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Full Idea
The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal.
Clarification
'Ordinals' are numbers with an order
Gist of Idea
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.13
A Reaction
This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel.
Related Idea
Idea 8663 Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 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] |