On 16 Oct 2010, at 23:45, Brian Tenneson wrote:

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If they are all elementary embeddable within it, then they are allelementary equivalent, given that the truth of first order formulaarepreserved.How would all structures be elementarily equivalent?

`If M1 is an elementarily substructure of M2, you cannot distinguish M1`

`and M2 elementarily, that is by a first order closed formula`

`(sentence). See Mendelson page 98, for example. Or Chang and Kiesler.`

`Only some higher order formula can distinguish them, by definition of`

`*elementary* embedding. So if you can embed elementarily all`

`structures in one structure, they will all verify the same first order`

`sentences and be elementarily equivalent (although not equivalent in`

`general). Think about the standard model of PA and its elementary`

`embedding in some non standard model.`

All mathematical theories would have the same theorems. So eventually there has to be something wrong in your theorem. My friend found the error. Your theorem page 12 is wrong, and the error is page 13 in he last bullet paragraph, when you do the negation inductionstep for your lemma. When you say: "Since it is not the case that Al=ψ(Fi (b)), it is not the case that Aj l= ψ(Fi (b)(j)) byinduction."This will be true for *some* j, not for an arbitrary j. If you negate "for all j Aj l= ψ(Fi (b)(j)) , it means that there is a j such that it is not the case that Aj l= ψ(Fi (b)(j)). After that your j is no more arbitrary.It took me a while to understand what you're saying but indeed I see the error of my proof at this point.

OK.

I'm going to try to prove it in a different way but my hope is quite limited.

`I suggest you find a far less strong notion than elementary embedding.`

`It is hard for me not to think about simulation, or recursive`

`injection, or something related to computations. Of course in that`

`case the 'universal structure' is the universal computer or`

`dovetailer. This follows from Church's thesis, which is a rather`

`unique statement of universality in mathematics.`

Although I have nothing to show for my efforts, I do feel like I learned a bit along the way. Thanks for your feedback.

`You are welcome. Sorry for not having seen the mistake at once. You`

`almost get me. I have to revise a bit of model theory.`

Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "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.