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This problem concerns the Peano Arthmetic. Px denotes: “x is prime”; that is: Show that

a) ∀x∀y [(Px ∧ Py ∧ x|y) --> x=y]

b) ∀x∀y∀z∀u∀v[Px ∧ Py ∧ Pz ∧ Pu ∧ Pu)-> x∙y∙z≠∙u∙v].

I am really clueless on how to solve both of these. Any help/hint will be appreciated thank u.

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    These are fairly straightforward theorems in ordinary arithmetic, but if you want formal Hilbert-style proofs, you should say so. (I doubt anyone will actually give you one though... it's just too tedious.)2012-12-05

1 Answers 1

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Here is a hint for the first one. I assume your definition of prime is "$x>1$ is prime if whenever $t|x$ (i.e. $\exists s . x=st$), either $t = 1$ or $t = x$."

Suppose $x$ is prime and $y$ is prime, and $x|y$. Since $y$ is prime, either $x=1$ or $x=y$. But since $x$ is prime, we know that $x>1$, so that $x = y$.