On 09 Jun 2015, at 01:26, Bruce Kellett wrote:
LizR wrote:
On 9 June 2015 at 05:31, meekerdb <[email protected] <mailto:[email protected]
>> 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. So does that make set
theory and its consequences real? Reality isn't defined by what
everyone agrees on. What makes ZFC (or whatever) real, or not, is
whether it kicks back. Is it something that was invented, and could
equally well have been invented differently, or was it discovered
as a result of following a chain of logical reasoning from certain
axioms?
Why do not those same arguments apply equally to arithmetic? What
axioms led to arithmetic? Could one have chosen different axioms?
Take RA, PA, PA+con(PA), PA + con(PA + con PA), etc. (con PA = "PA is
consistent"), DA, etc.
All those theories leads to the same arithmetical truth. Each theory
is just included in the next theory, but if one of them say that a
proposition is a theorem, the negation of it will not be a theorem in
any of them.
So there are many different theories of arithmetic, but they all
describes the same structure.
That's not the case in set theory, where many different theories leads
to different theorems.
Of course, by incompleteness, you could take the theory PA + ~con(PA).
That theory will lead to new theorem, which are false in the standard
model, but arithmetical truth is defined using the standard model. Non
standard models have some interest, but not for comp or for number
theory; unless when use indirectly, to make some argument non valid.
Bruno
I recall that RA = Robinson arithmetic: it has the following axioms
(on the top of predicate calculus):
0 ≠ s(x) (= 0 is not the successor of a number)
s(x) = s(y) -> x = y (different numbers have different successors)
x = 0 v Ey(x = s(y)) (except for 0, all numbers have a predecessor)
x+0 = x (if you add zero to a number, you get
that number)
x+s(y) = s(x+y) (if you add a number x to a successor of a number y,
you get the successor of x added to y)
x*0=0 (if you multiply a number by 0, you get 0)
x*s(y)=(x*y)+x (exercise)
PA is RA + the induction axiom (on first order sentence).
DA is Dedekind Arithmetic: it is like PA, except you can throw out
most axioms, as it has the very powerful second order full induction
axioms (on all set of numbers). DA defines categorically the standard
model, but is not an effective theory (you can't check all proofs, as
the notion of set is too vague).
Bruno
Bruce
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