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Full Idea
Non-standard natural numbers will yield non-standard rational and real numbers. These will include reciprocals which will be closer to 0 than any standard real number. These are like 'infinitesimals', so that notion is not actually a contradiction.
Gist of Idea
Infinitesimals are not actually contradictory, because they can be non-standard real numbers
Source
David Bostock (Philosophy of Mathematics [2009], 5.5)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.155
| 21382 | Things get smaller without end [Anaxagoras] |
| 18081 | Nature uses the infinite everywhere [Leibniz] |
| 18080 | A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz] |
| 18091 | Infinitesimals are ghosts of departed quantities [Berkeley] |
| 18085 | Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy] |
| 18086 | Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18083 | With infinitesimals, you divide by the time, then set the time to zero [Kitcher] |
| 18834 | Infinitesimals do not stand in a determinate order relation to zero [Rumfitt] |