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A theory T1 can be shown to be consistent by describing a model for it. But usually the model is also described in words, using terms from some other theory T2. So unless T2 is also consistent this method of constructing a model does not seem to help.

My question is, what is the "etiquette" of constructing models, what are considered best most reliable kinds of models? Set theoretical? How do mathematicians think about this issue in practice.

Is there some sort of natural "hierarchy" of relatively consistent theories that is used when constructing models for proving consistency?

Johannes
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1 Answers1

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The point is that when you're interested in the properties of a theory you are actually studying the theory inside a meta-theory, where you define what a theory is, what are is models and what it means that it is consistent.

Usually this meta-theory is a set theory and one use the deductive system of this meta-theory to prove the properties of the theory being studied.

Clearly this means that your consistency proofs rely on the fact that the formal system you're using to prove them (the meta-theory) doesn't prove contraddictions, that it is consistent. Of course you could prove that your meta-theory is consistent but in order to do that you would need a meta-theory in which you can prove that your theory is consistent, but that is just sweeping the dust under the carpet would bring to an infinite sequence of theories in which each element proves the consistency of the preceding theory.

One could contempt himself to being able to prove consistency of a meta-theory in itself, that is studying the meta-theory using itself as meta-theory. Unfortunatly this wouldn't be a great solution because ever if a meta-theory was able to prove its own consistency this wouldn't be enough to prove that it cannot prove contraddictions: because a formal system which is inconsistent is able to prove everything even its own consistency.

That should make clear that this problem is unavoidable, at some point we have to choose a meta-theory and we have to make an act of faith believing that it is consistent.

Usually mathematicians use set-theory (either ZFC or NBG or other kind of set theory) as meta-theories. The choice for set theory as meta-theory is due to many different reasons:

  • all the kind of set theory invented till now are provable equivalent (in a suitable sense) and no one has been able till now to prove a contraddiction in them
  • our minds are used to think in terms of collections and so set theory being a formal system for reasoning with collections is very appealing as a meta-theory
  • we have done set theory for many years so we are very strong with set theory and that make really reasonable to use this theory as meta-theory.

Of course there are also other reasons that justify the choice of set-theory as meta-theory here I've listed only some that came to my mind.

Hope this helps.

Giorgio Mossa
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  • Good answer! The first bullet point raises a question for me, because it doesn't seem to say anything: If someone did come up with an inconsistent set theory, you would just not count it to the "kind of set theory" that "we" came up with. So is there a historical case where some serious post-Russell mathematicians came up with a set theory the we found to be inconsistent, but that was published and survived for a non-trivial time? – Nikolaj-K Jun 23 '15 at 11:25
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    @NikolajK that's an interesting question. In my knowledge no one has build an inconsistent set theory after Russell paradox. History tells us that every kind of inconsistency is usually a variation of Russell paradox and is based on self-referentiality: a statement that asserts that is true or false. Since the foundational crisis mathematicians and logicians have learnt to develop systems that do not allow to derive statements of this kind, in this way they builded systems which hopefully should be consistent. – Giorgio Mossa Jun 23 '15 at 12:55
  • We actually did it a few times. Quine's New Foundation refutes the axiom of choice, but it was not known at first, so NF+AC is one of the inconsistent set theories after Russell's paradox. Girard's paradox is also one of these, because the system in question didn't involve an "X of all Xs", but had too much impredicativity and so became inconsistent. – Trebor Jan 14 '23 at 09:26