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Results tagged with axioms
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user 7206
4
votes
On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
Without the axiom of choice it is consistent that $\operatorname{cf}(\omega_1)=\omega$. This was shown by Feferman and Levy back in 1964, right after Cohen published his original results about forcing …
- 39.7k
1
vote
Accepted
Concrete mathematical statements in relation to Choice versus Reinhardt cardinals?
or consequences from Forcing Axioms ($\sf PFA$ or $\sf MM$) should motivate you to reject $\sf CH$? …
- 39.7k
7
votes
Accepted
Relation between AC and the axiom of foundation
Of course the axiom of choice is consistent with the failure of the axiom of foundation.
To get Fraenkel's model with the atoms, you usually start with a model of $\sf ZFA+AC$. You can find the relev …
- 39.7k
4
votes
Result that follows from ZFC and not ZF but are strictly weaker than choice
We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even …
- 39.7k
14
votes
Counterintuitive consequences of the Axiom of Determinacy?
To name a few,
There are no free ultrafilters on $\omega$.
There is no Hamel basis to $\Bbb R$ over $\Bbb Q$.
Under additional (but "reasonable") assumptions, every countable partial order embeds in …
- 39.7k