I've read Gödel's explanation and others but my understanding is unclear. Answers to the followup questions below would help:

  1. does ω-consistency have any relevance to methods or ideas not connected with Gödel numbering?
  2. Can we say that modus ponens and other rules of inference are ω-consistent?
  3. is there some philosophical significance to ω-consistency (for example, does it connect the idea of consistency to properties of natural numbers)?

Please explain in simple terms, if possible.

  • 1
    Conifold's answer is a good answer. To add a little bit more concrete information on your 3rd point, look up what Tarski calls 'ω-incompleteness', Hilbert's program, and the ω-rule. The upshot is that ω-incompleteness, which is very much so related to ω-consistency, has significance for the role that infinity plays in the foundation of mathematics and the philosophy of mathematics in general. Very important to note, even if the ω-rule is adopted, it corrupts the recursive aspects of formal systems which arguably makes the point moot.
    – Not_Here
    Commented Oct 31, 2018 at 12:01

2 Answers 2

  1. Yes, see ω-consistent theory. It plays a role in the study of formal theories in mathematical logic research, and is related to the so-called ω-logic needed to express the "true standard interpretation" of arithmetic, which Peano arithmetic, being first order, fails to capture.

  2. Yes, propositional logic by itself, i.e. not attached to any additional axioms, has a finite model (Boolean algebra on 0,1), so it is ω-consistent.

  3. Depends on which properties and what "connect" means. In Gödel's original proof it followed that if the arithmetic is ω-consistent then it is incomplete, i.e. some claims about natural numbers are undecidable. However, Rosser's trick showed that ω-consistency is not needed, and ordinary consistency suffices for the proof. So as far as Gödel's theorem and its philosophical consequences go one can forget about ω-consistency altogether.

  • Thank you. Regarding #3, does determining ω-consistency involve the use of natural numbers or their properties? E.g., as the unique factorization theorem and Chinese remainder theorem are used with Gödel numbers?
    – rgfuller
    Commented Apr 7, 2018 at 4:52
  • 3
    @rgfuller I may be wrong (Conifold will know), but I believe that the issue of ω-consistency only arises in non-standard models of arithmetic. The standard model is "trivially" ω-consistent since it does not contain any non-standard natural numbers.
    – nwr
    Commented Apr 7, 2018 at 16:20
  • @NickR Consistency and ω-consistency are properties of a theory, not of a model. If there is a model then the theory is ω-consistent but this is relative to ω-consistency of the theory used to build the model, namely a version of set theory. If the model is finite this is not a problem but for arithmetic one needs a lot more set theory, so one has to be careful how much such "ω-consistency" is non-circular. Gentzen did prove consistency of Peano arithmetic but it uses transfinite induction up to ordinal ε0.
    – Conifold
    Commented Apr 9, 2018 at 19:50
  • @Conifold I shall forever be a D-minus student. Hardly surprising. I just don't get enough time to read maths. An hour a night a few times a week. Still, I do enjoy it.
    – nwr
    Commented Apr 9, 2018 at 20:05

Look, if we allow ourselves the concept of truth (Gödel tried to avoid it), the assumption of ω-consistency of a theory T amounts simply to the assumption that T does not prove any false existential claims (i.e. is sound for them). (That the theory is simply consistent does not guarantee this.)

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