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

ISBN-10: 149870266X

ISBN-13: 9781498702669

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**Extra resources for Analysis with ultrasmall numbers**

**Example text**

This implies that r1 r2 , hence r1 = r2 , by Rule 3. Basic Concepts 15 A consequence of the Observable Neighbor Principle and Theorem 3 is that a real number x is not ultralarge if and only if it can be written as x = r + ε where r is observable and ε 0. This uniquely determined r is said to be the observable neighbor of x. In general, if we have x y, we say that x and y are neighbors. If one of the two is observable, then it is the observable neighbor of the other number. Intuitively, about each observable a there is a cluster of its neighbors, all of which are ultraclose to a.

By Stability, “For every x, x a implies 2x 2a, where is understood to be relative to a and q1 , . . , q ” is also true (for any q1 , . . , q ). Assume now that a is observable relative to q1 , . . , q . By Exercise 19, relative to a, q1 , . . , q is equivalent to relative to q1 , . . , q . Hence the statement “For every x, x a implies 2x 2a” is true when is understood to be relative to q1 , . . , q . That is, it is true in any context where a is observable. Arguably, this example is not very impressive, because the conclusion can be obtained directly from Rule 5, but it verifies the validity of Stability in this case.

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. Proof: Assume S = {n ∈ N : n is observable relative to p} is a set. Then (a) 0 ∈ S, because 0 ∈ N and 0 is observable relative to p. (b) If n ∈ S, then n ∈ N and n is observable relative to p, so n + 1 ∈ N and n + 1 is observable relative to p (the latter follows from the Closure Principle); hence n + 1 ∈ S.

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

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