Let $\NN \subseteq \FF$ be a nest.
We will show that $\bigcup \NN \in \FF$.
Let $F$ be a finite subset of $\bigcup \NN$.
- $\forall x \in F: x \in \map c x$
Then $c \sqbrk F$ is a finite subset of $\NN$.
- $F$ is a finite subset of $P$
- $P \in \FF$
Since $\FF$ has finite character:
- $F \in \FF$
We have thus shown that every finite subset of $\bigcup \NN$ is in $\FF$.
Since $\FF$ is of finite character:
- $\bigcup \NN \in \FF$
Axiom of Choice
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.
Source of Name
Also known as
The Tukey-Teichmüller Lemma is also known as:
- The Teichmüller-Tukey Lemma
- The Teichmüller-Tukey Theorem
- The Tukey-Teichmüller Theorem
- Tukey's Lemma.