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Math Help - Model theory - definiable sets 2

  1. #1
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    Model theory - definiable sets 2

    Here is more difficult exercise (for me of course) from definiable sets:

    Let A be the partial ordering (in a signature with \leq) whose elements are the positive integers, with m \leq ^A iff m divides n.
    Show that the set \{1\} and the set of primes are both \emptyset-definable in A.

    I don't have any idea how can I solve it. Could You give me some advices?
    Any help will be highly appreciated.
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  2. #2
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    Well, 1 is the only positive integer that divides all other positive integers. And a number other than 1 is prime if its divisors are 1 and itself. It's easy to write these properties as formulas.
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  3. #3
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    Could You show me how can I do this, for example in second case (set of primes)?
    This is probably easy but I'm really newbie in this subject so I'll be glad if You write some more details. Thanks.
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  4. #4
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    \mathrm{prime}(x) is \neg\mathrm{one}(x)\land\forall y.\,y\le x\leftrightarrow(\mathrm{one}(y)\lor y=x). Here \mathrm{one}(y) means y=1 (the first part of the problem). I also assume that = is in the language and is interpreted as identity.
    Last edited by emakarov; November 16th 2010 at 01:09 PM. Reason: Added parentheses.
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