Is there a formal system which is generally accepted, and how is it called?

Question

I read the question at Gödel's Incompleteness Theorem -- meta-reasoning "loophole"? about Gödel's incompleteness theorem. My question is little about the contents of that other question. Rather, it is about terminology.

How is the thinking system or formal system called, which is outside of the system described by Gödel's theorem? Are there terms for systems when we create formulations in one system about another system? I mean, are there coined words for "inner system" and "outside system" and a generally accepted "outermost system"?

score 1 · Answer 1 · edited Sep 04 '15 at 12:43

1

There is no absolutely strict terminology. You could say either "inner logic" or "inner system," and we'd know what you mean. But these terms should better mean the same thing! Similarly, "outer logic" or "meta logic" are interchangeable.

There can't be an outermost logic independent of context. You wouldn't even want two meta logics, as we (more or less) make up the rules in meta logic. So why do it twice and make your math that much more scattered? Better to encapsulate all of the meta voodoo in one place. Of course, I'm no logician, and there may be a use for it. I'd be interested to see any counter arguments.

That being said, a lot of very important and active work takes place on a meta logic (ZFC mumble mumble, quantification over first order formulas mumble mumble) that sits just on top of first order as the inner logic. This is a pretty normal place to study large cardinals or model theory.

edited Sep 04 '15 at 12:43

Daniel S.

530

answered Sep 03 '15 at 18:37

Zach Stone

5,701

i By analogy with set-theory, I would imagine that an outermost logic ,if it existed, would be inconsistent.... If it were not recursively definable there would be sentences that cannot be determined to be axioms or not axioms. If it were recursive and could encode itself it would be incomplete, hence not the outmost. – DanielWainfleet Sep 03 '15 at 20:14
@user254665 Why can the outermost not be incomplete? – Daniel S. Sep 04 '15 at 12:33
I have problems understanding the middle paragraph of this answer. First, you say that there can't be just one context-independent outermost logic and then you say that it is not useful to have many. So what's the case now - do we have many or just one? – Daniel S. Sep 04 '15 at 12:43

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

Any proof in mathematics actually lives in some kind of system. In practice mathematicians use informal language, which is what you see in all proofs of Godel's incompleteness theorem, since it is just too troublesome to write out a proof in a formal language. However, such informal proofs (with copious use of English and even meta-meta-reasoning) can in actual fact be systematically translated to a formal proof in some formal system called the meta-system. ZFC suffices as this meta-system for most proofs concerning mathematical logic. In this sense you can say that ZFC currently is the generally accepted 'outermost system'.

It is interesting to note that since ZFC can reason about ZFC itself, it proves various results about itself, such as that if it is consistent then there is no proof or disproof of Con(ZFC), which is an arithmetical sentence that (from the outer ZFC's point of view) the natural numbers satisfy iff ZFC is consistent. Note that the natural numbers in the meta-system are governed by the axioms of the meta-system, which fail to pin down a single structure. This can be seen from the incompleteness theorem, since there is some arithmetical sentence that ZFC itself cannot prove or disprove, and so if ZFC is consistent then it has two models that disagree on that sentence, which means that they must have non-isomorphic natural numbers! This is an inevitable property of any formal system $S$ (with decidable proof validity) that can interpret PA. So it is a curious fact that if we believe there is a collection of natural numbers (that satisfy PA), then we have no choice but to accept that no useful meta-system can prove all the true facts about the natural numbers, including in particular the consistency of the meta-system itself, which we ought to believe is 'true' because if it is 'false' then the meta-system is useless!

Is there a formal system which is generally accepted, and how is it called?

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