# User:Asalmon

Jump to navigation
Jump to search

The list on here previously is now largely out of date.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...

## Contents

## eqn with "ll"

\(\displaystyle \) | \(\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \) |

## eqn with equality

\(\displaystyle \) | \(=\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \) |

## eqn template

\(\displaystyle \) | \(\implies\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \) | |||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \) |

## 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 $