SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
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8 MENACHEM MAGIDORFeferman in [8] makes the clear distinction between structural axiomswhose role is to organize and expose a body of m<strong>at</strong>hem<strong>at</strong>ical work andthere is a gre<strong>at</strong> freedom in their form<strong>at</strong>ion and found<strong>at</strong>ional axiomswhich are very basic to the concepts studied. For the l<strong>at</strong>er he quotesas a requirement the definition of the word ”axiom” from the OxfordEnglish dictionaryA self-evident proposition requiring no formal demonstr<strong>at</strong>ionto prove its truth, but received and assented toas soon as mentioned.We think th<strong>at</strong> this is too restrictive because the process by which anaxiom is accepted as a fundamental and n<strong>at</strong>ural maybe a long processof reflection. The axiom can be found<strong>at</strong>ional but still not ”received andassented to as soon as mentioned”. I guess th<strong>at</strong> any axiom<strong>at</strong>iz<strong>at</strong>ion ofQuantum Mechanics will start from the axiom th<strong>at</strong> the set of st<strong>at</strong>es 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 th<strong>at</strong>an axiom in order to be adapted has to conform the concept understudy. So we put it r<strong>at</strong>her vaguely th<strong>at</strong>The 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 th<strong>at</strong> 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 underestim<strong>at</strong>e them as importantsource for insights on the basic concepts.Independence was the motiv<strong>at</strong>ing force for introducing new axiomstherefore it is n<strong>at</strong>ural th<strong>at</strong> we shall expect th<strong>at</strong>The new axiom should be strong enough to decide alarge class of st<strong>at</strong>ements which are undecidable on thebasis of the axioms adapted so far.When the axiom decides a class of problems we would prefer th<strong>at</strong> 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