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I added a brief note about the reason that the "intersection of the empty set" makes no sense in ZF; but I'm no set theorist, so please feel free to revert if I'm talking crap. Dmharvey 20:28, 21 April 2006 (UTC)
actually, the intersection of an empty set makes perfect sense with the stated definition, since the intersection is defined as a subset of the sum, and therefore, for an empty set, it is empty.-- 78.8.74.38 ( talk) 02:24, 14 March 2010 (UTC)
The first line in the interpretation section is:
"What the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A."
Shouldn't this be something like the following given the formal statement ?
"What the axiom is really saying is that, given a set A, we can find a set B whose members are contained in members of A." — Preceding unsigned comment added by Zair.M ( talk • contribs) 00:04, 11 December 2014 (UTC)
An intersection axiom can be made even more redundant if we set E = union X, we then have for the infinite intersection:
Jan Burse ( talk) 20:17, 6 March 2017 (UTC)
For background, see Zermelo–Fraenkel set theory, Von Neumann universe, Beth number, and Transitive model.
To show that the Axiom of union is independent of the other axioms of ZFC, we construct a transitive model which satisfies all the other axioms (that is: Axiom of extensionality, Axiom of regularity, Axiom of empty set, Axiom of pairing, Axiom of power set, Axiom schema of specification, Axiom schema of replacement, Axiom of infinity, and Axiom of choice) but does not satisfy the axiom of union. If the axiom of union could be proved from the other axioms, then this model would be impossible. So we will have proved independence.
For any ordinal α, define
Then one can show
and is a transitive set.
Furthermore
so that this hierarchy eventually becomes constant. [The superscript plus sign + means to take the successor cardinal. This will be a regular cardinal since we are assuming the axiom of choice holds in the universe in which we construct the model.]
We take
to be our model. It satisfies the other axioms because none of them can take sets in W (which thus have cardinality less than ) and make a set outside of W.
That it fails to satisfy the axiom of union is apparent from the fact that
but
OK? JRSpriggs ( talk) 08:16, 11 August 2019 (UTC)
Here are some more interesting details. W is a transitive set.
Any transitive model, such as W, will automatically satisfy the axioms of extensionality and regularity.
We can give a uniform verification for the axioms of separation and thus avoid dealing with the relativization to W of the quantifiers of the differentia. Similarly for the axioms of replacement.
So the powerset will not take us out of W.
It is also interesting to note that finite unions stay inside W.
This
level-5 vital article is rated Start-class on Wikipedia's
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I added a brief note about the reason that the "intersection of the empty set" makes no sense in ZF; but I'm no set theorist, so please feel free to revert if I'm talking crap. Dmharvey 20:28, 21 April 2006 (UTC)
actually, the intersection of an empty set makes perfect sense with the stated definition, since the intersection is defined as a subset of the sum, and therefore, for an empty set, it is empty.-- 78.8.74.38 ( talk) 02:24, 14 March 2010 (UTC)
The first line in the interpretation section is:
"What the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A."
Shouldn't this be something like the following given the formal statement ?
"What the axiom is really saying is that, given a set A, we can find a set B whose members are contained in members of A." — Preceding unsigned comment added by Zair.M ( talk • contribs) 00:04, 11 December 2014 (UTC)
An intersection axiom can be made even more redundant if we set E = union X, we then have for the infinite intersection:
Jan Burse ( talk) 20:17, 6 March 2017 (UTC)
For background, see Zermelo–Fraenkel set theory, Von Neumann universe, Beth number, and Transitive model.
To show that the Axiom of union is independent of the other axioms of ZFC, we construct a transitive model which satisfies all the other axioms (that is: Axiom of extensionality, Axiom of regularity, Axiom of empty set, Axiom of pairing, Axiom of power set, Axiom schema of specification, Axiom schema of replacement, Axiom of infinity, and Axiom of choice) but does not satisfy the axiom of union. If the axiom of union could be proved from the other axioms, then this model would be impossible. So we will have proved independence.
For any ordinal α, define
Then one can show
and is a transitive set.
Furthermore
so that this hierarchy eventually becomes constant. [The superscript plus sign + means to take the successor cardinal. This will be a regular cardinal since we are assuming the axiom of choice holds in the universe in which we construct the model.]
We take
to be our model. It satisfies the other axioms because none of them can take sets in W (which thus have cardinality less than ) and make a set outside of W.
That it fails to satisfy the axiom of union is apparent from the fact that
but
OK? JRSpriggs ( talk) 08:16, 11 August 2019 (UTC)
Here are some more interesting details. W is a transitive set.
Any transitive model, such as W, will automatically satisfy the axioms of extensionality and regularity.
We can give a uniform verification for the axioms of separation and thus avoid dealing with the relativization to W of the quantifiers of the differentia. Similarly for the axioms of replacement.
So the powerset will not take us out of W.
It is also interesting to note that finite unions stay inside W.