aleph numbers

The aleph numbers are infinite cardinal numbers defined by transfinite recursion, as described below. They are written ${\mathrm{\aleph }}_{\alpha }$, where $\mathrm{\aleph }$ is aleph, the first letter of the Hebrew alphabet, and $\alpha$ is an ordinal number. Sometimes we write ${\omega }_{\alpha }$ instead of ${\mathrm{\aleph }}_{\alpha }$, usually to emphasise that it is an ordinal.

To start the transfinite recursion, we define ${\mathrm{\aleph }}_0$ to be the first infinite ordinal. This is the cardinality of countably infinite sets, such as $\mathbb{N}$ and $\mathbb{Q}$. For each ordinal $\alpha$, the cardinal number ${\mathrm{\aleph }}_{\alpha +1}$ is defined to be the least ordinal of cardinality greater than ${\mathrm{\aleph }}_{\alpha }$. For each limit ordinal $\delta$, we define ${\mathrm{\aleph }}_{\delta }={\bigcup }_{\alpha \in \delta }{\mathrm{\aleph }}_{\alpha }$.

As a consequence of the Well-Ordering Principle (http://planetmath.org/ZermelosWellOrderingTheorem), every infinite set is equinumerous with an aleph number. Every infinite cardinal is therefore an aleph. More precisely, for every infinite cardinal $\kappa$ there is exactly one ordinal $\alpha$ such that $\kappa ={\mathrm{\aleph }}_{\alpha }$.

Title	aleph numbers
Canonical name	AlephNumbers
Date of creation	2013-03-22 14:15:39
Last modified on	2013-03-22 14:15:39
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	6
Author	yark (2760)
Entry type	Definition
Classification	msc 03E10
Synonym	alephs
Related topic	GeneralizedContinuumHypothesis
Related topic	BethNumbers