[FOM] The liar and the semantics of set theory (expansion)

[Rupert McCallum]

--- Roger Bishop Jones <rbj at rbjones.com> wrote:

> I think the usual definition of definablity in a theory suffices.

Yes.

> Conjecture 1: true@ set theory is definable in true@ set theory

This is correct in the sense that something's true@ if and only if it's
true@ that it's true at .

> Conjecture 2: the axiom of replacement is not decided by true@ set
theory.

Yup - there clearly exist arbitrarily large alpha such that the axiom
of replacement fails in V_alpha, and we can prove in KM, or ZF+"there
is no largest inaccessible", that there are arbitrarily large alpha
such that the axiom of replacement holds in V_alpha.

> Conjecture 3: true@ set theory is ccomplete in its claims about the
existence
of ordinals (we may need to discuss what this means!)

Yes, I'd say we should. 

Just had my mind boggled by the thought that it's ZF-provably true, and
also true@, that there's a V_alpha that models all of true@ set theory.


> The claim "there exists a pope" is in true@ set theory

Only if, say, the statement "alpha is a pope" could be expressed by a
statement in the theory of V_alpha+delta for some fixed absolutely
definable delta. Everything up to "measurable" would be fine with that,
but among things like "strongly compact", "supercompact", "extendible",
"huge" there might be some counterexamples - I'm not sure.

> Probably the substantive unresolvable ambiguity will be
in the extension of the concept of ordinal.

Isn't that also precisely the problem with supposing set theory can be
interpreted in V?

> Does this story do anything to dent your conviction that
infinite regress is unavoidable and that a unversal semantic
foundation is possible?

It's certainly given me pause for thought, yes. We need to examine the
completeness issue further, I think. It's certainly harder to come up
with an example of something you might want to say which you can't
convey by saying such-and-such is true at . I'll have a think about that.
I'll certainly be trying for an example which doesn't make reference to
outrageously humungous cardinals, but I wouldn't sneeze at the
usefulness of these cardinals, either - consider the use of Woodin
cardinals in Martin and Steel's work on projective determinacy, for example.

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