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So if n is a member of the set of natural numbers, such that n is greater than 1 and lesser than 3, then n=2 necessarily?

closed as off topic by Keelan, Annotations, iphigenie, Ben, Joseph Weissman Jun 18 '13 at 23:47

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  • In which system? If you specify the set of axioms, you may get a proof. – Memming Jun 13 '13 at 18:17
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    What does this have to do with philosophy? – Keelan Jun 13 '13 at 18:22
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    @Keelan If the question is (as the inclusion of "necessarily" suggests) about the modal status of mathematical claims, then it is very much on topic. The question of whether or not we must take all mathematical truths to be necessarily true (and what sort of modality is at issue) falls well within the purview of, and receives attention within, the philosophy of mathematics (and, philosophy of modality). – Dennis Jun 13 '13 at 21:06
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    That being said, the question could definitely use a bit more fleshing out (insert @JosephWeismann template request for clarification). – Dennis Jun 13 '13 at 21:07
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    @Kantian: that is no clarification, it's merely writing the same proposition in another written language. What in your philosophical investigations has motivated you to ask this question? What are you hoping to clarify --- merely this question, or some broader principle? – Niel de Beaudrap Jun 14 '13 at 12:24
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This is more on the lines of commentary on your question, rather than an answer.

If one uses the usual natural numbers then yes. But are there other systems of natural numbers?

  1. There are non-standard integers: The integers are axiomatically defined by the Peano Axioms. We actually obtain them in a model. The question here is are all such models isomorphic - that is categorial. If we use second-order logic, they are; if first-order then not. But the initial segment of any non-standard integer model is isomorphic to the usual ones. So this doesn't work.

  2. We could use topos rather than semantics for the PA axioms, this means the axioms are interpreted in a topos. This means that the topos must have a Natural Number Object. Whether there is a topos, within which the condition you specify fails, is for me, an open question.

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Yes, if "natural numbers" is the commonly agreed on term from arithmetic. When there is no other possibility, there can only be the distinct, unqualified answer.

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