Re: V
- From: Rupert <rupertmccallum@xxxxxxxxx>
- Date: Thu, 31 May 2007 19:36:19 -0700
On Jun 1, 12:20 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 31, 8:52 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jun 1, 11:41 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 31, 8:25 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jun 1, 11:09 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 31, 5:54 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jun 1, 4:52 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 31, 12:20 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On May 31, 1:40 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 30, 10:22 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On May 31, 12:28 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 30, 9:06 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On May 30, 11:07 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 30, 12:43 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On May 30, 6:13 am, zuhair <zaljo...@xxxxxxxxx> wrote:
I am claiming that oo-ML is equi-interpretable with ZF/ZFC+IC.
Personally I don't think that we can have an inaccessible ordinal in
ZF/ZFC+IC and not in oo-ML and vice a versa.
I'll have to have a look at your definition of the theory. I suspect
that there is an interpretation of your theory on which "There is an
inaccessible cardinal" comes out true, but I doubt that all the
Replacement axioms in unrestricted form come out true.
Of course all these become true.Since in every layer they are proved.
Just because they're true in every layer doesn't mean they're true.
If kappa_1, kappa_2, kappa_3 ... are inaccessibles, and lambda is the
limit of kappa_1, kappa_2, kappa_3, .... then V_lambda is a model for
your theory, and Replacement is true in each V_(kappa_i), but it is
not true in V_lambda. Because {kappa_1, kappa_2, ...} is an image of a
set under a function, but is not a set.
Your oo-ML can be interpreted in ZFC+"there are infinitely many
inaccessibles", and the latter proves the consistency of the former.
wait don't confuse me please. Before you were saying that
oo-ML is interpreted in ZFC+ one inaccessible ordinal exist.
Did I say that? I think I was talking about a different theory. If I
did say that it was because I was led astray by Aatu's claim that it
was equi-interpretable with omega-order ZFC. I hadn't actually seen
the definition of oo-ML before. Having looked at your theory I don't
think that's the case, I think it can prove the consistency of omega-
order ZFC. Sorry about the confusion.
Now you are saying it is interpreted in ZFC + there are infinitely
many inaccessibles.
Yes.
What happened did you change your statement or you mean something
else.
Yes, if I said before that it was interpretable in ZFC+"one
inaccessible exists" then I've changed my mind. But I don't know if I
said that, I think I only said that your other theory V is so
interpretable.
Oops. sorry for this misunderstanding.
So you think that oo-ML is stronger than
Omega order ZFC. OK
OK then, do you think that oo-ML
is interpretable in ZFC + wIC
or ZFC+(w+1)IC.
ZFC+wIC, I think. I'll have to check.
Remember that the theory oo-ML
that you saw is only ONE-dimensional
oo-ML, I will try to work on a definition
for w-dimensional oo-ML, which has
Vi1i2i3........ primitive constants
i.e it contain w^w primitive constants.
You'll have to be a bit careful defining this theory. If the indexes
range over all possible infinite sequences of natural numbers then
there's no definable well-ordering on them, so I don't think you'd be
able to define your theory properly. In any event the language would
be uncountable, so it wouldn't be a recursively enumerable theory.
Do you think that such a theory would
be interpretable in ZFC+w^wIC,
or might be ZFC+((w^w)+1)IC.
Well, if you really had a type w^w sequence of primitive constants
(which could be indexed by all the *finite* sequences of natural
numbers with a certain well-ordering), then I think you'd get
something interpretable in ZFC+w^wIC.
What I like about oo-ML is that
it is simple, it can always be
extended to define any inaccessible
ordinal we want.
Well, only the alpha-th inaccessible cardinal where alpha is a
recursive ordinal. I mean, consider Tarski's axiom of inaccessibles,
which says there's an inaccessible cardinal larger than any given
ordinal. That's fairly simple, and it yields a theory much stronger
than any of the theories you're talking about. Or you can add an axiom
saying that a hyperinaccessible cardinal exists (a cardinal kappa
which is the kappa-th inaccessible cardinal). Or an axiom which says
that a hyperinaccessible cardinal exists larger than any given
ordinal. Or a hyperhyperinaccessible cardinal. And so on. As I say,
you should study the theory of large cardinals.
Zuhair
Zuhair
Take your time, you'll see what I mean.
Meanwhile I will work on the Omega-dimensional version of oo-ML, which
I think it would be equi-interpretable with
ZFC+ wIC.
By the way Is there a theory which prove the consistency of
ZFC+wIC.
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