SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

More documents

Recommendations

Info

8 MENACHEM MAGIDORFeferman in [8] makes the clear distinction between structural axiomswhose role is to organize and expose a body of mathematical work andthere is a great freedom in their formation and foundational axiomswhich are very basic to the concepts studied. For the later he quotesas a requirement the definition of the word ”axiom” from the OxfordEnglish dictionaryA self-evident proposition requiring no formal demonstrationto prove its truth, but received and assented toas soon as mentioned.We think that this is too restrictive because the process by which anaxiom is accepted as a fundamental and natural maybe a long processof reflection. The axiom can be foundational but still not ”received andassented to as soon as mentioned”. I guess that any axiomatization ofQuantum Mechanics will start from the axiom that the set of states of aphysical system is the set of one dimensional subspaces of Hilbert space.This is universally accepted by physicists but it had to go through along process before being ”received and assented”. Still we believe thatan axiom in order to be adapted has to conform the concept understudy. So we put it rather vaguely thatThe new axiom should have intuitive or philosophicalappeal. It should conform to some mental image of thebasic concepts of Set Theory.Some semi-serious way of phrasing this principle is that a good axiomneeds a good slogan in order to be adapted. Of course the intuitivemental images are not always a reliable guide to fruitfulness or truthbut on the other hand we should not underestimate them as importantsource for insights on the basic concepts.Independence was the motivating force for introducing new axiomstherefore it is natural that we shall expect thatThe new axiom should be strong enough to decide alarge class of statements which are undecidable on thebasis of the axioms adapted so far.When the axiom decides a class of problems we would prefer that itgives a coherent structure to these problems.The Axiom should produce a coherent elegant theoryfor some important class of problems.A well known example of interplay of the last two principles is thethe decision between PD and V = L. Both them provides a very richstructure theory of the projective sets of reals but where the structureof the projective sets given by PD is much more elegant and coherent
SOME SET THEORIES ARE MORE EQUAL 9than the structure given by V = L. When PD was suggested for thefirst time I think that it lacked somewhat as far as the first principle :of fitting some intuitive mental image of the universe of sets. But thework of Martin-Steel and Woodin which derived PD and stronger determinacystatements from large cardinals increased its intuitive appealsubstantially .The next principle is along the lines of the passage form Gödel quotedabove.To the extent possible the axiom should have testableverifiable consequences.Here we have to explain what do we mean by ”testable verifiable”consequences. We are not talking about experimental science so the”verification” is a mathematical verification. So for instance if theaxiom has some new Π 1 0 consequences for number theory then the factthat so far we did not get a counter example is a verifying fact. (Ofcourse this could change with time but this is not different than anyscientific theory.) An evidence for the axiom could be a result that weintuitively believe that it is true and we were not able to derive withoutthe new axiom. I think that it is even possible that axioms could betested by their impact on fields outside of mathematics like physics. Itmay sound like an outrageous speculation and admittedly we do nothave any concrete example of such a possible impact , but in the nextsection we shall give an example where the set theory we use may havesome relevance to the mathematical environment in which a physicaltheory is embedded.The standard way in which we generate different set theories is byforcing . The forcing extensions of a given universe of ZFC can haveproperties which are very different from the properties of the originaluniverse. But some properties are resilient under forcing extensions.(We are talking about set forcing. As shown by Jensen followed by S.Friedman and others that using class forcing we can make dramaticchanges to the properties and the structure of the ground universe.) For instance if 0 # exists in the ground model it still exists in anygeneric extensions. If the ground model has a proper class of one ofthe standard large cardinals notions then a set forcing extension doesnot change this fact. A very remarkable generic absoluteness result isthe result of Woodin showing that the theory of L[R] is absolute underforcing set forcing extensions, provided the universe contains a properclass of Woodin cardinals. While we agree with Foreman in [13] thatthe generic absoluteness can not by itself be an argument for adapting anew axiom, we still think that having resilient axioms is a very desirable
Page 4: 4 MENACHEM MAGIDORpossibility ). No
Page 10 and 11: 10 MENACHEM MAGIDORproperty , so if
Page 12 and 13: 12 MENACHEM MAGIDOR. . . We can con
Page 14 and 15: 14 MENACHEM MAGIDORconstructed thus
Page 16 and 17: 16 MENACHEM MAGIDORcan be a good gu
Page 18 and 19: 18 MENACHEM MAGIDORforcing notions
Page 20 and 21: 20 MENACHEM MAGIDORdirection due to
Page 22 and 23: 22 MENACHEM MAGIDORIn the presence
Page 24 and 25: 24 MENACHEM MAGIDORSo not all set t
Page 26: 26 MENACHEM MAGIDOR[42] H. Hugh Woo

SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?