Nonstandard Models of Arithmetic on the Web

(Mathematics Subject Classification 2000:
 03H15, Nonstandard models of arithmetic)

Notes by Steven H. Cullinane on Oct. 23, 2005

The Mathematical Experience,
 by Philip J. Davis and Reuben Hersh:

"It was discovered by the Norwegian logician Thorolf Skolem that there are mathematical structures which satisfy the axioms of arithmetic, but which are much larger and more complicated than the system of natural numbers. These 'nonstandard arithmetics' may include infinitely large integers. In reasoning about the natural numbers, we rely on our complete mental picture of these numbers. Skolem's example shows that there is more information in that picture than is contained in the usual axioms of arithmetic."

"Non-Standard Models in a Broader Perspective,"
 by Haim Gaifman (pdf)

"Non-Standard Models of Arithmetic,"
 by Asher M. Kach (pdf)

MathWorld:

"The Löwenheim-Skolem theorem is a fundamental result in model theory which states that if a countable theory has a model, then it has a countable model. Furthermore, it has a model of every cardinality greater than or equal to aleph_0. This theorem established the existence of 'nonstandard' models of arithmetic."

"Nonstandard Models of Arithmetic,"
by Ali Lloyd (pdf)