Aleph Zero is less than Cardinality of Continuum

Theorem

$\aleph_0 < \mathfrak c$

where

$\aleph$ denotes the aleph mapping,

$\mathfrak c$ denotes the cardinality of the continuum.

Proof

By Power Set of Natural Numbers has Cardinality of Continuum:

$\mathfrak c = \card {\powerset \N}$

where:

$\powerset \N$ denotes the power set of $\N$

$\card {\powerset \N}$ denotes the cardinality of $\powerset \N$.

By Cardinality of Set less than Cardinality of Power Set:

$\card \N < \card {\powerset \N}$

Thus by Aleph Zero equals Cardinality of Naturals:

$\aleph_0 < \mathfrak c$

$\blacksquare$

Sources

• Mizar article TOPGEN_3:30