Search results
Results from the WOW.Com Content Network
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). [ 1 ] Existence
According to J. Mackenzie, the first edition of the book "deserves to become the standard textbook in its field", which he reiterated for the second edition. [10] [11] Reviewers particularly noted the book's utility as either a supplement to standard logic textbooks or as a primary text for courses on non-classical logic.
The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment.
These applications of nonstandard analysis depend on the existence of the standard part of a finite hyperreal r. The standard part of r, denoted st(r), is a standard real number infinitely close to r. One of the visualization devices Keisler uses is that of an imaginary infinite-magnification microscope to distinguish points infinitely close ...
Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations.
A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather that it has met with some degree of success, whether at the expense of giving significantly less meaningful proofs I do not know. My interest in non-standard analysis is that attempts are being made to introduce it into calculus courses.
Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51. "Abstract: This paper provides an explicit description of a model for intuitionistic nonstandard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice."
In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory. It is natural to ask whether a countable nonstandard model can be explicitly constructed.