Cantor–Bernstein–Schroeder Theorem
TheoremIf there exist injective functions $f: A \to B$ and $g: B \to A$ between sets $A$ and $B$, then there exists a bijection $h: A \to B$.
Set Theory
Field: Set Theory
Sequence of Expressions
Step 25
Step 69
Zorn's Lemma
TheoremIf every chain (totally ordered subset) in a partially ordered set $P$ has an upper bound in $P$, then $P$ contains at least one maximal element.
Step 115
Well-Ordering Theorem
TheoremEvery set can be well-ordered.
Step 172
Banach–Tarski Paradox
TheoremA sphere can be decomposed into a finite number of point sets and reassembled to form two spheres identical to the original.
Step 221
Cantor's Intersection Theorem
TheoremA decreasing sequence of non-empty compact closed sets in a metric space has a non-empty intersection.
Step 251
König's Theorem (Set Theory)
TheoremStrict inequality between the sum of cardinalities and product of cardinalities for infinite sets.
Step 288
Easton's Theorem
TheoremConsistency of powers of regular cardinals.