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Internal or standard view

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There are in fact two ways to view the axioms intuitively. In the internal view, advocated by Nelson, the numbers and sets of the theory are regarded as the usual sets and numbers we are all familiar with. In this view, no new objects are added to the usual mathematical universe; it is only the language that is being extended. The standardness (or, observability) predicate is a linguistic device that singles out some of the familiar objects for special attention. This idea is attractive to those who can reconcile their view of natural numbers with the existence of properties that do not satisfy the Principle of Mathematical Induction. Such properties can be expressed in the extended language; for example, ``\(x\) is standard'' is such a property: \(1\) is standard; if \(n\) is standard, then \(n+1\) is standard, but not all natural numbers are standard. It works quite well in the classroom, but it seems that many mathematicians find it incompatible with their ideas about natural numbers.

 

The alternative is the ``standard view''. In this view,  we identify the standard (observable in every context) sets with the familiar sets of traditional mathematics. But these sets are seen as having also a plethora of nonstandard, ideal elements, such as the ultrasmall and ultralarge elements of the set real numbers.

 

The book is written in the standard view. The teaching material is written in the internal view - for simplicity and also because students, having no prior knowledge about real numbers, the distinction appears irrelevant.

 

An important point is that the two views differ only philosophically. They are concerned with the intuitive interpretation; the actual mathematics is the same in either view.