vi21maobk9vp

If we are not inventive enough to find a menace not obviously shielded by lead+ocean, more complex tasks like, say, actually designing FOOM-able AI is beyond us anyway…

You say "presumably yes". The whole point of this discussion is to listen to everyone who will say "obviously no"; their arguments would automatically apply to all weaker boxing techniques.

How much evidence do you have that you can count accurately (or make a corect request to computer and interpret results correctly)? How much evidence that probability theory is a good description of events that seem random?

Once you get as much evidence for atomic theory as you have for the weaker of the two claims above, describing your degree of confidence requires more efforts than just naming a number.

I guess that understanding univalence axiom would be helped by understanding the implicit equality axioms.

Univalence axiom states that two isomorphic types are equal; this means that if type A is isomorphic to type B, and type C contains A, C has to contain B (and a few similar requirements).

Requiring that two types are equal if they are isomorphic means prohibiting anything that we can write to distinguish them (i.e. not to handle them equivalently).

Could you please clarify your question here?

Why, as you come to believe that Zermelo-Fraenkel set theory has a model, do you come to believe that physical time will never show you a moment when a machine checking for ZF-inconsistency proofs halts?

I try to intepret it (granted, I interpret it in my worldview which is different) and I cannot see the question here.

I am not 100% sure whether even PA has a model, but I find it likely that even ZFC has. But if I say that ZFC has a model, it means that this is a model where formula parts are numbered by the natural numbers derived from my notion of subsequent moments of time.

Link is good, but I guess direct explanation of this simple thing could be useful.

It is not hard to build explicit map between R and R² (more or less interleaving the binary notations for numbers).

So the claim of Continuum Hypothesis is:

For every property of real numbers P there exists such a property of pairs of real numbers, Q such that:

1) ∀x (P(x) -> ∃! y Q(x,y))

2) ∀x (¬P(x) -> ∀y¬Q(x,y))

(i.e. Q describes mapping from support of P to R)

3) ∀x1,x2,y: ((Q(x1,y)^Q(x2,y)) -> x1=x2)

(i.e. the map is an injection)

4) (∀y ∃x Q(x,y)) ∨ (∀x∀y (Q(x,y)-> y∈N))

(i.e. map is either surjection to R or injection to a subset of N)

These conditions say that every subset of R is either the size of R or no bigger than N.

ZFC is the universally, unequivocally best definition of a set

Worse. You are being tricked into believing that ZFC is at all a definition of a set at all, while it is just a set of restrictions on what we would tolerate.

In some sense, if you believe that there is only one second-order model of natural numbers, you have to make decisions what are the properties of natural numbers that you can range over; as Cohen has taught us, this involves making a lot of set-theoretical decisions with continuum hypothesis being only one of them.

Well Foundation in V_alpha case seems quite simple: you build externally-countable chain of subsets which simply cannot be represented as a set inside the first model of ZFC. So the external WF is not broken because the element-relation inside the models is different, and the inner WF is fine because the chain of inner models of external ZFC is not an inner set.

In the standard case your even-numbers explanation nicely shows what goes on — quoting is involved.

I need to think a bit to say what woud halt our attempts to build a chain of transitive countable models...

Ah, sorry, I didn't notice that the question is about model of ZFC inside a "universe" modelling ZFC+Con^∞(ZFC)

And you are completely right.

I meant that designing a working FOOM-able AI (or non-FOOMable AGI, for that matter) is vastly harder than finding a few hypothetical hihg-risk scenarios.

I.e. walking the walk is harder than talking the talk.