[FOM] The liar and the semantics of set theory (expansion)
Roger Bishop Jones
rbj at rbjones.com
Wed Oct 2 11:23:09 EDT 2002
On Tuesday 01 October 2002 11:32 pm, Rupert McCallum wrote:
<lots of interesting stuff skipped>
> Well, you're certainly getting closer and closer to "ordinary truth",
> aren't you? Why not just go there?
I'm not at all sure what you mean by "ordinary truth"!
Your discussion which follows involves various hypotheses
about the size of the universe, which suggests that you
are talking about truth in some V(alpha) but not sure which one.
Or do you mean truth in WF?
What I am pondering about is the foundations of (abstract) semantics.
It seems to me that there are three possibilities for where the
buck stops:
1. Under the Tarskian scheme the buck doesn't stop anywhere.
To give the semantics of a language you need a metalanguage
which is strictly more expressive (in some unspecified sense)
than the object language, and to give a semantics to the
meta-language you need a meta-meta-language,
and so on into infinite regress.
2. There exists a universal language (an abstract lingua characteristica)
in which an abstract semantics can be given to any coherent
language, including itself.
3. There exist one or more ultimately expressive languages whose
semantics cannot be defined in any language at all.
Option 2 corresponds to a set theoretic semantics along the
lines I have outlined, and is my preference if it can be made to
work well enough.
Option 3 corresponds to interpreting set theory in WF which I
don't like for reasons which I will give shortly.
Option 1, made more habiltable by the possibility of a family
of syntactically identical but semanticall progressive definitions
is my preferred fallback if 2 turns out not to work (which I had
expected to be the case when I started this thread).
My reasons for not liiking WF are as follows:
1. WF seems to me to be conceptually incoherent.
This is because it is part of the conception of the cumulative
heirarchy that whenever a collection of stages can be conceived
of, then there is a stage which follows all those stages.
And yet WF is conceived of as completed without any
further stages to follow.
It seems to me built into the conception of WF that its can never
be completed, i.e. that all there can be are the V(alpha) for the
various ordinals, but no WF, just as there can be no set of all
ordinals (and I don't think it helps to make it a class, you still
have to explain why it isn't an ordinal).
2. Putting aside my reservations about the conceptual coherence
of WF, if truth in set theory is identified with truth in WF then
it is bivalent and diagonalization shows that it is not definable
in set theory.
So it looks like we then have a language which has a semantics
which can only be described informally.
Well I'm happy that at some stage we have to fall back on
informal explanations, but I think it odd that there should be
some definite point at which we have no choice.
If you had in mind some meaning for "ordinary truth"
other than "true in WF" then I'd appreciate an explanation.
I agree that my best formulation to date of a self defining
notion of set theoretic truth is still not everything one might
hope for.
It does allow axioms asserting the existence of arbitrarily
large cardinals, but does not allow axioms the effect of
which is to strengthen the closure properties of the universe
of discourse (so that one never reaches a point beyond
which all inaccessibles yield models).
So even if these reflexive semantics work, we may indeed
still be looking at an endless sequence of some kind.
However, in this case, I think I prefer an endless
sequence of refinements to the semantics of a near universal
language capable of defining its own semantics to
the infinite regression through meta-languages.
Roger Jones
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