approximation one would take the notion involved in Jensen’soriginal covering lemma, that is, to say that L covers V is to say that forevery uncountable set of ordinals X there is an Y ∈ L such that |Y | = |X|and X ⊆ Y . The statement that L does not cover V implies, by the coveringlemma, that 0 # exists. A second application of width reflection would thenlead to the existence of 0 ## . In this way we proceed through the “sharp hierarchy”(using the same covering property in each application of with reflection)until we reach the Dodd-Jensen core model K. One more application of widthreflection yields an inner model with a measurable cardinal. From this stageonward the covering property used in the applications of width reflection isnecessarily weaker, by a result of Prikry. A basic consequence of current innermodel theory (in particular, the core model induction) is that successiveapplications of width reflection ultimately imply PD and AD L(R) and so definitelylead to a significant reduction in incompleteness. However, although theprinciple of width reflection may be intrinsically plausible it is hard to defendthe idea that it is intrinsically justified on the basis of the iterative conceptionof set.There may be other ways of intrinsically justifying principles which lead to asignificant reduction in incompleteness. Gödel certainly believed that a more25

profound analysis of the concept of set (following the lines of Husserl’s phenomenology)would lead to such principles. I am not optimistic but I do notwish to make a stronger claim than that.Let me close by discussing some applications of the above limitative results.The first concerns inner model theory. In the approach discussed above oneapproximates the hypothesis that “there exists a measurable cardinal” fromwithin via width reflection. One would like to approximate 0 # from belowin a similar fashion and the most natural way to do this is through heightreflection. However, the results of this paper make this approach seem doubtfulsince κ(ω) appears to be out of reach of reflection. The second concernsintrinsic justifications. The inconsistency result shows that serious problemscan arise even when one is embarked on the project of unfolding the contentof a conception. It should give us pause in placing too much confidence inthe security of intrinsic justifications. Third, the consistency result shows thatintrinsic justifications, insofar as they are exhausted by the general reflectionprinciples discussed above, will not take us very far. Finally, these results canbe used to provide a rational reconstruction of Gödel’s early view to the effectthat V = L, PU, and CH are “absolutely undecidable”. The idea is that if onehas a conception of set theory which admits only intrinsic justifications andif one thinks that these are exhausted by reflection principles then the aboveresults make a case for the claim that these statements really are “absolutelyundecidable”. 11 Fortunately, extrinsic justifications go a long way and I thinkthat one can make a strong extrinsic case for V ≠ L and PU. 12 Whether CHis “absolutely undecidable” is, of course, a more delicate question. 13ReferencesFeferman, S., 1991. Reflecting on incompleteness. Journal of Symbolic Logic56, 1–49.Gödel, K., 1947. What is Cantor’s continuum problem? In: Gödel (1990).Oxford University Press, pp. 176–187.Gödel, K., 1964. What is Cantor’s continuum problem? In: Gödel (1990).Oxford University Press, pp. 254–270.Gödel, K., 1990. Collected Works, Volume II: Publications 1938–1974. OxfordUniversity Press, New York and Oxford.Koellner, P., 2003. The search for new axioms. Ph.D. thesis, MIT.Koellner, P., 2006. On the question of absolute undecidability. PhilosophiaMathematica 14 (2), 153–188.11 See Sections 1 and 2 of Koellner (2006).12 See Section 3 of Koellner (2006) and the references therein.13 See Sections 4 and 5 of Koellner (2006) and Koellner and Woodin (2008) for moreon this topic.26

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