On 08 Jun 2015, at 19:31, meekerdb wrote:
On 6/8/2015 1:03 AM, Bruno Marchal wrote:
or that maths exists independently of mathematicians.
That even just arithmetical truth is independent of mathematician.
This is important because everyone agree with any axiomatic of the
numbers, but that is not the case for analysis, real numbers, etc.
Everyone agrees on ZFC in the same sense.
Not at all. There are many non isomorphic approach to set theory and
analysis. For the natural numbers, this does not occur. All theories
have a clear standard model on which we all agree. As Gödel saw, even
intuitionist arithmetic is isomorphic to classical arithmetic: it
changes only the vocabulary.
So does that make set theory and its consequences real?
It is a theory which explain too much. It is interesting for
logicians. Nobody use it, really. people refers to it when confronted
with possible paradoxes, but mathematicians avoid the paradoxes
naturally, and the "modern" one will use some category or elementary
toposes to fix the thing.
Read books on the subject. Arithmetic has a solidity status not
obtained by analysis, or even geometry.Some use ZF + ~AC, ZF + kappa,
or other will use NF (a very different set theory), or intuitionist ZF
(quite different from ZF), or NBG, etc.
Bruno
Brent
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