On 04 Dec 2010, at 18:50, Brian Tenneson wrote:

On Dec 4, 2:52 am, Bruno Marchal <marc...@ulb.ac.be> wrote:

I just said that if M1 < M2, then M1 [=] M2. This means that M2 needs
higher order logical formula to be distinguished from M1.
Elementary embeddings (<) are a too much strong notion of model
theory. It is used in context where we want use non standard notions,
like in Robinson analysis.

Doesn't the archemedian property show that R is not elementarily
equivalent to R*?  I mean the following 1st order formula true in only
one of R and R*:
for all X there is a Y such that (Y is a natural number and X<Y)

Note that you cannot define "natural number" in a first order theory of the reals. In the reals, natural numbers are second order notions, or you have to add a first order axiomatic of the sinusoïdal function.



This is true in R but not in R*.  This would appear to me to be an
example of why R is not [=] to R*.

That means that R (standard model of the first order theory of the reals + archimedian axiom, without the term "natural number") is not elementary embeddable in R*, given that such an embedding has to preserve all first order formula (purely first order formula, and so without notion like "natural number").

Bruno






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http://iridia.ulb.ac.be/~marchal/



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