Analysis with ultrasmall numbers by Hrbacek K., Lessmann O., O'Donovan R. PDF

By Hrbacek K., Lessmann O., O'Donovan R.

ISBN-10: 149870266X

ISBN-13: 9781498702669

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For every real number r there is a natural number n such that n ≤ r < n + 1 (of course, if r is “infinitely large,” then n is also “infinitely large”). Every nonempty set of natural numbers has a least element. Every continuous function defined on a closed bounded interval attains its maximum there. These are just a few facts of traditional mathematics; they all remain valid in our view. They justify the use of the familiar notation for the traditional mathematical concepts, in spite of the change of viewpoint.

But, as discussed in the Introduction, no (infinite) set can contain observable elements only. However, it is perfectly legitimate to make external statements and to use them in proofs, as long as one avoids collecting all objects that satisfy such a statement into a set. Example. (1) Consider the statement “n is observable relative to p,” for a fixed p. There is no set S such that n ∈ S if and only if n ∈ N and n is observable relative to p. In other words, the “collection” {n ∈ N : n is observable relative to p} is not a set.

Turning to mathematics, the standard set of natural numbers N has standard elements such as 0, 1, 2, 17, 324 and so on. In our view, it has also nonstandard, ideal elements. Let N ∈ N be such a nonstandard element. What can we say about N ? Well, certainly 0 < N , because N is assumed to have all the properties of natural numbers and there are no natural numbers less than 0; also N = 0 because N is not standard. Similarly, 1 < N , because the only natural number less than 1 is 0, and N = 0, 1. By the same argument it follows that 2 < N , 3 < N and, in general, n < N for any standard n.

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Analysis with ultrasmall numbers by Hrbacek K., Lessmann O., O'Donovan R.

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