Must try and explain transfinite post-capitalist economic theories to someone else one of these days, see if they think I'm talking out of my arse .... ❋ Unknown (2003)
They suck in infinite, recursively looping, transfinite time. ❋ Unknown (2009)
Indeed, between non-Euclidean geometry in the 1820s, abstract algebra in the mid-1800s, and transfinite numbers in the 1880s, it had begun to seem like mathematics was a kind of universal framework for abstraction. ❋ Unknown (2009)
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 epsilon-number. ❋ Unknown (2009)
Therefore transfinite induction to ε0 is not derivable. ❋ Unknown (2009)
Had I gone in with any assumptions about probable topics of conversation, they would have involved a discussion of neo-Platonism, arguments about Rousseau and a lecture on transfinite infinities, not how much of a shame it was that Firefly was cancelled. ❋ Unknown (2009)
Then he showed directly that transfinite induction up to the first epsilon-number ε0 is expressible but not provable in the system. ❋ Unknown (2009)
Hilbert had thought that the “transfinite” indirect existence proofs would be the part of arithmetic that needs to be secured of contradiction. ❋ Unknown (2009)
He extended Peano arithmetic through transfinite ordinals and made the transfinite induction principle part of this extended calculus. ❋ Unknown (2009)
Second-order number theory is used in Gentzen's last paper, published in 1943, in which it is briefly shown that the principle of transfinite induction up to ε0 is derivable in second-order number theory. ❋ Unknown (2009)
The use of transfinite induction is made clearly visible in the 1938 paper through an ordinal notation. ❋ Unknown (2009)
The idea of the proof is, in brief terms, as follows: first it is laid down what it means to derive transfinite induction to a specific ordinal number in the system. ❋ Unknown (2009)
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. ❋ Brian Clegg (2009)
It is then shown that if transfinite induction to an ordinal number is derivable, this ordinal number cannot be greater than the value of the derivation. ❋ Unknown (2009)
A real first-rate pomo would have been able to drop a mention of Godel here, or Hilbert's Hotel, or the transfinite. ❋ Unknown (2008)
Cantor in the 19th century pioneered transfinite mathematics through negative argumentation (again like ID's arguments). ❋ Unknown (2006)
For example, the Axiom of Choice allows the taking of maximal ideals in rings and other structures; it entails the kinds of maximality principles that even the analysts were already using; it simplifies transfinite arithmetic; and despite its suspicious abstractness it turns out to be equivalent to ❋ Paseau, Alexander (2008)
Dedekind also provided a proof of the Cantor-Bernstein Theorem (that between any two sets which can be embedded into each other one-to-one there exists a bijection, so that they have the same cardinality), another basic result in the modern theory of transfinite cardinals. ❋ Reck, Erich (2008)
