@Henry Cohn Thanks! Statement "Consistent theory T" has a clear substantial sense and it means that on any step of the proof we cannot prove the formula 1=0. But construction of proofs is real physical process and for example, if "10 ^ {10 ^ {10000} exists" has not the same substantial sense as well as "2 exists" then and statement Con (T) obviously has not any sense.

> I really think that "is $T $ consistent?" and "what do symbols mean?" are unrelated.) $T $ consistent has clear substantial sense and it means that on any step of the proof we cannot prove the formula 1=0. But construction of proofs is real physical process and if "10 ^ {10 ^ {10000} exists" has not the same substantial sense as well as "2 exists" then and statement Con (T) has not any sense.

Well. Then explain to me that you particularly mean when tell that number $10 ^ {10 ^ {10000}} $ exist. It something real or only a symbol?

Noah > In particular, can you please explain what you mean by "really exists?" "Really exists" means exist as real infinite physical object or infinite physical process. For example E.Nelson in fact asserts that such object or process does not exist, i.e. РА has no any standard models.

> then at some level you must be rejecting infinite sets, which is fine, In it there is no necessity. Nevertheless it is possible to assume that real-life infinite sets, are not obliged to correspond to the classical logic.

Andrej Bauer. >but "really exist" might mean "exists constructively". Well. Thus "really exist" completely depend on logic.

>Let me elaborate a bit on why having computationally simple models >is relevant. It's not just that such models are "less complicated" >than standard set-theoretic constructions, as I state above; it's >that we don't even need to talk about set theory, at all, to get >them! The models in question are uniformly It agree.But nevertheless there are known mathematicians which think that such constructive process will suffer contradictions web.math.princeton.edu/~nelson/papers/warn.pdf

Noah S. You rave. My question is put clearly and no any relation to my statements in other places has. Moreover I never addressed to whom or with intentions to check my proofs and the more so their sketches.

Joel David Hamkins.Yes of course.

I'm asking does it follow from ZFC+ there exists some model of ZFC

Timothy Chow >What you're asking is, as long as there exists some model of ZFC, does it follow that no models are standard? Yes of course.