Re: proven FROM WHAT AXIOMS??
- From: kleptomaniac666_@xxxxxxxxxxx
- Date: Thu, 4 Sep 2008 16:11:21 -0700 (PDT)
On Sep 4, 1:12 am, george <gree...@xxxxxxxxxxxxx> wrote:
On Sep 2, 12:39 pm, kleptomaniac6...@xxxxxxxxxxx wrote:
The arithmetic statement that encodes the consistency of Peano
Arithmetic is a proven mathematical theorem.
This is of no use to anyone.
There are a great many different proofs of the consistency of PA, but
the point is,
none of them IS IN (or from) PA itself. They ALL require a leap of
faith in the
consistency of the EVEN STRONGER, even MORE powerful axiom-system
in which THAT proof is conducted.
No, it's simple. The Diophantine statement Con(PA) can be proven from
axioms which are obviously true, therefore it is true.
It is quite complicated,
but is nevertheless a theorem.
This is only helpful if the MORE complicated axiom-system from which
this more-
complicated proof was derived IS CONSISTENT. THAT was PRECISELY the
question
that was in doubt TO BEGIN with!!
No, it's simple. The Diophantine statement Con(PA) can be proven from
axioms which are obviously true, therefore it is true.
Someone having objections (whatever
objections they may be) to it's truth is no different than someone
saying "Lagrange's theorem (in group theory) might be false" or "the
fundamental theorem of calculus might be false" or "the inclusion-
exclusion principle might be false".
Truth and falsity are not even legitimately invoked AT ALL in this
context.
Truth and falsity occur IN MODELS as OPPOSED to in theories.
Models, schmodels. Con(PA) is true, Lagrange's theorem in group theory
is true, I don't need to talk about "models" to say that. In fact I
don't even have to have ever heard the phrase before.
Godel's theorem proves that there are models of PA in which the
statement
commonly taken to mean "Con(PA)" is itself false. You therefore
ALWAYS
have to CLARIFY, BEFORE you say "true" or "false", which axiom-set is
producing your models and which of the produced models you are
preferring
(since different models will give different truth-values for important
questions).
Hmm, I'm tempted to say "models, schmodels" again. My phrasebook is
limited, you see. Actually, I think I'll use: "No, it's simple. The
Diophantine statement Con(PA) can be proven from axioms which are
obviously true, therefore it is true".
I'm not sure what it means to say the set of numbers N is "definite"
or "indefinite". As a gripe, it's so vague as to mean nothing. As for
unique, that does have a fixed meaning, but in mathematics when we
Oh, shut up.
You can't generalize over what every mathematician always does.
Uhh, what? The word "unique" has a generally used mathematical
meaning, which I just described.
You need to PICK a PARTICULAR foundation or axiomatization and
start from there. Historically it has usually been ZFC. In this
context
people will often advocate for other things, but some of those things
are
second-order, and second-order logic is so silly that most of the
time,
people are CALLING something second-order while it IS ACTUALLY a
first-order system.
As given, the declaration of skepticism towards the "definiteness" and
"uniqueness" of N makes no sense. Further, Nelson's main view makes no
sense. It's not wrong, it just makes no sense. His "reason" for taking
his various views is that "numbers are made or created by us". For
example, he would say that 5 got created by us.
You're right; he's wrong.
If Math is about anything, it is about at least speaking AS THOUGH
abstractions actually "existed".
I didn't say he was wrong, I said what he said didn't make sense.
.
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