aleph numbers

The aleph numbers are infiniteMathworldPlanetmath cardinal numbersMathworldPlanetmath defined by transfinite recursion, as described below. They are written α, where is aleph, the first letter of the Hebrew alphabet, and α is an ordinal numberMathworldPlanetmath. Sometimes we write ωα instead of α, usually to emphasise that it is an ordinal.

To start the transfinite recursion, we define 0 to be the first infinite ordinal. This is the cardinality of countably infiniteMathworldPlanetmath sets, such as and . For each ordinal α, the cardinal number α+1 is defined to be the least ordinal of cardinality greater than α. For each limit ordinalMathworldPlanetmath δ, we define δ=αδα.

As a consequence of the Well-Ordering Principle (, every infinite set is equinumerous with an aleph number. Every infinite cardinal is therefore an aleph. More precisely, for every infinite cardinal κ there is exactly one ordinal α such that κ=α.

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