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

Rupert McCallum rupertmccallum at yahoo.com
Wed Oct 2 18:53:05 EDT 2002

--- JoeShipman at aol.com wrote:
> Here is a notion of set-theoretic truth that does not require
> inaccessibles.
> Let's expand the language of set theory to include a constant /kappa,
> and the 
> axiom scheme 
> phi iff V_/kappa satisfies phi
> , for ALL sentences phi in the original language (not including the
> symbol 
> /kappa)..
> /kappa shall henceforth be referred to as an "infallible cardinal"
> (being a 
> Catholic, I find this a congenial concept).

If it's only the first-order sentences you want reflected down to
V_kappa, then in Kelly-Morse set theory we can actually prove there is
a countable ordinal with this property. 

In Kelly-Morse set theory plus "the universe is totally indescribable",
we can prove there's a V_kappa that reflects all the second-order
properties of the universe (without parameters), I believe. I think you
can probably prove that even in KM plus "the universe is Mahlo", but
I'm not quite sure how and I am a little uncertain about this

I have an idea you probably would like to allow parameters. That is,
whenever phi is a second-order formula with free second-order variables
X_1, ..., X_n, you'd like the axiom phi<->V_kappa satisfies phi(V_kappa
intersect X_1,...,V_kappa intersect X_n) as part of your axiom schema. 

Saying the universe is totally indescribable precisely says that
there's such a V_kappa where kappa depends on the formula, you want to
strengthen that to "kappa *doesn't* depend on the formula". That's
definitely a substantial strengthening, for example your kappa would
clearly be the kappa-th totally indescribable cardinal, and stationary
limit of totally indescribable cardinals, and so forth.

If kappa is omega-indescribable then V_kappa is a model of your theory.

> If an infallible cardinal exists, it provides a nice semantics for
> set 
> theory, but in order to avoid circularity we need to say something
> else about 
> infallible cardinals.  Here are some questions:
> 1) If an infallible cardinal exists, what non-ZFC-provable sentences
> in the 
> language of set theory must be true?

Most notably that there is a standard model for ZFC (even one which is
a V_kappa), and a standard model for ZFC plus "there is a standard
model for ZFC", etc., etc.

If a V_kappa which reflects all second-order properties (without
parameters) of the universe exists, then it's clearly a kappa which is
hyper-kappa-inaccessible, for example, but I don't see how to prove
it's Mahlo (and I suspect one can show that it's impossible to prove
that, provided the assumption is consistent - though I don't see how to
prove that either).

If a V_kappa which reflects all second-order properties (with
parameters) of the universe exists, then as I pointed out it's a
stationary limit of totally indescribables, etc.

> 2) If no infallible cardinal exists, what consequences follow?

If kappa is an omega-indescribable cardinal, then V_kappa is a model of
ZFC+an infalliable cardinal. I don't think too much follows from the
nonexistence of an infallible, but a lot would follow from its
inconsistency - the skyscrapers of the large-cardinal hierarchy would
collapse and only a few small buildings would remain standing. (Of
course omega-indescribables are consistent, we all know that, and
therefore infallibles are). ;)

> 3) Suppose the Universe is Mahlo, and there is a stationary class of 
> inaccessibles such every sentence is either eventually true or
> eventually 
> false in the corresponding set of V_j. 

As follows from a discussion in my previous post, that's not consistent
- the liar paradox applies.

> What is the relation between
> this 
> assumption, and the assumption that an infallible cardinal exists? 
> (Can a 
> college of cardinals always identify an infallible one? Actually,
> since the 
> cardinals referred to form a proper class one could call them a
> University of 
> cardinals -- maybe a University can pick an infallible cardinal even
> if a 
> college, which is a term that could be reserved for mere sets of
> cardinals, 
> cannot.)
> -- Joe Shipman

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