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Science Forum Index » Logic Forum » Intersections of transitive models
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| Daniel Waggoner |
 Posted: Mon Dec 22, 2003 11:30 am |
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I feel sure that others have looked at this, but I do not have a
reference. I believe that the intersection of transitive models* will
be a transitive model. If so, then there is a unique smallest
transitive model (in the sense that there is a unique smallest set
containing the empty set and successor sets). Any insights on what this
model looks like?
Daniel Waggoner
* By transitive model I mean a set M such that
(1) If x is an element of M, then x is a subset of M (transitive)
(2) That M, with the inherited element of binary relation, satisfies the
axioms of set theory, say ZF or NGB. In either case, choice is not
included as one of the axioms.
A quick look at these axioms (i.e. the ones I could recall on my morning
commute) convinced me that the intersection of such models will be a
such a model, but I could easily be wrong. |
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| H. Enderton |
| Posted: Mon Dec 22, 2003 11:30 am |
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Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> wrote:
Quote: I feel sure that others have looked at this, but I do not have a
reference. I believe that the intersection of transitive models* will
be a transitive model. If so, then there is a unique smallest
transitive model (in the sense that there is a unique smallest set
containing the empty set and successor sets). Any insights on what this
model looks like?
Right, if ZF has any transitive epsilon-models at all, then it
has a minimum such model. What does it look like? Well, it is
L_alpha for a certain countable ordinal alpha. Sure, alpha is
large, but not nearly as large as the least ordinal that is not
Delta^1_2.
Reference: Paul J. Cohen. A minimal model for set theory. BAMS,
vol. 69 (1963), pp. 537-540.
--Herb Enderton |
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| Daniel Waggoner |
| Posted: Tue Dec 23, 2003 10:16 am |
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H. Enderton wrote:
Quote: Right, if ZF has any transitive epsilon-models at all, then it has a
minimum such model. What does it look like? Well, it is L_alpha for
a certain countable ordinal alpha. Sure, alpha is large, but not
nearly as large as the least ordinal that is not Delta^1_2.
Reference: Paul J. Cohen. A minimal model for set theory. BAMS,
vol. 69 (1963), pp. 537-540.
Thanks for the info. Your answer raises more questions. The phrase,
"if ZF has any transitive epsilon-models at all" is telling. For some
reason, I had always assumed that transitive epsilon-models abounded.
Of course the existance of a minimal such model says that could not be true.
I was also under the impression that Cohen used the existance of
countable transitive epsilon-models in his proof that the CH was
independent of the other axioms. In particular, I thought that he
constructed a model of ZF+(~CH), and in this construction used countable
transitive epsilon-models. Does this mean that in ZF+CH one can prove
that there exist transitive epsilon-models, which implies that in a
minimal transitive epsilon-model that ~CH holds?
One last question, in ZF+AC can one construct transitive epsilon-models,
which would imply that ~AC holds in minimal transitive epsilon models?
Thanks again,
Daniel Waggoner |
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| H. Enderton |
| Posted: Mon Dec 29, 2003 1:35 pm |
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Earlier I wrote:
Quote: Right, if ZF has any transitive epsilon-models at all, then it has a
minimum such model. What does it look like? Well, it is L_alpha for
a certain countable ordinal alpha. Sure, alpha is large, but not
nearly as large as the least ordinal that is not Delta^1_2.
Reference: Paul J. Cohen. A minimal model for set theory. BAMS,
vol. 69 (1963), pp. 537-540.
Maybe a better place to read about the minimal model is pp. 104-106
of Cohen's 1966 monograph, "Set Theory and the Continuum Hypothesis."
Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> then wrote:
Quote: I was also under the impression that Cohen used the existance of
countable transitive epsilon-models in his proof that the CH was
independent of the other axioms.
Yes and no. He shows how to eliminate the use of transitive epsilon
models, and work simply from Con ZF.
Note that inside the minimal model, "There exists a transitive
epsilon model" is false, and yet ZF + CH is true, and Con ZF is
true. (Of course my phrase "inside the minimal model" conceals
some assumptions.)
--Herb Enderton |
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| H. Enderton |
| Posted: Tue Dec 30, 2003 8:42 am |
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Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> wrote:
Quote: When you say that ZF + CH is true inside the minimal model, do you mean
that CH is provable in ZF + "There does not exist a transitive epsilon
model" or do you mean something else?
Something else.
Theorem: Assume ZF has a transitive epsilon model. Then there
is a transitive epsilon structure (the minimal model) in which
the following are true: ZFC, V = L, CH.
That is what I meant.
Quote: Also, I asked this before, but I will ask again.
Is AC provable, or ~AC provable or are these independent in
ZF + "There does not exist a transitive epsilon model"?
