nonstandard analysis
Quick Summary
Nonstandard analysis is a branch of mathematics that, contrary to its name, is a rigorous variant of real analysis. Developed by Abraham Robinson, it introduces infinitesimals and infinities that genuinely exist without paradoxes.1
In nonstandard analysis, systems of nonstandard real numbers ((\mathbb{R}^)) and nonstandard natural numbers ((\mathbb{N}^)) are used. These systems include numbers that are infinitesimal relative to standard real numbers, meaning they are smaller than any nonzero standard real number. Conversely, they also include numbers that are infinite relative to standard real numbers.1
This approach allows for a different way of thinking about concepts like limits. For instance, the limit (a) of a sequence (a_n) can be defined as the real number such that (|a_n - a|) is infinitesimal for all infinite integers (n). While outside the mathematical mainstream, nonstandard analysis offers new methods and has shown promise, particularly in areas like stochastic differential equations.1
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metalogicElementary logic…what is known as “nonstandard analysis” that yields an unambiguous interpretation of the classical concept of infinitesimals—the division into units as small as one pleases. They have also been applied by two mathematicians, James Ax and Simon B. Kochen, to problems in the field of algebra (on p-adic fields).
Read More…what is known as “nonstandard analysis” that yields an unambiguous interpretation of the classical concept of infinitesimals—the division into units as small as one pleases. They have also been applied by two mathematicians, James Ax and Simon B. Kochen, to problems in the field of algebra (on p-adic fields).
Read More- metalogicUltrafilters, ultraproducts, and ultrapowers
…is in the introduction of nonstandard analysis, which was originally instituted by other considerations. By using a suitable ultrapower of the structure of the field ℜ of real numbers, a real closed field that is elementarily equivalent to ℜ is obtained that is non-Archimedean—i.e., which permits numbers a and b…
Read More…is in the introduction of nonstandard analysis, which was originally instituted by other considerations. By using a suitable ultrapower of the structure of the field ℜ of real numbers, a real closed field that is elementarily equivalent to ℜ is obtained that is non-Archimedean—i.e., which permits numbers a and b…
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