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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

[infinity as an unending ordered series]

9 ideas
Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
     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.
     From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
     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.
     From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1
     A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them?
The number of natural numbers is not a natural number [Frege, by George/Velleman]
     Full Idea: Frege shows that the number of natural numbers is not identical to any natural number. This is because, while no natural number is identical to its successor, the number of natural numbers is.
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
     A reaction: Frege is notorious for the lack of respect shown in his writings for the great Cantor, and this seems to have blocked him from a more sophisticated account of infinity, but this idea seems a nice one.
ω names the whole series, or the generating relation of the series of ordinal numbers [Russell]
     Full Idea: The ordinal representing the whole series must be different from what represents a segment of itself, with no immediate predecessor, since the series has no last term. ω names the class progression, or generating relation of series of this class.
     From: Bertrand Russell (The Principles of Mathematics [1903], §291)
     A reaction: He is paraphrasing Cantor's original account of ω.
Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
     Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered.
     From: Shaughan Lavine (Understanding the Infinite [1994], III.4)
     A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'!
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
     Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one.
     From: Shaughan Lavine (Understanding the Infinite [1994], III.5)
Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
     Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
     A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone.
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
     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.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
     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.
Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
     Full Idea: The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1)
     A reaction: Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm.