[FOM] The liar and the semantics of set theory
Roger Bishop Jones
rbj at rbjones.com
Fri Sep 20 06:27:44 EDT 2002
My first posting on this topic was I think misunderstood
by those from whom I have so far seen a response,
and so I propose to expand on my explanation of the
problem.
First let me re-iterate the concession I made in the
original message, that if the semantics of set theory
is given as truth in some single interpretation (however
specified) then truth will not be definable in that set
theory.
Further to that it seems probable that any definition
of truth which is total, i.e. under which every sentence
is either true or false, can be shown to be impossible
using the liar construction.
The question is therefore about how we can rule
out the possibility that a semantics in which some
sentences have no truth value (or have a third truth
value) can be defined in a set theory with that
semantics.
The idea is of course that we make it possible for the
liar sentence and its negation to be neither true nor false.
Note here that I am not proposing any change to
the definition of "truth in an interpretation", but
am considering alternative ways of defining
"truth in set theory" either using "truth in an interpretation"
or otherwise.
Now let me give a partial answer to my own question,
which effectively transforms the nature of the problem.
The answer is that there is no completely general proof
that the semantics of set theory cannot be defined in
set theory without specifying some constraint on what
is to count as an acceptable semantics.
This can be seen by considering the consequences of a
position which might be taken on the semantics either
by a moderate foramalist (such as Hilbert) or even a
(very) moderate intuitionist.
Hilbert believed (at least prior to the incompleteness results)
that semantics should be given exclusively formally, so that
the meaning of the sentences in an axiomatic theory
is to be understood in terms of the models of the axioms,
with no further informal explication (e.g. disregarding
the informal account of the cumulative heirarchy, and
presumably regarding as unacceptable the standard semantics
of higher order logics).
Given that first order logic is complete, this is similar in
consequences (for present purposes) to the identification
of truth with provability (a small part of some intuitionist philosophies)
Someone with those beliefs might wish to define truth
in set theory either as truth in all models of ZFC (and
falsity as falsehood in all models of ZFC, leaving some
sentences without truth value) or equivalently as
provability in ZFC (i.e. a sentence is true if provable in ZFC,
false if refutable in ZFC and otherwise
has no truth value).
Since under this semantics the truths and falsehoods of
set theory are r.e., and provable sentences are provably
provable, I can't see any problem in defining this notion
of set theoretic truth in the set theory which has this
truth definition.
I don't myself regard this account of the semantics of
set theory as acceptable, and I don't believe that an
acceptable semantics will turn out to definable in its
own language.
However, its not at all clear to me how to rule out the
possibility that there is an acceptable semantics for set
theory which is definable in the same set theory.
The challenge is then to come up with some (partial)
criterion of acceptability for a semantics of set theory
together with a proof that a semantics satisfying the
acceptability criteria is not definable in a set theory
having that semantics.
Bivalence would probably be a sufficient criterion.
But I myself regard this as too strict.
Can anyone come up with a weaker provably sufficient constraint?
Roger Jones
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