is this theory consistent?
Rupert
"Consider the theory
ZFC+V=L+there is a transitive model of 'ZFC+0# exists' "
Is there an upper bound for the consistency strength of this? If there
is a measurable cardinal then there must be a transitive model of ZFC
+0# exists in L[U] where L[U] is the inner model with one measurable
cardinal, but it doesn't seem obvious that there must be such a model
in L.
David Libert
This theory is equiconsistent with ZFC + there is a transitive model of
ZFC + 0# exists.
Ie, the V=L part does not affect consistency strength in this case.
To see better what this consistency stength is, it is strictly above
ZFC + Con(ZFC + 0# exists).
In fact it is strictly above all reasonable iterations of
Con^alpha(ZFC + 0#) exists for small concrete ordinal notations.
But it is strictly below the consistency strength of
ZFC + 0# exists and there is an inaccessible cardinal.
So it is well below the consistency strength of ZFC + there is a measurable
cardinal.
To see the opening claimed equiconsitency of adding V=L, suppose we are
in ZFC + there is a transitive model of 'ZFC + 0# exists' .
So by downward Lowenhiem Skolem there is countrable elementary submodel
of that transitive model. So this model is well-founded, being a submodel
of a transitive model.
We can now pull back the universe of this model isomorphically to a model
with underlying universe the real omega. So this model's interpretation of
epsilon is a subset of omega, ie a real.
So the property we have our universe is there is a real interpretting
epsilon over omega making a well-founded model of
ZFC + 0# exists.
To say the model is well-founded (we have AC) is equivalent to saying
there is no omega sequence in our universe making an infinite descending
sequence through the epsilon interpetation.
So the existence of an epsilon interpeation on omega making a well
founded model is equiocalent to exists a real (the epsilon interpretaion)
such that for all reals (omega sequences) the omega sequence is not
infinite descending through the epsilon interpretation.
This amounts to (exists a real) (for all reals) a property expressible
with wuantification over omega.
So this is a Sigma^1_2 property in the universe.
So by the Shoenfield Absoluteness Thoerem, since this holds in our
possible large universe, it also holds in L of that universe.
So in L, that statement true in L gives a real in L making
a well founded model of the theory.
Yes, I should hae said, we out into that sentence that the resulting
model also models 'ZFC + 0# exists', but that is just number
quantification to say that, so still stays in Sigma^1_2 ,
So the L inside our universe is a model of
ZFC + V=L + there is a transitive model of ZFC + 0# exists.
Some other interesting points about this.
This got a countable well-founded model of ZFC + 0# exists into L.
Suppose in our outer working universe we assume more: we assume there
is an uncountable transtivie model of ZFC + 0# exists.
We can't use the extra strength of this assumption to get a similar
uncountable well-founded model of ZFC + 0# exists in L.
Because if an uncountable well founded model ZFC believes a real is
0#, then it really is 0#. This from the theory of 0#.
So in L, if inside there was an uncountable well-founded model
inside of ZFC + 0# exists, then that real translating back to
transitive would really be 0#, so we would have 0# in L.
Which is impossible, by the theory of 0#.
--
David Libert ah...@FreeNet.Carleton.CA
Rupert
Yes, fantastic, Shoenfield Absoluteness Theorem. I think I must have
forgotten the correct statement of that.
> So in L, that statement true in L gives a real in L making
> a well founded model of the theory.
>
> Yes, I should hae said, we out into that sentence that the resulting
> model also models 'ZFC + 0# exists', but that is just number
> quantification to say that, so still stays in Sigma^1_2 ,
>
> So the L inside our universe is a model of
> ZFC + V=L + there is a transitive model of ZFC + 0# exists.
>
> Some other interesting points about this.
>
> This got a countable well-founded model of ZFC + 0# exists into L.
>
> Suppose in our outer working universe we assume more: we assume there
> is an uncountable transtivie model of ZFC + 0# exists.
>
> We can't use the extra strength of this assumption to get a similar
> uncountable well-founded model of ZFC + 0# exists in L.
>
> Because if an uncountable well founded model ZFC believes a real is
> 0#, then it really is 0#. This from the theory of 0#.
>
> So in L, if inside there was an uncountable well-founded model
> inside of ZFC + 0# exists, then that real translating back to
> transitive would really be 0#, so we would have 0# in L.
>
> Which is impossible, by the theory of 0#.
>
Yes, that's right, Maddy makes this observation.
Thanks a lot.