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Following what is written at the 10th chapter of the "Teoria de Cojuntos, una introduccion" by Fernando Herandez (For the sake of completeness I should say that this is also present at the 11th chapter of "The Axiom of Choice" by Thomas Jech).

For every infinite cardinal number $\kappa$, let $\aleph(\kappa)$ be the Hartogs number of $\kappa$, i.e., the least ordinal which cannot be embedded by a one-to-one mapping in a set of cardinality $\kappa$. For every $\kappa$, $\aleph(\kappa)$ is an aleph, viz. the least aleph $\aleph$ such that $\aleph\not\le\kappa$.

Lemma 10.5

If $\kappa$ is an infinite cardinal and $\aleph$ is ana aleph, and if

10.1.1 $\quad\quad\quad\quad\quad\quad\kappa+\aleph=\kappa*\aleph$,

then either $\kappa\ge\aleph$ or $\kappa\le\aleph$. In particular, if

10.1.2$\quad\quad\quad\quad\quad\kappa+\aleph(\kappa)=\kappa*\aleph(\kappa)$

then $\kappa$ is an aleph.

Theorem 10.6

The Axiom of Choice is equivalet to say that for two cardinal $\kappa$ and $\lambda$ it's result $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\kappa+\lambda=\kappa*\lambda$.

Proof. We will show that under the hypothesis of the theorem any infinite cardinal is an aleph. Let $\kappa$ be an infinite cardinal, by Lemma 10.5 it follows that $\kappa\le\aleph(\kappa)$, which says that $\kappa$ is an aleph.

So I don't understand the proof of theorem 10.6: could someone explain to me why what is here written prove the theorem?

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The statement "every infinite cardinal is an aleph" means that every infinite set can be put in bijection with an aleph (which is a particular ordinal). Thus, every set can be well-ordered. The well-ordering principle is well-known to be equivalent to the axiom of choice.

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