Axiom of Extensionality - in Set Theory With Ur-elements

In Set Theory With Ur-elements

An ur-element is a member of a set that is not itself a set. In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory. Ur-elements can be treated as a different logical type from sets; in this case, makes no sense if is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require to be false whenever is an ur-element. In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set. To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

That is:

Given any set A and any set B, if A is a nonempty set (that is, if there exists a member C of A), then if A and B have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to define itself to be the only element of whenever is an ur-element. While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

Read more about this topic:  Axiom Of Extensionality

Famous quotes containing the words set and/or theory:

    It is ... despair at the mutability of all created things that links the Artist and the Ascetic—a desire to purify and preserve—to set oneself apart—somehow—from the river flowing onward to the grave.
    Michele Murray (1933–1974)

    The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.”
    George Bernard Shaw (1856–1950)