Transfinite numbers, also known as infinite numbers, are numbers that are not finite. These numbers were discovered by Georg Cantor.
As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.
The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the real numbers (the "continuum"): that is to say, aleph-one is the cardinality of the set of real numbers.
In both the cardinal and ordinal number systems, the transfinite numbers can keep on going forever, with progressively more bizarre kinds of number.
Beyond all these, Georg Cantor's conception of the Absolute Infinite surely represents the absolute largest possible concept of "large number".
This section of Descriptive Mathematics explores the paradoxes of transfinite theory and suggests that we would be better served by stopping the modern habit of treating paradoxes as a foundational issue and treat them as an interesting aside.
Transfinite theory begins with Bolzano's interpretation of Galileo's paradox to assert that the rational numbers are denumerable, the concludes with the liars paradox to show that the reals are not.
Transfinite theory is extremely seductive in that it gives mathematicians a feeling that they can derive the fundamental nature of the universe from contemplating paradoxes.
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