Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Going beyond the finite.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. Beyond finite.
 adj. Relating to transfinite numbers.
 n. A transfinite number.
from The Century Dictionary and Cyclopedia
 Beyond the finite.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Must try and explain transfinite postcapitalist economic theories to someone else one of these days, see if they think I'm talking out of my arse ....

They suck in infinite, recursively looping, transfinite time.

Indeed, between nonEuclidean geometry in the 1820s, abstract algebra in the mid1800s, and transfinite numbers in the 1880s, it had begun to seem like mathematics was a kind of universal framework for abstraction.
Wolfram Blog : Stephen Wolfram on the Quest for Computable Knowledge

Had I gone in with any assumptions about probable topics of conversation, they would have involved a discussion of neoPlatonism, arguments about Rousseau and a lecture on transfinite infinities, not how much of a shame it was that Firefly was cancelled.

Whether it was St Augustine contemplating the nature of creation, Newton and Leibniz battling over ownership of calculus, or Cantor struggling to publicize his vision of transfinite numbers, infinity's fascination was as much with the characters involved as the maths they were wrestling with.

Therefore transfinite induction to Îµ0 is not derivable.

Anyway, the result of the criticism was that Gentzen changed without further ado the proof into a third proof that uses the now famous principle of transfinite induction up to the first epsilonnumber.

Then he showed directly that transfinite induction up to the first epsilonnumber Îµ0 is expressible but not provable in the system.

Hilbert had thought that the “transfinite” indirect existence proofs would be the part of arithmetic that needs to be secured of contradiction.

He extended Peano arithmetic through transfinite ordinals and made the transfinite induction principle part of this extended calculus.
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