# Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum/Corollary

## Corollary to Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum

Let $\MM$ be an infinite $\sigma$-algebra on a set $X$.

Then $\MM$ is uncountable.

## Proof

From Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum, $\MM$ has at least the cardinality of the set of real numbers $\R$.

The result follows from Real Numbers are Uncountable.

$\blacksquare$

## Axiom of Choice

This theorem depends on the Axiom of Choice, by way of Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

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.

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*: Exercise $1.3$