Skip to main content
MathOverflow

Return to Answer

Post Made Community Wiki by Stefan Kohl♦
added 20 characters in body
Source Link
Joel David Hamkins
Joel David Hamkins
  • 236.3k
  • 44
  • 777
  • 1.4k

Another classic example is the Schröder-Cantor-Bernstein theorem.

Theorem. If a set $A$ injects into $B$ and $B$ injects into $A$ then there is a bijection.

If AC holds, then this is nearly trivial, since if $A$ and $B$ are well orderable, then the minimal ordinals would have to be the same, giving a bijection.

But there is a more constructive proof (one not using AC), stitching together pieces of the injections as illustrated in the following figure.

enter image description here

Historically, the distinction between the proofs is important, and actually the history of the theorem and who proved what when is quite complicated. See the Wikipedia entry.

Another classic example is the Schröder-Cantor-Bernstein theorem.

Theorem. If a set $A$ injects into $B$ and $B$ injects into $A$ then there is a bijection.

If AC holds, then this is nearly trivial, since if $A$ and $B$ are well orderable, then the minimal ordinals would have to be the same, giving a bijection.

But there is a constructive proof, stitching together pieces of the injections as illustrated in the following figure.

enter image description here

Historically, the distinction between the proofs is important, and actually the history of the theorem and who proved when is quite complicated. See the Wikipedia entry.

Another classic example is the Schröder-Cantor-Bernstein theorem.

Theorem. If a set $A$ injects into $B$ and $B$ injects into $A$ then there is a bijection.

If AC holds, then this is nearly trivial, since if $A$ and $B$ are well orderable, then the minimal ordinals would have to be the same, giving a bijection.

But there is a more constructive proof (one not using AC), stitching together pieces of the injections as illustrated in the following figure.

enter image description here

Historically, the distinction between the proofs is important, and actually the history of the theorem and who proved what when is quite complicated. See the Wikipedia entry.

Source Link
Joel David Hamkins
Joel David Hamkins
  • 236.3k
  • 44
  • 777
  • 1.4k

Another classic example is the Schröder-Cantor-Bernstein theorem.

Theorem. If a set $A$ injects into $B$ and $B$ injects into $A$ then there is a bijection.

If AC holds, then this is nearly trivial, since if $A$ and $B$ are well orderable, then the minimal ordinals would have to be the same, giving a bijection.

But there is a constructive proof, stitching together pieces of the injections as illustrated in the following figure.

enter image description here

Historically, the distinction between the proofs is important, and actually the history of the theorem and who proved when is quite complicated. See the Wikipedia entry.