Re: help with Godel's
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 8 Mar 2007 15:39:33 -0800
The answer is, nothing in general is ever MERELY "proven",
just by-itself "proven". Everything is instead rather proven FROM
something. Godel's theorem was about "recursive axiom-sets"
for the natural numbers. Given any such set of axioms, there
will be things true about the natural numbers that you cannot
prove FROM *THOSE* AXIOMS.
But you can still know that they are true because you can prove
them FROM OTHER bigger better axiom-sets, that you trust
for other deeper reasons.
On Mar 5, 8:38 pm, maina...@xxxxxxxxx wrote:
When you say "bigger and better", you mean more general right?
No, not really. Since the original holy grail was to talk about
the natural numbers specifically, and since every recursive
axiomatization will wind up ALSO talking about its non-standard
models, the actual goal here is the OPPOSITE of generality:
as your axiom-set gets bigger, as you add new axioms, more
models get EXCLUDED from consideration, and your theory
becomes more SPECIFIC, more closely matched to, the ONE
model that you are ultimately trying to be about.
But your intuition is right in that achieving significant progress on
this front may well require migrating to a different language. But
the new language is not so much more "general" as it is more
detailed and nuanced, or "lower"-level, as we would say, from
computer science, like going down from a high-level programming
language to an assembly language.
Specifically, in this case, we could start with the axioms in the
language of Peano Arithmetic and then migrate to using the
language of set theory (ZFC) instead. In set theory, it is possible
to state criteria for distinguishing between finite and infinite sets.
Since the standard model has only finite numbers, while all the
non-standard ones have a lot of infinite ones, being able to make
this distinction enables us to match our theory more closely to
the standard model of the naturals.
"better" is rather subjective. I think, in math, you more you prove,
the better you know about it's properties. But switching axioms...Is
that even possible?
You can't "switch" if it changes the truth value of something
that you were previously able to prove, no. But as long as the
new axioms leave all the previously-decided sentences decided
the same way, you can switch to something that also now decides
sentences that were not decided before.
As soon you as do that, how do know you the new
model is "closer" to what natural numbers mean
The issue is NOT the new model: the issue IS the new THEORY.
You can now prove things you couldn't prove before.
This means you can throw away models that you couldn't
exclude before. Before, to every sentence that you could
not prove or disprove, there corresponded TWO models, one
with it true and one with it false. You will never throw away
"all" the unintended models (as long as you keep a recursive
axiom-set over a first-order language) but you can throw away
more and uglier ones, and keep ones that are closer to the
right one. Knowing that anything is "closer to the right one"
required being able to actually evaluate the truth of the sentence
IN the right/standard model. As a general rule, we can't do that
either; all we can do is add more axioms.
.... I don't even want
think about what natural means...
It just means finite, in this context.
However, you can have infinite sets of finite numbers.
.
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