[FOM] The liar and the semantics of set theory (expansion)
Rupert McCallum
rupertmccallum at yahoo.com
Wed Oct 2 19:17:38 EDT 2002
--- Roger Bishop Jones <rbj at rbjones.com> wrote:
> 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?
>
Truth in V (which is the direct limit of all the V_alpha's). Not sure
what WF is.
The idea is that V is what you get when you extend the hierarchy of
V_alpha's "as far as it can possibly go", whatever that means exactly.
I believe this ginormous structure called V exists, is the intended
referent of our talk about sets, is fully determinate up to
isomorphism, and has the properties it has and always has had them and
always will regardless of the state of knowledge of any human mind,
etc., etc.. And I'm very confident that (at least) there is a proper
class of extremely indescribable cardinals in V and that V itself is
totally indescribable (or at least all the first-order consequences of
that, I'm not sure how meaningful it is really to ascribe second-order
properties to V), but I'm not sure what happens after that, if anything
- it's not for my finite human mind to know, I don't think.
Since, e.g. I'm agnostic about measurables and some aren't, I throw in
a hypothesis indicating how many large cardinals I have assumed for a
theorem, for the benefit of those who are agnostic about the large
cardinals I believe in. I try to only state things that are theorems of
ZFC, even though I believe in more. And this sometimes puts me in a
frame of mind where I feel like I'm pretending that I only believe in
ZFC and I'm agnostic about all the rest. But this doesn't mean I'm
changing my mind about which model I'm talking about - just that I'm
entertaining different conjectures about what it looks like.
(Oh, and by the way, I gave "there are infinitely many inaccessibles"
as the hypothesis for a result at the end of a previous post when it
should actually have been "there is no largest inaccessible").
> 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.
>
I think the considerations we have encountered tend to reinforce the
idea that this is inescapable. You suggested, defining TRUE to mean
"true in some V_alpha with alpha inaccessible, and true in all V_beta
with beta an inaccessible greater than alpha". Then I said, just like
you always can, "Okay, what about the sentence 'I am TRUE'?" This is
not TRUE, but it is TRUE*, TRUE* meaning "true in some V_alpha with
alpha the gamma'th inaccessible where gamma is a limit ordinal, and
true in all V_beta with beta a larger such inaccessible." The set of
TRUE sentences is a proper subset of the set of TRUE* sentences (which
is a proper subset of the set of true sentences, assuming the universe
is Mahlo). And one can similarly obtain a notion of TRUE**, TRUE***,
and so on, never fully exhausting the notion of truth. (Or never
achieving bivalence, as I might put it to someone who professed not to
know what I mean by truth).
You agreed that your earlier suggested "alternative semantics" were
unsatisfactory because they missed sentences which clearly ought to be
reckoned true. I put it to you that these considerations from the liar
paradox show that this will always happen for a non-bivalent
"alternative semantics" which is accepted as at least partially
characterizing truth.
> 2. There exists a universal language (an abstract lingua
> characteristica)
> in which an abstract semantics can be given to any coherent
> language, including itself.
>
Not consistent, I don't think.
> 3. There exist one or more ultimately expressive languages whose
> semantics cannot be defined in any language at all.
>
Same as (1), really, isn't it? Given an infinite regress of languages
you can always take the union of all those languages...
[...]
> My reasons for not liiking WF are as follows:
>
> 1. WF seems to me to be conceptually incoherent.
Okay, okay. I sympathize to some extent. There is a sense of
difficulty.
[...]
> It seems to me built into the conception of WF that its can never
> be completed,
Yep, yep. I know what you mean.
But have a think about this. There's a second-order sentence which is
satisfiable if and only if a measurable cardinal exists. Either there
can exist a structure satisfying this second-order sentence, or there
can't. If there can, then the levels of the set-theoretic hierarchy do
at some point eventually bump into a measurable cardinal, if not they
don't. So the proposition "there is a measurable cardinal" has a
truth-value - either there is one out there or there isn't.
And I think there is a certain difficulty about constructing a
semantics for set theory which does justice to intuitions like that,
and yet falls short of the full-blooded sort of conception of truth
that I am advocating. These are the sorts of considerations which
clinch the matter for me.
So let me know what you think about that.
I think I'll stop here and just send this off.
[...]
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