Lagmental Vicfred

Cantor's Theorem Makes Every Power Set Larger by Vicfred

tags: mathematics, set theory, cardinals, choice

No function from a set onto its power set can be surjective. I will separate the object being defined from the consequence being claimed.

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.

$$ f:A\to\mathcal P(A) $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ D=\{a\in A:a\notin f(a)\}\Longrightarrow D\notin\operatorname{im}f $$

Compute before generalising

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ D=f(d)\Longrightarrow\bigl(d\in D\Longleftrightarrow d\notin f(d)=D\bigr) $$

The global view

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f:A\to\mathcal P(A),\\[5pt] \mathsf{C}\;&:\quad D=\{a\in A:a\notin f(a)\}\Longrightarrow D\notin\operatorname{im}f. \end{aligned} $$

Edge conditions

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] D=\{a\in A:a\notin f(a)\}\Longrightarrow D\notin\operatorname{im}f \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 17 January 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.