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Full Idea
Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite.
Gist of Idea
Cantor's theory concerns collections which can be counted, using the ordinals
Source
report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.3
Related Idea
Idea 15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
| 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] |