Aleph Zero is less than Cardinality of Continuum
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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