Without the Axiom of Choice, must two nonempty sets have a surjection between them?
32081
The dual version of trichotomy of cardinality without AC
124847
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
49385
Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
88736
How does the Axiom of Extensionality prove the uniqueness of Specified sets?
77548
Given an arbitrary infinite set $S$, how to construct an injection $\mathbb{N} \to S$ with the Axiom of Choice?
97878
Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$?
156161
Banach-Tarski theorem without axiom of choice
53101
Is it true that the relation |A| < |B| is a sufficient condition for claiming that $f$ is a bijection?
23125
Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.
154578
Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?
100134
Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
1117
Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
105815
Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
146356
To prove two sets are isomorphic is it sufficient to find 2 injections or 2 surjections?
152272
Can you explain the "Axiom of choice" in simple terms?
5313
Existence of how many sets is asserted by the axiom of choice in this case?