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Math Help - Formula with parameters

  1. #1
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    Formula with parameters

    Hi everybody. Can someone help me finding a solution for this problem:

    "Let M be a model and phi(x) be a formula with parameters in M, x being a single variable. Prove that if phi(M) contains the parameters of phi(x), then phi(M) is not an elementary substructure of M."

    I'm not even fully understanding the problem, since I can't figure it out what the "parameters of phi(x)" are.

    Thank you so much
    bye!
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  2. #2
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    Hi, and welcome to the forum.

    Let M=\langle D,\Sigma,I\rangle where D is a domain, \Sigma is a signature, and I is an interpretation of \Sigma on D.

    I can't figure it out what the "parameters of phi(x)" are.
    This probably means elements of D that occur in phi.

    Prove that if phi(M) contains the parameters of phi(x), then phi(M) is not an elementary substructure of M.
    What is phi(M)? I'll assume it is \langle D',\Sigma,I'\rangle where D'=\{a\in D\mid M\models\varphi(a)\}, and I' is the restriction of I on D'. However, it is not clear that this is a structure, namely, that D' is closed with respect to functions. (It may happen that for a functional symbol f\in\Sigma and a\in D', I(f)(a)\notin D'.) So, I'll assume that there are no functional symbols in \Sigma.

    Next, what happens if D'=D, e.g., when \varphi(x) is x=x? Then phi(M) = M and so phi(M) is an elementary substructure of M. So, I'll assume that D'\subset D, but D'\ne D, i.e., there exists an a\in D such that M\not\models\varphi(a).

    In this case, what about \exists x\,\neg\varphi(x)?

    Maybe I made too many assumptions here...
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  3. #3
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    Hi and thank you for the quick reply! You made the right assumptions, I had a reply from a classmate of mine that clarified me the meaning of "parameters"... Apparently (I can't follow the lectures) our professor has a slightly uncommon approach to the subject, so the "parameters" are elements of a set that in this case, but not necessarily, coincide with D.

    However, this:

    Quote Originally Posted by emakarov View Post
    Next, what happens if D'=D, e.g., when \varphi(x) is x=x? Then phi(M) = M and so phi(M) is an elementary substructure of M.
    is the exact same thought I had, that made me submit the problem to the community, since the definition of elementary substructure the professor gave to us implies D'\subseteq D.

    Only a bit of a correction to your statement... in the hypotesis of the exercise it says "a formula with parameters" and x=x doesn't have any. But the core of the reasoning is correct, I think it is sufficient putting D=\{0,1,2\}, \Sigma=\{<\} with its normal interpretation, and \varphi(x)=x<2 \vee x=2. Am I right?
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  4. #4
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    I believe so.

    I tend to understand "a formula with parameters" as "a formula that is allowed to contain parameters". But to change my example, you could take phi to be x = x /\ a = a for some a in D, or x = a -> x = a.

    Maybe your instructor wanted to claim that phi(M) is not a proper elementary substructure of M.
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  5. #5
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    Quote Originally Posted by emakarov View Post
    Maybe your instructor wanted to claim that phi(M) is not a proper elementary substructure of M.
    This could be, I'll inspect this possibility!

    Thank you!!
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