Just to be clear on this:

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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 thereals + archimedian axiom, without the term "natural number") isnotelementary embeddable in R*, given that such an embedding has topreserve all first order formula (purely first order formula, andsowithout 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-- You received this message because you are subscribed to the Google Groups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com . For more options, visit this group athttp://groups.google.com/group/everything-list?hl=en .http://iridia.ulb.ac.be/~marchal/--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-l...@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

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