5 ms·

The integers can't be captured by a formal system of any kind either.

• naasking 8y ago
  Well that's not true. You can't prove every true statement about the integers, but that's not the same thing.
  • yorwba 8y ago
    For every unprovable true statement about the integers, you can find a set of "non-integers" that has all the properties of integers that are provable, and in addition the unprovable statement is false. I don't think a formal system can be said to capture the integers if it also allows such "non-integers".
    • naasking 8y ago
      I honestly don't follow your first claim or how it has any bearing on this question. Perhaps you could provide a link to the actual theorem you're describing to clarify.
      • danharaj 8y ago
        GP is alluding to the fact that any unprovable true statement about the integers has a counter-model for which the statement is false which the formal system cannot distinguish from the integers. Therefore no formal system characterizes the integers, there are always these false integers lingering. These objects are highly pathological artifacts of the formal system in question, but that just goes to show how slippery the integers are.
        • naasking 8y ago
          > GP is alluding to the fact that any unprovable true statement about the integers has a counter-model for which the statement is false which the formal system cannot distinguish from the integers. Again, do you have a link or a name for the theorem you're describing?
          • danharaj 8y ago
            This is a consequence of Goedel's completeness theorem: a statement in a first order theory is provable if and only if it is true in every model. The counter models obtained for arithmetic true unprovable statements are called non-standard integers.
            • naasking 8y ago
              Ok, now we're getting somewhere [1]. So various theorems can be used to prove the existence of non-standard models of arithmetic. I'm not sure why I should find this compelling. There are infinitely many possible models of arithmetic, each of which has different properties. Some of these are isomorphic to standard models in various ways, some of which are not. But, if the world is governed by mathematics, then at least one such model would best match reality. How would we even speak of a world which was not internally consistent such that it could be formalized in such a fashion? [1] https://en.m.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.m.wikipedia.org/wiki/Non-standard_model_of_arithm...
              • danharaj 8y ago
                I'm not sure what you're asking. Any formalization of the integers won't be "the" integers, the totality of which perhaps isn't even a reasonable idea. Mathematics is a part of the world and does not stand apart from it. There is no reason why such a "best" model should exist, and by any reasonable definition, it doesn't. Mathematics is irrefragably incomplete. It is perhaps the only human discipline that demonstrates its own limits.
                • drdeca 8y ago
                  What? Every model of arithmetic has every standard integer though? Are there nonstandard models of second order ... Ok Wikipedia says that "the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics.", so...
                  • 8y ago
    • drdeca 8y ago
      I thought that was just for first order systems?
      • deleted 8y ago
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