Just to be clear on this:

On 09 Dec 2010, at 20:43, Brian Tenneson wrote:

Is there any first order formula true in only one of R and R*?

So yes, there is one: the weak pure archimedian formula AF:

AF:     for all x there is a y such that (x<y)

(not your: "for all X there is a Y such that (Y is a natural number and X<Y)", because this is a second order formula. You cannot defined "natural number" in first order logic (actually you cannot defined "finite" in first order logic).



I would think that if the answer is NO then R < R*.

You would be right. But AF is true in R, and false in R*

In R* there is an object "infinity" which is such that there is no y such that "infinity" < y, making AF false.


Bruno






What I'm exploring is the connection of < to [=], with the statement
that < implies [=].

Are there any other comparitive relations besides elementary embedding
that would fit with what I'm trying to do?  What I'm trying to do is
one major "leg" of my paper: there is a "superstructure" to all
structures.  What super means could be any comparitive relation.  But
what relation is 'good'?

On Dec 9, 8:12 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
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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