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I'm curious which axiomatic frameworks Gödel's incompleteness theorems apply. Do the theorems apply to any axiomatic system that adopts potential infinity? Or just to those that adopt completed infinity? Are the "potential infinity only" systems safe from the incompleteness theorems?

Let me be more precise.

If one were to adopt the framework that there is no largest natural number, but does not accept that ALL the natural numbers can be gathered into one unified whole, collection, or "set". Do the incompleteness theorems apply to this?

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  • $\begingroup$ Actually in this case the sentence "$\omega \text{ exists}$" would be independent. $\endgroup$
    – Ldddd
    Commented May 5, 2019 at 16:11
  • $\begingroup$ Godel's theorem is usually proved in the case of (first-order) Peano arithmetic, which is a theory of natural numbers alone - no sets involved at all. So no, completed infinity is not relevant to Godel. In fact, it's harder to prove Godel for ZFC (for example), because it takes more busywork to code sentences in the rather limited language of set theory. $\endgroup$ Commented May 5, 2019 at 17:10
  • $\begingroup$ @NoahSchweber I'd say it was easier to encode sentences in set theory. After the slightly fiddly definition of ordered pair it's easy to define "ordered list", which is all a sentence is. Whereas in PA you have to define exponentiation and then prime numbers or a base system. $\endgroup$ Commented May 6, 2019 at 18:29
  • $\begingroup$ @OscarCunningham Ultimately it's a matter of preference. Personally I find the definition of exponentiation in PA less difficult (and it's really trivial to define the primes). The point is that set theory plays no essential role. $\endgroup$ Commented May 6, 2019 at 19:29

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Godel's Incompleteness Theorems hold in any system that's rich enough to code formulas and satisfaction via natural numbers. You don't need $\omega$ to be a set to do that, so you don't need the Axiom of Infinity. (You also don't need the Power Set Axiom.) --Bob

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    $\begingroup$ In fact you don't need to speak of sets at all. Natural numbers (and quantification over same) with addition and multiplication is plenty. $\endgroup$ Commented May 5, 2019 at 16:05

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