Nonstandard analysis

class=skin-invert-image|450px|thumb|Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω)

a totally ordered proper class containing the real numbers as well as hyperreal numbers such as infinity and infinitesimals.

element of a nonstandard model of the reals, which can be infinite or infinitesimal

alternative formulation of calculus using a logically rigorous notion of infinitesimal numbers

algebra over a field

mathematical notation

thumb|Hasse diagram of the [[divisors of 210, ordered by the relation is divisor of, with the upper set ↑14 colored dark green. It is a , but not an , as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements. Since ↑2 cannot be extended any further, it is an ultrafilter.]] In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.

class of extensions of the real numbers

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is the quotient set of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.

system of numbers with non-finite quantities

In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.

In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence in the ultrapower construction of the hyperreals.

principle that whatever succeeds for the finite also succeeds for the infinite