[FOM] ZFC

[Harvey Friedman]

>1. Axiom of Choice: I believe the right context in which to view its 
>history (see Moore's excellent ZERMELO'S AXIOM OF CHOICE, Springer 
>1982, for a detailed blow-by-blow account) is a conflict for the 
>"soul" of mathematics among three contender's:
>1. a purely extensional mathematics (Cantor, Dedekind, Frege,Hilbert 
>- but with a bow to 2)
>2. a constructivist mathematics (Kronecker, Brouwer, Weyl)
>3. a predicative mathematics (Poincare, Weyl -again but earlier, 
>Lebesgue, Borel)
>Although the battle was fought with loud philosophical proclamations 
>on all sides, in my opinion what was really decisive was the 
>question of the extent to which the widest view could be reliably 
>used in mathematical practice. After being profoundly shaken by the 
>paradoxes, calm and assurance was gradually restored, and, except on 
>the fringes, view 1 has prevailed. Of course, 2 and 3 have each 
>given rise to interesting and important foundational studies.

I agree with this, but would also add the following about the 
mathematicians' views.

For results to be stated in expositionally natural generality, a 
liberal viewpoint admitting all kinds of nonconstructive and 
impredicative constructions, such as in full blown set theory, is 
very useful and appropriate.

However, there is the general feeling that in context that really 
matter, far less is needed, and one can dispense with the luxury of 
the previous paragraph if need be.

This explains the present day indifference to substantial set 
theoretic issues. This will change according to the level of 
penetration of substantial set theoretic methods into contexts that 
"really matter" from the mathematician's point of view.

>
>I do not believe it to be true that Cantor ever sought a proof of AC 
>(as Harvey said); he simply assumed well ordering as an obvious 
>consequence of his extensional viewpoint. It was part of Hilbert's 
>1st problem (along with CH). I believe that the first explicit 
>statement of something like AC as a separate principle was by Peano, 
>who regarded it as obviously false. It was Zermelo who isolated AC 
>and showed how to prove well-ordering from it. Criticism of his 
>proof led him to formulate his axioms for set theory to make it all 
>explicit.

I cannot find any evidence in the few historical books I have that 
Cantor tried to prove the axiom of choice as we know it today. 
However, it DOES appear that he had tried to prove that the continuum 
is of cardinality an aleph, in response to Konig's claim of the 
opposite:

 From Dauben, page 248-250.

"Then [Konig] launched his proof that the power of the continuum was 
not an alpeh by employing a general theorem Bernstein had presented 
... Despite Cantor's eagerness to produce the complete and absolute 
refutation of Konig's results, he was unable to do so. Even before 
the congress was over, everyone was fully aware that Konig's proof 
had failed, but that did not dispel the possibility that at any 
moment he might find a new way to repair his proof. The only 
conclusive answer to Konig would be the proof that the continuum 
could be well-ordered or that every transfinite cardinal number was 
in fact an alpeh. but these were results which Cantor was unable to 
establish himself."

So apparently Cantor had a perfect opportunity to forcefully declare 
that Konig was refuting something obvious, like trying to refute that 
any two sets have an unordered pair. He apparently didn't take that 
view, but instead tried to prove at lest this special case of 
Zermelo's well ordering theorem.

>
>2. "Completeness" of ZFC: Harvey has proposed that in some sense ZFC 
>is complete. Without an explicit indication, at least roughly, of 
>what he has in mind, this is hard to discuss. But I do want to 
>register my profound skepticism. In one sense, Harvey's own 
>brilliant work on combinatorial results not obtainable from ZFC is 
>an obvious obstacle such a result would need to overcome.

I thought it might be quite clear what I have in mind from various 
FOM postings.

There is a notion of simplicity in the primitives. The axioms of ZFC 
are simple in the primitives. They also have some coherent picture 
associated with them.

CONJECTURE: Any sentence in the primitives of ZFC that has a simple 
coherent picture is either inconsistent with a weak fragment of ZFC 
or provable in ZFC.

CONJECTURE: Any sentence in the primitives of ZFC that has an simple 
explicitly coherent picture is either inconsistent with a weak 
fragment of ZF or provable in ZF.

>
>I see the ZF axioms as "closure" principles under suitable 
>operations, as in algebraic structures. The axioms are of the two 
>forms:
>a) Such and such is a set.
>b) If this and that is a set, then so is this other.
>Such a situation leads one naturally to form the minimal structure 
>closed under these operations.
>In the case of ZF this leads one to accept (strongly) inaccessible 
>cardinals. This is the sort of thing G\"odel had in mind when he 
>said (in 1933):

The existence of a (strongly) inaccessible cardinal is technical, and 
so does not refute my conjectures.

Harvey Friedman