FOM: The logical, the set-theoretical, and the mathematical: Reply to Ketland
V. Yu. Shavrukov
vys1 at mcs.le.ac.uk
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
>"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
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