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

[Rupert McCallum]

--- Richard Heck <heck at fas.harvard.edu> wrote:
> 
> Roger Bishop Jones wrote:
> 
> >On Monday 23 September 2002  1:47 am, Rupert McCallum wrote:
> >  
> >
> >>We can re-interpret the diamond of modal propositional logic to
> mean "true in some V_kappa with kappa inaccessible" and the box to
> mean "true in all V_kappa with kappa inaccessible". As Boolos
> discusses in Chap. 13 of the cited work, Solovay proved in 1975,
> assuming infinitely many inaccessibles, that the following is a
> complete axiomatization of the modal logic of these notions:
> >>(1) all tautologies
> >>(2) box(a implies b) implies (box a implies box b)
> >>(3) box(box a implies a) implies box a
> >>(4) box(box a implies b) or box((box b and b) implies a)
> >>Rules of inference: modus ponens and necessitation
> >>    
> >>
> >This is certainly of great interest and I will spend some time on
> it. On the briefest perusal however, its not obvious how this helps
> to settle the question at hand.
> >
> Even if "true in all V_kappa for kappa inaccessible" is definable in
> ZFC 
> (I expect Solovay did resolve the question of definability, but I
> don't 
> have Boolos's book with me), Solovay's result shows that there is 
> something strange about treating this notion as a truth-predicate.

I would say "true in all V_kappa for kappa inaccessible" is definitely
definable in ZFC. The notion "true in the model M", where M is a set,
is definable in ZFC (as discussed in, e.g., Chap. 5 of Drake, "Set
Theory: An Introduction to Large Cardinals"). And of course the notions
"V_kappa" and "kappa inaccessible" are definable in ZFC. So I can't see
any problem there.

Agreed, the work on treating notions like these with modal logics
naturally suggests that the notion is more like "necessary truth" than
"truth". Or possibly even more like "provable", as you suggest below.

> For 
> the conditions mentioned already imply that "~[]A" can never be a 
> theorem. If it were, then of course "[]A --> A" would be a theorem;
> by 
> necessitation, "[]([]A --> A)" would be a theorem; and so, by (3),
> "[]A" 
> would be a theorem, and so the theory would be inconsistent. So
> nothing 
> can ever be proven not to be true.
> 

You could of course have "[]~A" as a theorem.

But yes, certainly the fact that []A->A can never be proven is quite
startling and led me to look back at the statement of the theorem,
which is as follows ("Logic of Provability", p. 173, notes in square
brackets are mine):

"A universe is a set V_kappa where kappa is inaccessible.... define A*
as before [note - i.e. by induction on the complexity of the modal
sentence A, with the propositional connectives all unchanged but the
box operator getting 'translated' ], except that now we redefine ([]A)*
as the sentence of the language of set theory that translates 'A* holds
in all universes'.

A finite strict linear ordering is a frame <W,R>, where W is finite and
R is a transitive and irreflexive relation that is connected on W,
i.e., for all w, x, in W, either wRx or w=x or xRw. [ note - this will
be used as a structure for interpreting modal sentences using Kripke's
semantics; the elements of W are worlds and wRx means 'x is a world
accessible from w, or a possible world with respect to w' ]

Let J be the system that results when all sentences

[]([]A->B) or []([.]B->A)

are added to GL as new axioms. [ note - [.]A means []A or A ]

Theorem 2 (Solovay). Let A be a modal sentence. Then (A), (B), and (C)
are equivalent:
(A) For all *, ZF proves A*
(B) A is valid in all finite strict linear orderings
(C) J satisfies A."

Okay, so note the relativization to ZF. We could ask "Never mind when
A* is ZF-provable, when does it actually hold?" But this would depend
on controversial questions about large cardinals. If we assume that the
universe is Mahlo, then []A->A would always hold. This issue gets
discussed a page later in the cited work and a more extensive modal
logic (where necessitation is not a rule of inference) is given as a
plausible candidate for the logic describing when A* actually holds.

> Obviously, the argument here is similar to that by which we infer the
> 
> unprovability of consistency from Loeb's theorem, which is roughly
> the 
> import of (3). That suggests that "true in all V_kappa for kappa 
> inaccessible" is a notion more closely related to notions like 
> provability than it is to semantic notions like truth: Indeed, the 
> axioms governing the box, so interpreted, simply include those that 
> govern the box, interpreted as just meaning provability: The modal
> logic 
> GL, which characterizes provability, just omits (4). (Note that the 
> condition "[][]A --> []A", familiar from Loeb's version of the 
> derivability conditions, is itself derivable from the others.)
> 

Yes, and note that "truth in all V_kappa with kappa inaccessible" is
coextensive with "second-order logical consequence of the axioms of
BG", which brings out the analogy with "provability". And of course in
the course of the book Boolos shows how deep that analogy runs. 

[ Aside: In a previous post, I characterized the Goedel sentence as "I
am *false" - i.e., in the quirky terminology I was using at the time,
"I am refutable". Of course this is incorrect, the sentence Goedel
actually used was "I am not provable". ]

> Richard Heck
> 
> 
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