> 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?

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

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

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