A set Y is at least as big as a set X if there is an injective (one-to-one) mapping from the elements of X to the elements of Y. A one-to-one mapping identifies each element of the set X with a unique element of the set Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size we would observe that there is a mapping:
1 → a
2 → b
3 → c
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Anonymous2013-08-31 8:35
For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.
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Anonymous2013-08-31 9:21
Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries.
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Anonymous2013-08-31 10:07
The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything," relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.
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Anonymous2013-08-31 10:52
Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.