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Results tagged with independence-results
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user 138274
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.
7
votes
3
answers
431
views
Dedekind-"finiteness" for arbitrary limit cardinals
In $\mathbf{ZF}$, it is possible for a set $A$ to be infinite but not to admit a countable set. In other words, for any $\alpha\in\omega$, there is an injection from $\alpha$ into $A$, but there is no …
- 1,799
5
votes
1
answer
509
views
Smallest size of a non-measurable set of reals
The question is pretty much the title. I'm wondering if anything is known about the smallest size $\kappa$ of a non-measurable subset of the real numbers (regarding the Lebesgue measure). Since we hav …
- 1,799
5
votes
0
answers
211
views
Questions about very fat sets
If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ordertype $ …
- 1,799
9
votes
1
answer
474
views
Relationship between AC, WO, and Zorn's lemma in ZF-Powerset
In regular ZF, AC, WO, and Zorn's Lemma are equivalent, but every proof I know (of the implication AC -> WO and AC -> Zorn) uses the axiom of choice on the powerset of X (where X is the Set which is t …
- 1,799