more from this thinker | more from this text
Full Idea
Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities.
Gist of Idea
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities
Source
report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.17
A Reaction
Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them?
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 10034 | The number of natural numbers is not a natural number [Frege, by George/Velleman] |
| 14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |