The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R. Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...
Calculus is an important branch of mathematics. ...
Sir Isaac Newton, FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, regarded by many as the greatest figure in the history of science. ...
Leibniz redirects here. ...
In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a...
In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...
In mathematics, the real numbers may be described informally in several different ways. ...
In mathematics particularly in nonstandard analysis, the transfer principle is a rule which transforms assertions about standard sets, mappings etc. ...
It has been suggested that Predicate calculus be merged into this article or section. ...
An important property of *R is that it has infinitely large as well as infinitesimal numbers, where an infinitely large number is a number that is larger than all numbers representable in the form In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...
The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Kanovei and Shelah shows that there is a definable, countably saturated (meaning ωsaturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ...
Saharon Shelah (×©×”×¨×Ÿ ×©×œ×—, born July 3, 1945 in Jerusalem) is an Israeli mathematician. ...
In mathematical logic, and in particular model theory, a saturated model M is one which realizes as many complete types as may be reasonably expected given its size. ...
In model theory, given two structures and in the same language , we say that is an elementary substructure of (notated sometimes ) if 1. ...
The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin. In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every nonnegative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F...
The superreal numbers compose a more inclusive category than hyperreal number. ...
W. Hugh Woodin is a set theorist at University of California, Berkeley. ...
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis. When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilondelta definition of a limit by Cauchy, Weierstrass and others, they were largely abandoned. Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...
In the most restricted sense, nonstandard analysis or nonstandard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of...
Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
Sir Isaac Newton, FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, alchemist, and natural philosopher, regarded by many as the greatest figure in the history of science. ...
Leibniz redirects here. ...
Euler redirects here. ...
Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ...
Bishop George Berkeley George Berkeley (British English://; Irish English: //) (12 March 1685 â€“ 14 January 1753), also known as Bishop Berkeley, was an influential Irish philosopher whose primary philosophical achievement is the advancement of what has come to be called subjective idealism, summed up in his dictum, Esse est percipi (To...
Beginning of the Napoleonic Wars (1805  1815). ...
Calculus is an important branch of mathematics. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
Karl WeierstraÃŸ Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Biography Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...
However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from the use of them in nonstandard analysis, have no necessary relationship to model theory or first order logic. Abraham Robinson Abraham Robinson (October 6, 1918  April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ...
The transfer principle
Historically, the concept of number has been repeatedly generalized. At each step in this process of generalization, mathematicians knew that they wished to retain as many properties as possible from the earlier concepts of numbers. However, some properties always had to be given up. In the case of the hyperreals, a long historical delay in their development was caused by uncertainty among mathematicians as to exactly which properties could be retained, and which would have to be given up. The selfconsistent development of the hyperreals turned out to be possible if every true firstorder logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it: It has been suggested that Predicate calculus be merged into this article or section. ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a nonnegative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

