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

Aatu Koskensilta wrote:
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.

Have you seen the following link?
http://plato.stanford.edu/entries/dedekind-foundations/

To quote:

As noted, Dedekind has gone beyond considering only sets of numbers in his essay on the natural numbers. This is a significant extension of the notion of set [...]
[...] his set theory is subject to the set-theoretic antinomies, including Russell's antinomy. [...]
The discovery shocked Dedekind. Not only did he delay republication of Was sind und was sollen die Zahlen? because of it; pending a resolution, he even expressed doubts about 'whether human thinking is fully rational'.

(end of quote)

Or was 'Was sind und was sollen die Zahlen?' not a mathematical work?

--
Cheers,
Herman Jurjus
.

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