G Cantor Quotes

.....but let's emphasize once more here that G. Cantor is, like R. Dedekind, a mathematical Platonist; i.e., he believes that both infinite sets and transfinite numbers really exist, as in metaphysically, and that they are "reflected" in actual real-world infinities.....

But what G. Cantor posits as the defining formal property of an infinite set is that such a set can be put in a 1-1C with at least one of its proper subsets. Which is to say that an infinite set can have the same cardinal number as its proper subset, as in Galileo's infinite set of all positive integers and that set's proper subset of all perfect squares, which latter is itself an infinite set.

It's specifically this Z = 2^(Aleph0) that he couldn't prove. Ever. Despite years of unimaginable doodling. Whether it's what unhinged him or not is an unanswerable question, but it is true that his inability to prove the C.H. caused Cantor pain for the rest of his life; he considered it his great failure. This too, in hindsight, is sad, because professional mathematicians now know exactly why G. Cantor could neither prove nor disprove the C.H. The reasons are deep and important and go corrosively to the root of axiomatic set theory's formal Consistency, in rather the same way that K. Godel's Incompleteness proofs deracinate all math as a formal system. Once again, the issues here can be only sketched or synopsized (although this time Godel is directly involved, so the whole thing is probably fleshed out in the Great Discoveries Series' Godel booklet).

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

several notable works containing Thetian stories have been penned through the centuries. Grenville's work, Ancient Warriors of Scandinavia (1884), and Addleson's, The Lost Cities of Prehistoric Europe (1921), both contain several stories of Theta's exploits. The Warlords (1408), by Chuan Chien contains two tales of Theta's adventures in Asia during the Neolithic Age. While there is no complete English translation of Chien's text, the accounts contained therein serve as independent evidence of the existence of Theta as a historical figure. The essay, Forgotten Empires by Charles Sawyer (1754), and Da Vinci's manuscript, Of Prehistory (1502), also contain story fragments and references to the historical Theta. The voluminous treatise, Prehistoric Cities of Europe and the Near East, by Cantor (1928), presents noteworthy, though inconclusive evidence of the historical existence of the city of Lomion in what is now southwestern England.

tive certeza sobre o homem na Lua antes de ter certeza sobre a Lua. Isso está em harmonia com a tradição popular. Os obscuros poetas modernos são naturalistas e falam de arbustos e riachos; mas os cantores dos poemas épicos e fábulas da antiguidade eram sobrenaturalistas e falavam dos deuses dos riachos e arbustos. É isso que os modernos querem dizer quando afirmam que os antigos não “apreciavam a natureza”, porque diziam que ela era divina. As antigas babás não falavam às crianças sobre a relva, mas sobre fadas que dançam sobre a relva; e os antigos gregos não conseguiam ver as árvores devido às dríades.