FOM: The logical, the set-theoretical, and the mathematical: Reply to Ketland

[V. Yu. Shavrukov]

Joe Shipman writes:

>[...] Various forms
>of AC are more or less "logical" in flavor but I want to research these
>some more before saying whether I think it can properly be considered a
>"logical" axiom.

>[...]  My favorite form of AC is Russell's
>"Multiplicative Axiom" but there may be others which appear even more
>"logical".

Recall that Bourbaki smuggles AC into their foundational system syntactically,
by allowing (formulas with) epsilon terms with free variables into
Comprehension.

I suppose this at least tells us that some people tend to view AC as a logical
thing.

This trick was much derided by Adrian Mathias on FOM some two years ago.
If I remember correctly, somebody referred to a proof-theoretic assessment,
possibly predating Bourbaki, of the above device.

It would be nice if a knowledgeable person could summarize the conclusions by
e.g. answering the following questions:

What is the result of allowing free variable epsilon terms into various
set existence schemas?  Is it always, or at least "in most cases" equivalent to
adding AC?


sincerely,
V.Shavrukov