On 09 Dec 2010, at 05:12, Brian Tenneson wrote:
On Dec 5, 12:02 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 04 Dec 2010, at 18:50, Brian Tenneson wrote:
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").
I'm a bit confused. Is R < R* or not? I thought there was a fairly
natural way to elementarily embed R in R*.
I would say that NOT(R < R*).
*You* gave me the counter example. The archimedian axiom. You are
confusing (like me when I read your draft the first time) an
algebraical injective morphism with an elementary embedding. But
elementary embedding conserves the truth of all first order formula,
and then the archimedian axiom (without natural numbers) is true in R
but not in R*.
Elementary embeddings are *terribly* conservator, quite unlike
algebraical monomorphism or categorical arrows, or Turing emulations.
Bruno
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