We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder-Bernstein theorem, this is equivalent to there being both a one-to-one mapping from X to Y and a one-to-one mapping from Y to X. We then write |X| = |Y|. The cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
Name:
Anonymous2013-08-31 23:43
The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless.
Name:
Anonymous2013-09-01 0:28
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation.
Name:
Anonymous2013-09-01 1:13
An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.
Name:
Anonymous2013-09-01 1:59
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty". More explicitly, it states that for every indexed family (S_i)_{i \in I} of nonempty sets there exists an indexed family (x_i)_{i \in I} of elements such that x_i \in S_i for every i \in I.
Name:
Anonymous2013-09-01 2:43
It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of Zermelo–Fraenkel set theory (ZF), regardless of the truth or falsity of the axiom of choice in that particular model.
Name:
Anonymous2013-09-01 3:29
If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite).