Consequences
The Power Set Axiom allows a simple definition of the Cartesian product of two sets and :
Notice that
and thus the Cartesian product is a set since
One may define the Cartesian product of any finite collection of sets recursively:
Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.
Read more about this topic: Axiom Of Power Set
Famous quotes containing the word consequences:
“There is not much that even the most socially responsible scientists can do as individuals, or even as a group, about the social consequences of their activities.”
—Eric J. Hobsbawm (b. 1917)
“The medium is the message. This is merely to say that the personal and social consequences of any medium—that is, of any extension of ourselves—result from the new scale that is introduced into our affairs by each extension of ourselves, or by any new technology.”
—Marshall McLuhan (1911–1980)
“We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.”
—Alice Miller (20th century)