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

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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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