FOM: Re: Query on AC

[Joe Shipman]

> Joe Shipman writes:
>  > Dear Steve and Harvey,
>  > Consider the weak form of AC "the collection of nonempty sets of real
>  > numbers has a choice function".  Is there any significant consequence of
>  > AC in "ordinary mathematics" (to be specific, let's say a sentence is
>  > "ordinary mathematics" if it is provably equivalent in ZF without Choice
>  > to a sentence of second-order arithmetic) that is not already a
>  > consequence of this weak form of AC?
>  > -- Joe
>  >

Stephen G Simpson wrote:

> I think your specific question boils down to: Is ZFC conservative over
> ZF +
>
>     P: "there exists a choice function for nonempty sets of reals"
>
> with respect to sentences of 2nd order arithmetic?  The answer is yes.
> First, note that P is equivalent over ZF to the existence of a well
> ordering of R, the set of real numbers.  Now, given a well ordering W
> of R, the inner model L(RU{W}) (i.e. the smallest transitive model of
> ZF containing R and W and all the ordinals) has the same reals as V
> and satisfies ZFC.  This gives the desired conservation result.

Dear Steve,
  Thanks very much!  I hoped that this was true.  My motivation here is to see
how far logicism can be rehabilitated.  Regarding the question whether AC
should be regarded as a mathematical statement or as a "law of thought" (a
logical statement), I wanted to see whether
mathematically restricting AC to sets of reals made it any more plausible.  As
far as ordinary math is concerned, it does not.  This tends to support the
logicist case for AC -- as a law of thought AC ought to apply to arbitrary
collections and this shows extending it from sets of reals to arbitrary sets
doesn't get you into any trouble.
   Justifying the Axiom of Infinity as a law of thought is another issue.  I
have one point to add to the ongoing discussion here.  It would be nice to say
that the Infinity Axiom in the context of ZFC is just the old question of the
existence of "the completed infinite": all the philosophical argument
supporting this concept would then also justify the logicist program.  But of
course to get "all the way up" from Infinity requires the Replacement Axiom,
which is much harder to justify as a law of thought than the Separation Axiom.
We are left with the alternatives

1) There is no "completed infinite" and the Infinity Axiom is just a
mathematical fiction; mathematics is not grounded in logic alone although the
arithmetical part of it is.
2) There is a "completed infinite" but the Choice Axiom is not a law of
thought; the parts of mathematics which require AC are inventions of the
mathematicians and not logically necessary.
3) Logicism justifies Zermelo Set Theory and practically all ordinary
mathematics, but a
statement like "Borel sets are determined" which requires Replacement is not so
justified.
4) The Replacement Axiom is to be regarded as a logical rather than a
mathematical principle;
then all mathematics can be seen as "advanced logic" and logicism is a complete
success (unless and until large cardinals start becoming essential for ordinary
mathematics).

Can anyone provide arguments to support 4)?

-- Joe