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

Roger Bishop Jones rbj at rbjones.com
Thu Oct 3 02:45:34 EDT 2002

On Thursday 03 October 2002 12:17 am, Rupert McCallum wrote:

> Truth in V (which is the direct limit of all the V_alpha's). Not sure
> what WF is.

I think V and WF usually both mean the same thing,
the well-founded sets, the union of V(alpha) for all ordinals alpha.
Is that what you mean by "direct limit"?
My definition of TRUTH is in effect a different kind of limit which
does not depend upon ever forming V or WF.

Since your central objection to my "universal langauge" scenario
is that it is always possible to extend the definition of truth
I have given some consideration to whether the definition can
be stabilised.

I believe it can.

The reason for this is that the extensions we are contemplating
are required to improve agreement with certain kinds of large
cardinals axioms.
(and, not being suficiently familiar with the details of
large cardinal axioms I don't know whether the kind of large
cardinal axiom which would require an extension are ever used.)

However, the extensions do not extend the expressiveness of
the language, the language is universal without them.

To clarify this point I am going to retrench on the semantics
and offer a version which leaves even the axiom of replacement
undecided, and then explain why this suffices.

"True@ set theory"
is the set S of sentences of first order set theory such that
for every sentence s in S there exists an ordinal alpha such that
s is true in V(beta) for every beta>=alpha.

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

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

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

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

In relation to large cardinal axioms I believe the situation is as follows.

Suppose we have a description of a kind of large cardinal (say, a pope),
and suppose that there exists a pope.
The claim "there exists a pope" is in true@ set theory, but
the claim "every set is a member of a pope" is undecided
by true@ set theory.
However, the claim that there exists a set which is a model for
ZFC + "every set is a member of some pope" is in true set theory.

A universal language does not have to agree with everyone's
favourite large cardinal extension of ZFC.

In conclusion I claim for this definition of true@ set theory
that if offers a semantic foundation preferable to an infinite
regress of metalanguages, that it is arguably "universal",
and that it avoids the tenuos assumption that set theory
can be interpreted in V.

Of course, the whole thing is highly circular, and it is therefore
desirable, if this kind foundation were to be taken seriously,
to formalise the definition (which is presumably easy), to
examine the extent to which the definition is ambiguous
and to consider ways of disambiguating it.
Probably the substantive unresolvable ambiguity will be
in the extension of the concept of ordinal.

It is important to note here that it is INTENDED in this
language ("true set theory") that not all sentences have truth values,
and that these are not just unresolved problems.
The unresolved problems are about which ordinals there
are, which affects which sentences are true under
this semantics, not about what the truth value should
be of sentences which are given no truth value by this

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

Roger Jones

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