# Set Theory

DRAFT

Aleph null:- The smallest Aleph defined as the cardinality of the positive integers, and also that of the rationals and of the algebraic numbers, but not of the reals. The usual symbol is the Hebrew letter subscript 0.

Aleph:- Any infinite Cardinal number, usually denoted by the Hebrew letter Aleph.

Cardinal Number:- A measure of the size of a set that does not take into account the order of its members. This can be defined in terms of the cardinality of a recursively generated sequence of classes and is a wider concept than Natural number.

Any particular number having this function. For example, 1, 0 and Aleph null are cardinals.

Precisely, the smallest ordinal number that is equipollent to a given set.

Ordinal:- A set of which every member is also a subset (a transitive set) that contains only transitive elements. This can be used to generate the transfinite sequence

Phi, {Phi}, {Phi, {Phi}}, {Phi, {Phi}, {Phi, {Phi}}}, ….

Ordinal Number:- A measure of a set that takes account of the order as well as the number of its elements, defined to be the set of all well ordered sequences that are ordinally similar.

Ordinally similar (of two relations):- Such that there is a one to one correspondence between their domains that preserves order under the given relations.

Omega:- The smallest infinite ordinal. The ordinal of the natural order of the natural numbers represented by the Greek letter omega.