On 09 Jun 2015, at 19:15, meekerdb wrote:
On 6/9/2015 12:46 AM, Bruno Marchal wrote:
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.
So what? That just makes my point that Platonia implies many
different "realities". First order predicate logic is also "a clear
standard model". So it must be as "real" as arithmetic. And
arithmetic isn't so complete as you imply - that's why negative
numbers and fractions and reals were invented.
The first orrder theory of the real is complete. real numbers are an
oversimplification of the natural numbers (integeres and rationals add
nothing, with respect to computation). Robinson arithmetic is sigma_1
complete (not complete). Arithmetical truth is trivially complete
about arithmetic. No other notion of completeness is used.
Bruno
Brent
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