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from PART I - ELEMENTS OF REAL ANALYSIS
Published online by Cambridge University Press: 05 January 2013
Introduction Traditional treatments of analysis are based on the ordinary real number system (R, +, ·, <), and their set-theoretical framework is ZFC. In this book, the set-theoretical framework is upgraded to IST and the real number system is upgraded to a system denoted by (ℝ, +, ·, <), where ℝ, +, ·, and < are standard sets in IST. It is useful to think of the system (ℝ, +, ·, <) as one to which the following informal remarks apply.
The properties of the elements of (ℝ, +, ·, <) are like the properties of the elements of anultrapower (*R, +, ·, <) of the ordinary real numbers, which we introduced in Definition 0.5.6.
The set ℝ, like *R, has both standard and nonstandard elements, and the standard elements of ℝ are none other than the ordinary real numbers. We shall use the symbols σℝ and R interchangeably to denote the class of the standard elements of ℝ. The presence of nonstandard numbers in ℝ endows the system (ℝ, +, ·, <) with external properties.
As far as the internal properties of the elements of (ℝ, +, ·, <) are concerned, the theory of (ℝ, +, ·, <) as an entity in IST is formally identical with the theory of (R, +, ·, <) as an entity in ZFC.
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