Re: non-Archimedean models of Euclidean geometry?

On Mon, 9 Mar 2009 09:27:14 -0700 (PDT), Gc <Gcut667@xxxxxxxxxxx>
wrote:

On 9 maalis, 12:22, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sun, 8 Mar 2009 07:44:18 -0700 (PDT), Gc <Gcut...@xxxxxxxxxxx>
wrote:



On 7 maalis, 14:17, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 06 Mar 2009 20:31:43 -0800, Ben Crowell

<crowel...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
I've been trying to absorb a few ideas from Tarski's work leading up
to the proof that elementary Euclidean geometry is complete and
consistent. I'm guessing that the full proof is hopelessly huge
and technical, but I at least want to start by understanding
the interpretation of Tarski's axiomatization.

He has an axiom schema of continuity. It's basically the Dedekind
cut construction, but because he wants a first-order theory he defines
the partitions (A,B) not by quantifications over sets A and B but by
making an axiom schema in which A and B are defined by first-order
formulae. The axiom basically says that if every point in A is to the
left of every point in B, then there exists a point between A and B.

Now non-Archimedean models are generally going to violate this
axiom. E.g., A could be the set of infinitesimals and B the positive
reals. However, I'm not clear on whether this still applies when A
and B have to be defined by first-order logical formulae. The first-
order language in which the formulae have to be constructed can't,
for example, define the set of all infinitesimals I used in the
counterexample above.

So I can tell that this axiom definitely rules out the rationals
as a model, but does it actually rule out the hyperreals or the
surreal numbers?

I don't know about them specifially, but it certainly can't
rule out non-Archimedean models, by the compactness
theorem: Add a new constant symbol c, and consider
the theory plus the sentences c > 0, c < 1, c < 1/2,
c < 1/3, ... .

I think for the schema you have to use symbols of the language in
question or definable in that language.

What are you talking about?

I don`t mean to disrespect you, but I understand that we are speaking
about a schema which contains FOL formulas which are formulas of a
spesific language.

Why in the world would you imagine that? I was not talking
about any schema at all.


David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: non-Archimedean models of Euclidean geometry?
    ... but because he wants a first-order theory he defines ... The axiom basically says that if every point in A is to the ... order language in which the formulae have to be constructed can't, ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: non-Archimedean models of Euclidean geometry?
    ... but because he wants a first-order theory he defines ... The axiom basically says that if every point in A is to the ... order language in which the formulae have to be constructed can't, ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: Finite schema?
    ... Supposing you're working with a language with a different ... over classes and this axiom being a finite schema? ... The comprehension schema restricted to formulas without bound class variables ...
    (sci.math)
  • Re: A set theory equivalent to ZFC.
    ... to the language of ZFC. ... language of your theory in the language of ZFC. ... Axiom schema of Relation 7) Axoim schema of Size Comprehension 8) ...
    (sci.math)
  • Re: AC and Completeness
    ... a new theory T' by adding choice function *symbols* as follows: ... It seems to me that if we restrict whatever axiom schemas ... Not if the language of the theory is countable, ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)