# Tagged Questions

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### Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?

Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition. View ...
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### Reference Request: Non-Standard Models of PA

I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...
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### Weakest subsystems of second order arithmetic for mathematical logic

It is known that to prove completeness of first-order logic for countable languages WKL0 is enough. But, is it the weakest subsystem where one can prove it? What about the incompleteness theorems? Is ...
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### Composite pairs of the form n!-1 and n!+1

It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$. Is ...
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### Incompleteness and nonstandard models of arithmetic

The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome. Reading Peter Smith's "GĂ¶del Without (Too Many) ...
On page 164 of his book Models of Peano Arithmetic, Kaye states Friedman's Theorem: Let $M{\vDash}PA$ be nonstandard and countable, let $a\in M$ and let $n\in {\mathbb N}$. Then there is a proper ...
An old theorem of A. J. Wilkie (Some results and problems on weak systems of arithmetic, Logic Colloquium '77) asserts that a discretely ordered ring $R$ can be extended to a model of open induction ...