The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. 2, ω2, ωω number p in alphabet ε0 are among the countably infinite sets.

This ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Using the axiom of choice we can show one of the most useful properties of the set ω1: any countable subset of ω1 has an upper bound in ω1. This follows from the fact that a countable union of countable sets is countable, one of the most common applications of the axiom of choice. 1 is actually a useful concept, if somewhat exotic-sounding. The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. For example, for any successor ordinal α this holds.

There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Each finite set is well-orderable, but does not have an aleph as its cardinality.

The assumption that the cardinality of each infinite set is an aleph number is equivalent over ZF to the existence of a well-ordering of every set, which in turn is equivalent to the axiom of choice. The method of Scott’s trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. Earliest Uses of Symbols of Set Theory and Logic”. Georg Cantor:His Mathematics and Philosophy of the Infinite. His new numbers deserved something unique. Department of Mathematics, University of Michigan.

This page was last edited on 7 April 2018, at 20:00. The NATO phonetic alphabet is a way of using words to replace letters. The first letter of the word is the letter the word stands for. Numbers are also in the phonetic alphabet. The English numbers 0 through 3 and 5 through 8 are written and spoken the same. The number 4 is written the same, but pronounced fower to avoid confusion with the word “for”. This page was last changed on 30 March 2017, at 06:29.

See Terms of Use for details. This article is about sets of letters used in written languages. The Proto-Canaanite script, later known as the Phoenician alphabet, is the first fully phonemic script. Thus the Phoenician alphabet is considered to be the first alphabet. Many languages use modified forms of the Latin alphabet, with additional letters formed using diacritical marks. Alphabets are usually associated with a standard ordering of letters. This makes them useful for purposes of collation, specifically by allowing words to be sorted in alphabetical order.

Knowing one’s ABCs”, in general, can be used as a metaphor for knowing the basics about anything. The history of the alphabet started in ancient Egypt. In the Middle Bronze Age, an apparently “alphabetic” system known as the Proto-Sinaitic script appears in Egyptian turquoise mines in the Sinai peninsula dated to circa the 15th century BC, apparently left by Canaanite workers. The Proto-Sinaitic script eventually developed into the Phoenician alphabet, which is conventionally called “Proto-Canaanite” before ca.