[FOM] On Existence of Mathematical Objects: reply to Taranovsky

JoeShipman@aol.com JoeShipman at aol.com
Sat Oct 4 14:21:25 EDT 2003

Taranovsky writes:
>>>The issue of multiple universes of sets does explain the fact that so
many important questions are undecidable in ZFC.  Figuratively speaking,
the deductive apparatus of ZFC does not know that ZFC is meant to be the
theory of all sets and considers a statement to be a theorem only if it
is true in every universe that satisfies the basic closure conditions
enumerated in ZFC.  The solution is to find new set existence axioms and
closure properties:  Since the universe consists of all sets, it should
satisfy strong closure properties,  and since every set that exists in
some universe actually exists, it should also satisfy strong existence
axioms.  The universe, being unique rather than random, should also be
canonical.  The difficulty is that we can know which statements are
subtle approximations to the claim that every set exists or that the
structure of the universe is canonical only after we carefully research
their consequences and their subtle relationship with other
propositions.  Only recently did it become clear that projective
determinacy is the correct existence axiom for second order arithmetic. 
The vast realms of the set theoretical universe are yet to be explored.<<<

This is thought-provoking and seems largely right.  But can you please clarify why "projective
determinacy is the correct existence axiom for second order arithmetic"?

-- JS

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