Re: non-Archimedean models of Euclidean geometry?

On Tue, 10 Mar 2009 04:36:17 -0700 (PDT), Gc <Gcut667@xxxxxxxxxxx>
wrote:

On 10 maalis, 13:13, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Mon, 9 Mar 2009 09:27:14 -0700 (PDT), Gc <Gcut...@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.

So now I understand that you were speaking about Tarski axioms with
you added axioms c > 0, c > 1,.....and constant
c, 1, 2.... Now you must understand that 0, 1... can mean anything if
you can`t define the naturals in the original theory. In particular
they can be elements of a converging sequence in R. So your proof
works if we can define naturals one by one in the theory and we have
relation <, without added axioms you didn`t mention.. Suppose we would
have 1, so you can define naturals by 1, 1 + 1, 1 + 1 +1...etc. In
present theory we don`t have 1,

Ok, here's a purely "geometric" version of the argument:

Assume we have a definition of "c is a line segment"
(meaning a non-degenerate line segment, not a
single point.) Assume that given two line segments
c and d we have a way to define a line segment of
length equal to the length of c plus the length of d
(you can certainly do that with ruler and compass,
for example). Below I'll use the abbreviation
"c + d" for such a segment. Assume that we have
a definition of "segment c is shorter than
segment d"; below I'll abbreviate this as
"c < d".

Now take your theory, add two constant
symbols c and d to the language, and add
the axioms

c is a line segment
d is a line segment
c > d
c > d + d
c > (d + d) + d
....

The compactness theorem shows that _that_
theory has a model, and a model of that
theory is a non-archimedean model of the
original theory.

You could have figured this out for yourself...

or any obvious way to define naturals.
Moreover I suspect that if we have recursion definition of the sum,
constant 1 and a archimedian model, we can prove that any FOL theory
is undeciable.

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.)
.

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