Role of Axiom of Choice
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. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable.
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. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF.
Read more about this topic: Aleph Number
Famous quotes containing the words role of, role, axiom and/or choice:
“The role of the writer is not simply to arrange Being according to his own lights; he must also serve as a medium to Being and remain open to its often unfathomable dictates. This is the only way the work can transcend its creator and radiate its meaning further than the author himself can see or perceive.”
—Václav Havel (b. 1936)
“Women’s battle for financial equality has barely been joined, much less won. Society still traditionally assigns to woman the role of money-handler rather than money-maker, and our assigned specialty is far more likely to be home economics than financial economics.”
—Paula Nelson (b. 1945)
“The writer who neglects punctuation, or mispunctuates, is liable to be misunderstood.... For the want of merely a comma, it often occurs that an axiom appears a paradox, or that a sarcasm is converted into a sermonoid.”
—Edgar Allan Poe (1809–1845)
“In this choice of inheritance we have given to our frame of polity the image of a relation in blood; binding up the constitution of our country with our dearest domestic ties; adopting our fundamental laws into the bosom of our family affections; keeping inseparable and cherishing with the warmth of all their combined and mutually reflected charities, our state, our hearths, our sepulchres, and our altars.”
—Edmund Burke (1729–1797)