The cardinality of the first infinite ordinal, omega (the number of natural numbers).
Aleph 1 is the cardinality of the smallest ordinal whose cardinality is greater than aleph 0, and so on up to aleph omega and beyond. These are all kinds of infinity.
The Axiom of Choice (AC) implies that every set can be well-ordered, so every infinite cardinality is an aleph; but in the absence of AC there may be sets that can’t be well-ordered (don’t posses a bijection with any ordinal) and therefore have cardinality which is not an aleph.
These sets don’t in some way sit between two alephs; they just float around in an annoying way, and can’t be compared to the alephs at all. No ordinal possesses a surjection onto such a set, but it doesn’t surject onto any sufficiently large ordinal either.
the cardinal number of the set of all positive integers; the smallest transfinite cardinal number. noun the smallest infinite cardinal number; the cardinal number of the set of positive integers א0 aleph-null (ä’lěf-nŭl’) The first of the transfinite cardinal numbers, corresponding to the number of elements in the set of positive integers.
noun the Hebrew alphabet Contemporary Examples When I learned my ABCs, they taught us our aleph-bet at the same time. How I Write: Nathan Englander Noah Charney March 26, 2013
- Aleppo boil
aleppo boil Aleppo boil A·lep·po boil (ə-lěp’ō) n. The lesion found in cutaneous leishmaniasis. Also called Bagdad boil, date boil. Historical Examples This is a disgusting eruption, which is called the aleppo boil, or Date-mark. A Woman’s Journey Round the World Ida Pfeiffer
- Aleppo gall
a nutlike gall produced by gall wasps on certain oaks in western Asia and eastern Europe, used as a source of astringent and tannic and gallic acids. noun a type of nutgall occurring in oaks in W Asia and E Europe