On 16 Oct 2010, at 23:45, Brian Tenneson wrote:
If they are all elementary embeddable within it, then they are all
elementary equivalent, given that the truth of first order formula
are
preserved.
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 induction
step for your lemma. When you say: "Since it is not the case that A
l=
ψ(Fi (b)), it is not the case that Aj l= ψ(Fi (b)(j)) by
induction."
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/
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