Transfer Principle
Between the first and second edition of the Elementary Calculus, much of the theoretical material that was in the first chapter was moved to the epilogue at the end of the book, including the theoretical groundwork of non-standard analysis.
In the second edition Keisler introduces the extension principle and the transfer principle in the following form:
- Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.
Keisler then gives a few examples of real statements to which the principle applies:
- Closure law for addition: for any x and y, the sum x + y is defined.
- Commutative law for addition: x + y = y + x.
- A rule for order: if 0 < x < y then 0 < 1/y < 1/x.
- Division by zero is never allowed: x/0 is undefined.
- An algebraic identity: .
- A trigonometric identity: .
- A rule for logarithms: If x > 0 and y > 0, then .
Read more about this topic: Elementary Calculus: An Infinitesimal Approach
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