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2. Non-standard model of arithmetic - Wikipedia

   en.wikipedia.org/wiki/Non-standard_model_of...

   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.

3. Non-standard model - Wikipedia

   en.wikipedia.org/wiki/Non-standard_model

   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

4. Nonstandard analysis - Wikipedia

   en.wikipedia.org/wiki/Nonstandard_analysis

   The standard way to resolve these debates is to define the operations of calculus using limits rather than infinitesimals. Nonstandard analysis [1] [2] [3] instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. [4] [5 ...

5. Elementary equivalence - Wikipedia

   en.wikipedia.org/wiki/Elementary_equivalence

   Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model. Elementary substructures and elementary extensions

6. Tennenbaum's theorem - Wikipedia

   en.wikipedia.org/wiki/Tennenbaum's_theorem

   The first step in the proof is to show that, if M is any countable nonstandard model of PA, then the standard system of M (defined below) contains at least one nonrecursive set S. The second step is to show that, if either the addition or multiplication operation on M were recursive, then this set S would be recursive, which is a contradiction.

7. Nonstandard calculus - Wikipedia

   en.wikipedia.org/wiki/Nonstandard_calculus

   In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic .