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

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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. However, in general Stability provides information that is not obtainable otherwise. ” By our convention about contexts, is to be taken Basic Concepts 29 relative to the context of the statement, that is, f, a.

In our view, the universe of mathematical objects is a much richer place than is the standard view, full of ideal elements of all sorts. But the presence of the ideal elements in the standard sets does not change the properties of these sets. Every fact (be it axiom or theorem) of traditional mathematics remains true. Thus the arithmetic operations +, − and × are defined for all real numbers, whether these are standard or not, and satisfy the usual axioms. Division is defined whenever the denominator is not 0; in particular, it is defined for infinitesimal denominators.

4) The set C = { x, y ∈ R × R : x2 + y 2 = a2 }. 7 Show that the assertion in the preceding exercise need not hold when n is not observable. Hint: Use Exercise 9. 8 If x is ultrasmall [respectively, ultralarge] relative to p1 , p2 , then x is ultrasmall [respectively, ultralarge] relative to p1 . If x y relative to p, p1 , . . , pk , then x y relative to p1 , . . , pk . 9 Show that (relative to a fixed context): (1) {x ∈ R : x is not ultralarge} is not a set. (2) {h ∈ R : h is ultrasmall} is not a set.

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