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Full Idea
The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals.
Gist of Idea
Completed infinities resulted from giving foundations to calculus
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.3)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.51
A Reaction
Effectively, completed infinities just are the real numbers.
| 12489 | If there were real infinities, you could add two together, which is ridiculous [Locke] |
| 19406 | I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz] |
| 13190 | I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz] |
| 10855 | Actual infinities are not allowed in mathematics - only limits which may increase without bound [Gauss] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 15923 | Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine] |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| 11025 | Infinite cuts and successors seems to suggest an actual infinity there waiting for us [Read] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |