FOM: Relevance and mathematical modeling: a case in point

Lou van den Dries vddries at
Wed Nov 19 13:17:41 EST 1997

Barwise refers to "mathematicians who do not know of a principled
response to, say, Russell's paradox".

Why is it not a principled response to consider the idea of defining
"the set of all things satisfying a given property" as preposterous
on its face? The notions involved here are simply so vague that
applying precise reasoning to such vague ideas is bound to lead to
logical disasters. The remedy of Zermelo (and Skolem) to first
of all consider only things lying in some given (supposedly welldefined)
set, and considering first-order properties in the language of set-theory
seems completely sensible to me. In fact, Frege should have adopted
this in his own system, and it's hard for me to understand why he
didn't do so. Anyway, I am always amazed that paradoxes based on
vague notions combined with precise logical reasoning are still
being discussed seriously in connection with mathematics. 
  (To avoid misunderstandings, I consider it quite possible that ZFC
is inconsistent, in which case the iterated power set axiom will
probably be to blame, not comprehension, and I also think that in
that case all worthwhile parts of math will hardly be effected.)

-Lou van den Dries-

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