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Math Help - first order language in logic

  1. #1
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    first order language in logic

    Let L = {f } be a first-order language containing a unary function
    symbol f , and no other non-logical symbols.
    1.Write down a sentence χ of L which is satisfiable in some structure
    with an infinite domain but is false in every structure with a finite domain.
    What can you say about the size of the domains of the models of the sentence
    2.Write down a sentence ρ such that whenever A |= ρ and A is finite,
    then A contains an even number of elements and, further, every finite set
    with an even number of elements is the domain of some model of ρ. What
    can you say about the size of the domains of the models of the sentence ρ?

    Could anyone please give me some hints how to deal with this problem? I'm not sure where to start. Any help is appreciated!
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  2. #2
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    Let L = {f } be a first-order language containing a unary function
    symbol f , and no other non-logical symbols.
    1.Write down a sentence χ of L which is satisfiable in some structure
    with an infinite domain but is false in every structure with a finite domain.
    What can you say about the size of the domains of the models of the sentence
    You probably have equality symbol in the language as well. (This differs from one definition to the next.)

    Describe f as a successor function in arithmetic. For 0 use the element that that does not have a predecessor (i.e., which is not a successor of anything). Then say that f is injective.
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