The irrationals are the infinite decimals that don’t repeat. ❋ Unknown (2009)
The irrationals and delusionals insist on having a vote. ❋ Unknown (2010)
From Cantor's proof, however, set theorists erroneously conclude that “the set of irrational numbers” is greater in multiplicity than any enumeration of irrationals (or the set of rationals), when the only conclusion to draw is that there is no such thing as the set of all the irrational numbers. ❋ Rodych, Victor (2007)
As with his intermediate views on genuine irrationals and the Multiplicative Axiom, Wittgenstein here looks at the diagonal proof of the non-denumerability of “the set of transcendental numbers” as one that shows only that transcendental numbers cannot be recursively enumerated. ❋ Rodych, Victor (2007)
Complex and imaginary numbers have grown organically within mathematics, and they have proved their mettle in scientific applications, but pseudo-irrationals are inorganic creations invented solely for the sake of mistaken foundationalist aims. ❋ Rodych, Victor (2007)
Second, and more fundamentally, the intermediate Wittgenstein wrestles with irrationals in such detail because he opposes foundationalism and especially its concept of a “gapless mathematical continuum,” its concept of a comprehensive theory of the real numbers, and set theoretical conceptions and ˜proofs™ as a foundation for arithmetic, real number theory, and mathematics as a whole. ❋ Rodych, Victor (2007)
Inexact also are all nonterminating decimal representations of rationals, not to also mention irrationals and transcendentals. ❋ Unknown (2007)
Wittgenstein's discussion of irrationals is one with his critique of set theory, for, as he says, “[m] athematics is ridden through and through with the pernicious idioms of set theory,” such as ❋ Rodych, Victor (2007)
As we shall see in the next section, on Wittgenstein's account, if we do not understand irrationals rightly, we cannot but engender the mistakes that constitute set theory. ❋ Rodych, Victor (2007)
All things are one thing but the irrationals are something else. ❋ Lemon Hound (2007)
We simply do not need these aberrations, because they are not sufficiently comparable to rationals and genuine irrationals. ❋ Rodych, Victor (2007)
One of Wittgenstein's main aims in his lengthy discussions of rational numbers and pseudo-irrationals is to show that pseudo-irrationals, which are allegedly needed for the mathematical continuum, are not needed at all. ❋ Rodych, Victor (2007)
Ross is to be congratulated for smoking out the irrationals. ❋ Unknown (2007)
Though Wittgenstein stresses differences where others see similarities (LFM 15), in his intermediate attacks on pseudo-irrationals and foundationalism, he is not just emphasizing differences, he is attacking set theory's “pernicious idioms” (PR §173) and its “crudest imaginable misinterpretation of its own calculus” (PG 469-70) in an attempt to dissolve “misunderstandings without which [set theory] would never have been invented,” since it is “of no other use” (LFM 16-17). ❋ Rodych, Victor (2007)
MS and TS passages on irrationals and Cantor's diagonal, which were not included in PR or PG, Wittgenstein says: ❋ Rodych, Victor (2007)
Russell preferred to reconstruct talk of irrationals this way rather than infer or postulate the existence of a new species of mathematical entity not already known to exist; complaining that the method of ❋ Klement, Kevin (2005)
In much the same spirit, Karl Weierstrass, Richard Dedekind and Georg Cantor had also all developed methods for founding the irrationals in terms of the rationals. ❋ Irvine, A. D. (2003)
Richard Dedekind (18311916) gave arithmetic definitions of irrational numbers (the Dedekind cut), constituting the first rigorous theory of irrationals. ❋ Unknown (2001)
Still, from the Middle Ages until then, most mathematicians and scientists ignored the fact that the irrationals seemed not to exist, and happily, if awkwardly, used them anyway. ❋ Leonard Mlodinow (2001)
"You continue to [disagree] with me, despite having heard all the [facts] of the matter, which indicates you are either [daft] or irrational." ❋ Kplugrafa (2009)
Guy: That's a very [poor] [decision]...
Girl: I'm [irrational]. ❋ BigSlice (2008)
[I am] so irratted today that [I want] to [scream]! ❋ Carl Lundgren (2008)
"Yo dude, how are the Mets gonna do this season?"
"World series in six! You can't beat a core offense with Wright, [Reyes], [Beltran], and Bay!"
(Five months later, 17 games back in the [NL] East, irrational hope still exists) ❋ Yagottabelieve (2010)
[Juliette] is completely [irrational] ❋ FYR3 CRACKR (2020)
[i.e] Buying [the mug] was [irrational] as he never used it. ❋ 0110001101100100 (2019)
[My sister] is being [irrational] ❋ My Sister Is Annoying (2019)
That [TV show] gives me the irrates.
[I've been] waiting [3 hours] for you, I was really starting to get the irrates. ❋ Tool On Bass (2012)
Brian was being very irratical when he bought [TF2] on [steam], and not on [the orange box] ❋ Mr. Jakobitz (2011)
[John] is such an irratic [skateboarder] ❋ Aphirozialix (2009)
