# User:Asalmon

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The list on here previously is now largely out of date. Use the list on the book page for Introduction to Axiomatic Set Theory.

I have now finished all the proofs in $\S 1 - \S 9$. Basis Representation Theorem for Ordinals is just proven. $\S 9$ is done! Skipped Well-Founded Recursion, though. May come back to it later.

Judging by the length of the proof, I expect Sierpinski's proof that $\operatorname{GCH} \implies \operatorname{AC}$ to break the record for page length on this site.

Template for me...

## eqn with "ll"

\(\ds \) | \(\) | \(\ds \) | ||||||||||||

\(\ds \) | \(\) | \(\ds \) | ||||||||||||

\(\ds \) | \(\) | \(\ds \) |

## eqn with equality

\(\ds \) | \(=\) | \(\ds \) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \) |

## eqn template

\(\ds \) | \(\implies\) | \(\ds \) | ||||||||||||

\(\ds \) | \(\implies\) | \(\ds \) | ||||||||||||

\(\ds \) | \(\implies\) | \(\ds \) | ||||||||||||

\(\ds \) | \(\implies\) | \(\ds \) |

## Proofs by Transfinite Induction Template

The proof shall proceed by Transfinite Induction on $$.

### Basis for the Induction

This proves the basis for the induction.

$\Box$

### Induction Step

This proves the induction step.

$\Box$

### Limit Case

This proves the limit case.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S $