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Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so and so" can be expressed in FOL. Also these need to be expressed as uninterpreted functions.

"For every natural number n, there are infinitely many other natural numbers such that the greatest common divisor of n and each of these other numbers is 1"

As I said if you can't express this sentence in FOL at least suggest links/material where I can get leads.

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up vote 2 down vote accepted

If $f(n)$ is a predicate in the first order language of arithmetic, then "there are infinitely many $n$ such that $f(n)$ holds" can be expressed as $$\forall m\in\mathbb{N}\ \exists k\in\mathbb{N}:f(m+k+1).$$

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Or: For all m there is a k larger than m such that f(k). – Andrés E. Caicedo Aug 12 '10 at 22:18
Okay. I was a bit confused as per who the "infinitely many.." is represented. – Akshar Prabhu Desai Aug 13 '10 at 5:58

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