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Results tagged with independence-results
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user 7206
This tag is for questions about proving that some statement is independent from a theory, meaning it is neither provable nor refutable from that theory. Common examples are the continuum hypothesis from the axioms of ZFC, and the axiom of choice from the axioms of ZF.
14
votes
Accepted
Relationship between AC, WO, and Zorn's lemma in ZF-Powerset
This is a classic theorem of Zarach, that it is consistent that ${\sf ZF}^-$ holds with the Axiom of Choice, but not every set can be well-ordered.
Zarach, Andrzej, Unions of ${\sf ZF}^-$models wh …
- 39.8k
8
votes
Accepted
Relationship between fragments of the axiom of choice and the dependent choice principles
The idea is to mimic the permutation models as given in Jech. One can then ask, "Well, in Jech he chooses some set of objects in the full universe, and shows it has a support. But in forcing we don't …
- 39.8k
5
votes
Accepted
Implications of the existence of a pair of surjective functions, without Axiom of Choice
No, and here is a counterexample.
Suppose that $|\Bbb R|<|[\Bbb R]^\omega|$, that is, there are more countable subsets of reals than reals. This is indeed possible, e.g. if all sets of Lebesgue measu …
- 39.8k
5
votes
Accepted
Dedekind-"finiteness" for arbitrary limit cardinals
Start with your favourite model of $\sf ZFC$, your favourite regular cardinal $\mu$, and your favourite limit cardinal $\lambda>2^\mu$.
Now consider the ${<}\mu$-support product $\prod_{\alpha<\lambda …
- 39.8k
4
votes
Independence of the countable axiom of choice
First of all, the easy answer.
We can prove that the axiom of choice implies the axiom of countable choice, quite easily. So by showing that the axiom of choice is consistent with the axioms of $\sf …
- 39.8k