By Hrbacek K., Lessmann O., O'Donovan R.
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Additional info for Analysis with ultrasmall numbers
There are also numbers unobservable relative to 24 Analysis with Ultrasmall Numbers these, and so on. The guiding principle is that all levels of observability should have the same properties. In particular, each level satisfies the Closure Principle, and so it is closed under traditional mathematical operations. This is a strong, uniform version of the principle that can be traced to Leibniz, namely, that the ideal elements have the same properties as the standard ones. An analogy with physics may be helpful in visualizing relative observability.
The last example is of great importance, and applies generally. By our convention, if a theorem does not specify the context of the relative concepts used in it, then we understand this context to be that of its parameters. By Stability and Exercises 18, 19, the theorem is then true in every context where the parameters are observable. Conversely, if the theorem is true in some context where its parameters are observable, then it is true also in the context specified by the parameters. Similar remarks apply to definitions.
That is, we assume that the universe of mathematical objects (including both the standard and the ideal ones) is stratified into levels of observability. The standard objects are always observable. If, say, h is ultrasmall (relative to the standard objects), then it is not observable relative to the standard objects; but the standard objects, as well as h itself and other objects uniquely definable from h (such as 2h, h3 /2, 1/h) are observable relative to h. However, there are also numbers that are not observable relative to h.