First, ~AC is unprovable, because the minimal model is (given
the necessary assumption) a counterexample. Secondly, AC
is surely also unprovable, but somebody else will need to
supply details or a reference.
--Herb Enderton |
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| Daniel Waggoner |
| Posted: Tue Dec 30, 2003 1:28 pm |
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H. Enderton wrote:
Quote: Earlier I wrote:
Right, if ZF has any transitive epsilon-models at all, then it has a
minimum such model. What does it look like? Well, it is L_alpha for
a certain countable ordinal alpha. Sure, alpha is large, but not
nearly as large as the least ordinal that is not Delta^1_2.
Reference: Paul J. Cohen. A minimal model for set theory. BAMS,
vol. 69 (1963), pp. 537-540.
Maybe a better place to read about the minimal model is pp. 104-106
of Cohen's 1966 monograph, "Set Theory and the Continuum Hypothesis."
Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> then wrote:
I was also under the impression that Cohen used the existance of
countable transitive epsilon-models in his proof that the CH was
independent of the other axioms.
Yes and no. He shows how to eliminate the use of transitive epsilon
models, and work simply from Con ZF.
Note that inside the minimal model, "There exists a transitive
epsilon model" is false, and yet ZF + CH is true, and Con ZF is
true. (Of course my phrase "inside the minimal model" conceals
some assumptions.)
--Herb Enderton
Thanks, I will check out both of these references on my next trip to the
library.
When you say that ZF + CH is true inside the minimal model, do you mean
that CH is provable in ZF + "There does not exist a transitive epsilon
model" or do you mean something else? Also, I asked this before, but I
will ask again. Is AC provable, or ~AC provable or are these
independent in ZF + "There does not exist a transitive epsilon model"? |
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| Robert E. Beaudoin |
| Posted: Mon Jan 05, 2004 8:12 pm |
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H. Enderton wrote:
Quote: Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> wrote:
When you say that ZF + CH is true inside the minimal model, do you mean
that CH is provable in ZF + "There does not exist a transitive epsilon
model" or do you mean something else?
Something else.
Theorem: Assume ZF has a transitive epsilon model. Then there
is a transitive epsilon structure (the minimal model) in which
the following are true: ZFC, V = L, CH.
That is what I meant.
Also, I asked this before, but I will ask again.
Is AC provable, or ~AC provable or are these independent in
ZF + "There does not exist a transitive epsilon model"?
First, ~AC is unprovable, because the minimal model is (given
the necessary assumption) a counterexample. Secondly, AC
is surely also unprovable, but somebody else will need to
supply details or a reference.
--Herb Enderton
Let T be the theory ZF + ~AC, and let phi be the sentence "there is
no transitive epsilon model of ZF". Now Con(ZF) -> Con(T) is known,
and furthermore this statement is provable in ZF. Also,
~phi -> Con(ZF) is provable in ZF. Were ZF inconsistent it would
yield AC trivially as a consequence, so it is kosher to assume Con(ZF).
But then Goedel's second incompleteness theorem implies that
T + ~Con(T) is consistent, and as ZF proves that ~Con(T) -> phi we
have consistency of ZF + phi + ~AC. I.e., if ZF is consistent then
ZF + phi does not suffice to prove AC.
Robert E. Beaudoin |
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| Robert E. Beaudoin |
| Posted: Mon Jan 05, 2004 8:12 pm |
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H. Enderton wrote:
Quote: Daniel Waggoner <DanielWaggonerNo@Spam.mindspring.com> wrote:
When you say that ZF + CH is true inside the minimal model, do you mean
that CH is provable in ZF + "There does not exist a transitive epsilon
model" or do you mean something else?
Something else.
Theorem: Assume ZF has a transitive epsilon model. Then there
is a transitive epsilon structure (the minimal model) in which
the following are true: ZFC, V = L, CH.
That is what I meant.
Also, I asked this before, but I will ask again.
Is AC provable, or ~AC provable or are these independent in
ZF + "There does not exist a transitive epsilon model"?
First, ~AC is unprovable, because the minimal model is (given
the necessary assumption) a counterexample. Secondly, AC
is surely also unprovable, but somebody else will need to
supply details or a reference.
--Herb Enderton
Let T be the theory ZF + ~AC, and let phi be the sentence "there is
no transitive epsilon model of ZF". Now Con(ZF) -> Con(T) is known,
and furthermore this statement is provable in ZF. Also,
~phi -> Con(ZF) is provable in ZF. Were ZF inconsistent it would
yield AC trivially as a consequence, so it is kosher to assume Con(ZF).
But then Goedel's second incompleteness theorem implies that
T + ~Con(T) is consistent, and as ZF proves that ~Con(T) -> phi we
have consistency of ZF + phi + ~AC. I.e., if ZF is consistent then
ZF + phi does not suffice to prove AC.
Robert E. Beaudoin |
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