Lagmental Vicfred

Choice, Well-Ordering, and Zorn Are Equivalent by Vicfred

tags: mathematics, set theory, cardinals, choice

Three differently phrased principles select representatives, order arbitrary sets, or produce maximal objects. This is a compact note, but the quantifiers and hypotheses stay on the page.

Start locally

Cardinality compares sets through bijections rather than geometry. The notation \(|A|\le|B|\) means an injection \(A\hookrightarrow B\) exists, while equality requires a bijection.

$$ \prod_{i\in I}X_i\ne\varnothing\quad\text{whenever every }X_i\ne\varnothing $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \mathrm{AC}\Longleftrightarrow\text{well-ordering theorem}\Longleftrightarrow\text{Zorn's lemma} $$

Compute before generalising

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \begin{array}{ccc}\text{choice function}&\Longrightarrow&\text{well-order}\\\Uparrow&&\Downarrow\\\text{maximal element}&\Longleftarrow&\text{chain upper bounds}\end{array} $$

The global view

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad \prod_{i\in I}X_i\ne\varnothing\quad\text{whenever every }X_i\ne\varnothing,\\[5pt] \mathsf{C}\;&:\quad \mathrm{AC}\Longleftrightarrow\text{well-ordering theorem}\Longleftrightarrow\text{Zorn's lemma}. \end{aligned} $$

Edge conditions

Infinite cardinal arithmetic does not follow finite intuition. Removing one element or doubling a countably infinite set does not change its cardinality.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathrm{AC}\Longleftrightarrow\text{well-ordering theorem}\Longleftrightarrow\text{Zorn's lemma} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sun 14 October 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.