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Thread: phi_x

  1. #1
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    model

    $\displaystyle \forall x_1\forall x_2\dots\forall x_{n-1},\exists x_{n}{{\neg{x_{n}=x_1}}\land\neg{x_n=x_2}\land\dot s,\land{\neg{x_n=x_{n-1}}\$
    i would like to know how to show form the above sentence that every model of phi_n must have at least n elements in the underlying set
    Last edited by Mike12; Mar 21st 2011 at 06:32 AM.
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  2. #2
    MHF Contributor
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    If $\displaystyle \phi_n$ is $\displaystyle \forall x_1,\dots,x_{n-1}.\,x_n\ne x_1\land\dots\land x_n\ne x_{n-1}$, then $\displaystyle \phi_n$ is not a sentence since $\displaystyle x_n$ is a free variable.

    A remark concerning notation. Subscripts consisting of more than one symbol must be enclosed in parentheses or braces. Otherwise, it is not clear whether X_n-1 means $\displaystyle X_n-1$ or $\displaystyle X_{n-1}$. You can use /\, \/ and ~ to denote conjunction, disjunction and negation, respectively, in ASCII. It is also not too hard to write LaTeX code for these formulas. The formula above is produced by [tex]\forall x_1,\dots,x_{n-1}.x_n\ne x_1\land\dots\land x_n\ne x_{n-1}[/tex]. Also, \lor produces $\displaystyle \lor$, \neg produces $\displaystyle \neg$ and \exists gives $\displaystyle \exists$.
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