Adding the extra predicate of observability can, in principle, allow for statements which contradict classical theorems. In nonstandard analysis, these are called *external statements* and are to be avoided at introductory level since they introduce what would be considered pathological objects. Internal statements are the ones that we should refer to. In most nonstandard approaches, determining whether a statement is internal is a crucial and sometimes complicated issue. Here, thanks to multiplicity of levels of observability and the concept of context, even statements that do refer to observability are internal, as long as they take it relative to their context. So while in most nonstandard approaches the usual definitions of continuity, derivative, etc., are external, in our approach they are internal.

The “≃” symbol is the only new symbol introduced. By defining “ultraclose” to refer to the context, the notation is “rigged” in such a way that it is almost impossible to write an external statement without inventing an extra notation.

In short: an internal statement is either a classical statement of mathematics or one which can be shown to be equiconsistent with these classical statements. A statement which uses only the usual symbols or the additional symbol “≃” is internal.

We give here an example of an external statement. This question would not be addressed in class unless a student asked deeper questions–and this has not happened yet.

In nonstandard approaches with two levels (standard and nonstandard) (see [5–8]) there is a **st**() predicate for the standard part. It is an external predicate but is used in the definition of the derivative thus making the derivative an external statement. Consider the rule *x* ↦ **st**(*x*). If this defined a function, then zooming on the graph we would see a horizontal line on any ultrasmall neighbourhood (all points on an ultrasmall interval have the same observable neighbour.) There is no value where we could point to a discontinuity yet this everywhere horizontal “continuous” graph (if it exists) is increasing!

These nonstandard approaches must therefore deal with the difficult problem of justifying when external concepts can be used (in defining the derivative) and when they cannot.

In this approach, where observability is relative to the context, the derivative for instance, is given by the observable neighbour of the quotient (*f*(*a* + Δ*x*) − *f*(*a*)) /Δ*x*, where the observability depends on *f* and on *a*. The ultrasmallness of Δ*x* also refers to that context. (Because Δ*x* is bounded by a quantifier, it is not part of the context: this is not *about* Δ*x*: as in any classical explanation, it is a dummy variable.) The derivative is now an internal object. The “new rule” does not allow the construction of a function such as the one above where **st** would be “observable relative to 1”.

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