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Ideas for 'fragments/reports', 'The Tarskian Turn' and 'Grundlagen (Foundations of Theory of Manifolds)'

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2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
     Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence).
     From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
15364 English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]
     Full Idea: The number of English expressions is denumerably infinite. But Cantor's theorem can be used to show that there are nondenumerably many real numbers. So not every real number has a (simple or complex name in English).
     From: Leon Horsten (The Tarskian Turn [2011], 06.3)
     A reaction: This really bothers me. Are we supposed to be committed to the existence of entities which are beyond our powers of naming? How precise must naming be? If I say 'pick a random real number', might that potentially name all of them?