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6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity

[infinity as an unending ordered series]

8 ideas
The number of natural numbers is not a natural number [George/Velleman on Frege]
ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Maddy]
Cantor's theory concerns collections which can be counted, using the ordinals [Lavine]
Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
Raising omega to successive powers of omega reveal an infinity of infinities [Friend]