more from this thinker     |     more from this text

Single Idea 10112

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities ]

Full Idea

Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity.

Gist of Idea

The naturals won't map onto the reals, so there are different sizes of infinity

Source

report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.99

The 5 ideas with the same theme [infinities beyond the bounds of natural numbers]:

15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
18959 Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam]
10483 Mathematics and science do not require very high orders of infinity [Boolos]
15909 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]