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

V. Yu. Shavrukov vys1 at
Wed Sep 13 03:46:32 EDT 2000

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

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

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

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?


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