Re: Mathematicians are in deep *** for 2 reasons

On 2008-04-22, in sci.logic, David C Ullrich wrote:
Whether any set theorist has every worked with it or not, I do
think it deserves a name. Because it does seem to me that a
lot of (ignorant unwashed) mathematicians do work in such
a theory.

The set theoretic arguments and thinking we find in the works of
mathematicians usually concern just the naturals, sets of naturals,
the reals, sets of reals, functions on reals, and so on -- essentially
the few infinite levels of the cumulative hierarchy. This reasoning
and the set theoretic constructs used are, by set theoretic standards,
quite innocuous and never do we meet such things as the set of all
sets or of all ordinals. Even proofs about structures in general, such
as we might meet in group theory, have the property that, relative to
a given arbitrary structure exhibiting the relevant characteristics,
we meet only a few iterations of the "set-of" operation.

Not that they actually give any thought to what sort of set
theory they're using, but bare {x : P(x)}'s come up all the time.
When they come up in actual mathematics it's typically
trivial to some up with an S so that the above is the same
as {x in S : P(x)}, but I don't get the impression that most
mathematicians are aware this is required in "official" set
theories.

What they are or are not aware of is not at issue, but rather what
sort of set theoretic arguments we find in their mathematical work.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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