The same will then also hold for hyperreals: 
Another example is the statement that if you add 1 to a number you get a bigger number: 
which will also hold for hyperreals: 
The correct general statement that formulates these equivalences is called the transfer principle. Note that in many formulas in analysis quantification is over higher order objects such as functions and sets which makes the transfer principle somewhat more subtle than the above examples suggest. In mathematics particularly in nonstandard analysis, the transfer principle is a rule which transforms assertions about standard sets, mappings etc. ...
The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element w such that 
but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like w is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals. The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology. In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a settheoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah, in the paper linked to at the end of this article, have found a method that gives an explicit construction, at the cost of a significantly more complicated treatment.) In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
The ultrapower construction We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example, In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, ...) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be selfconsistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say, 7 + ε, where ε is a certain infinitesimal number. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In mathematics, an algebra over a field K, or a Kalgebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...
Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion: but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters which do not contain any finite sets. (The good news is that the axiom of choice guarantees the existence of many such U, and it turns out that it doesn't matter which one we take; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that "matter": We write (a_{0}, a_{1}, a_{2}, ...) ≤ (b_{0}, b_{1}, b_{2}, ...) if and only if the set of natural numbers { n : a_{n} ≤ b_{n} } is in U. In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A, and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent. In the mathematical area of order theory, a total preorder over a set X is a preorder ≤ over X that is total; that is, for all a and b in X, it holds that a ≤ b or b ≤ a. ...
In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
The field A/U is an ultrapower of R. Since this field contains R it has cardinality at least the continuum. Since A has cardinality An ultrapower is an important special case of the ultraproduct construction. ...
it is also no larger than , and hence has the same cardinality as R. As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R. Thus in some sense of "larger" we do not need to go to a larger field to do nonstandard analysis. In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every nonnegative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F...
One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the continuum hypothesis false we can prove that there are nonorderisomorphic pairs of fields which are both countably indexed ultrapowers of the reals. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
The ZermeloFraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
For more information about this method of construction, check out ultraproducts and ultrapowers. An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. ...
An ultrapower is an important special case of the ultraproduct construction. ...
An intuitive approach to the ultrapower construction The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt (see the references below). Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense, the true infinitesimals are the classes of sequences that contain a sequence converging to zero. Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply add and subtract them, but not always divide by nonzero. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is, a_{n} = 0 for all n. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
In our ring of sequences one can get ab = 0 with neither a = 0 nor b = 0. Thus, if for two sequences a,b one has ab = 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As result, the classes of sequences that differ by some sequence declared zero will form a field which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, they will be represented by the sequences converging to infinity). Also every hyperreal which is not infinitely large will be infinitely close to an ordinary real, in other words, it will be an ordinary real + an infinitesimal. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, let us consider the zero sets of our sequences, that is, the z(a) = {i:a_{i} = 0}, that is, z(a) is the set of indexes i for which a_{i} = 0. It is clear that if ab = 0, then the union of z(a) and z(b) is N (the set of all natural numbers), so: Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ...
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...
 (i) one of the sequences that vanish on 2 complementary sets should be declared zero
also  (ii) if a is declared zero, ab should be declared zero too, no matter what b is.
and  (iii) if both a and b are declared zero, then a^{2} + b^{2} should also be declared zero.
Now the idea is to single out a bunch U of subsets X of N and to declare that a = 0 if and only if z(a) belongs to U. From the conditions (i), (ii) and (iii) one can see that  (i) From 2 complementary sets one belongs to U
 (ii) Any set containing any set that belong to U, belongs to U.
 (iii) An intersection of any 2 sets belonging to U belongs to U.
Also  (iv) we don't want an empty set to belong to U
because then everything becomes zero because every set contains an empty set. Any family of sets that satisfies (ii)(iv) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (i) holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers (exercise). Any ultrafilter containing a finite set is trivial (exercise). It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter lemma can be added as an extra axiom, it's weaker than the axiom of choice (that says that for any bunch of nonempty sets there is a function that picks an element from any of them, f(X) is an element of X). In mathematics, a filter is a special subset of a partially ordered set. ...
In mathematics, FrÃ©chet filter is an important concept in order theory. ...
In mathematics, the Ultrafilter Lemma states that every filter is a subset of some ultrafilter, i. ...
Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter, exercise) and do our construction, we get the hyperreal numbers as a result. The infinitesimals can be represented by the nonvanishing sequences converging to zero in the usual sense, that is with respect to the Fréchet filter (exercise). In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
If f is a real function of a real variable x then f naturally extends to a hyperreal function of a hyperreal variable by composition:  f({x_{n}}) = {f(x_{n})}
where means "the equivalence class of the sequence relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter. All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. One can prove that any finite (that is, such that  x  < a for some ordinary real a) hyperreal x will be of the form y + d where y is an ordinary (called standard) real and d is an infinitesimal. It is parallel to the proof of the BolzanoWeierstrass lemma that says that one can pick a convergent subsequence form any bounded sequence, done by bisection, the property (i) of the ultrafilters is again crucial. In real analysis, the Bolzanoâ€“Weierstrass theorem is an important theorem characterizing sequentially compact sets. ...
Now one can see that f is continuous means that f(a) − f(x) is infinitely small whenever x − a is, and f is differentiable means that is infinitely small whenever x − a is. Remarkably, if one allows a to be hyperreal, the derivative will be automatically continuous (because, f being differentiable at x, is infinitely small when x − a is, therefore is also infinitely small when x − a is).
Polynomial Ratio Construction Polynomial ratios (see rational function provide an alternative to the ultrapower construction with no requirement for an ultrafilter and only uses familiar concepts of high school algebra. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
Let any relation of real polynomials in a single variable and their ratios hold when and only when they hold for all but a finite number of natural number values of the variable. The proof that firstorder statements about polynomial ratios have the same truth value as corresponding firstorder statements about standard real numbers is much the same as the proof for the ultrapower model, but requires only the use of a cofininite or Fréchet filter not ultrafilters or the Axiom of Choice. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
In mathematics, FrÃ©chet filter is an important concept in order theory. ...
In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
Infinitesimal and infinite numbers A hyperreal number r is called infinitesimal if it is less than every positive real number and greater than every negative real number. Zero is an infinitesimal, but nonzero infinitesimals also exist: take for instance the class of the sequence (1, 1/2, 1/3, 1/4, 1/5, 1/6, ...) (this works because the ultrafilter U contains all index sets whose complement is finite). A hyperreal number x is called finite (or limited by some authors) if there exists a natural number n such that n < x < n; otherwise, x is called infinite (or illimited). Infinite numbers exist; take for instance the class of the sequence (1, 2, 3, 4, 5, ...). A nonzero number x is infinite if and only if 1/x is infinitesimal. The finite elements of F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an orderpreserving homomorphism and hence wellbehaved both algebraically and order theoretically. However, it is orderpreserving but not isotonic, which means implies , but it is not the case that x < y implies In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...
In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds. ...
In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...
The word kernel has several meanings in mathematics, some related to each other and some not. ...
 We have, if both x and y are finite,

 If x is finite and not infinitesimal.


The map st is locally constant, which entails that its derivative is identically zero and that it is continuous with respect to the order topology on the finite hyperreals. In mathematics, a function f from a topological space A to a set B is called locally constant, iff for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
Hyperreal fields Suppose X is a Tychonoff space, also called a T_{3.5} space, and C(X) is the algebra of continuous realvalued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can have the cardinality of the continuum, in which case F is isomorphic as a field to R, but is not order isomorphic to R. In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...
An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory. We give a particular example, commonly used in nonstandard analysis, below. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
Aleph0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...
An ultrapower is an important special case of the ultraproduct construction. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
Compare with: In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ...
The superreal numbers compose a more inclusive category than hyperreal number. ...
In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every nonnegative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F...
References  H. Garth Dales and W. Hugh Woodin, SuperReal Fields, Clarendon Press, 1996.
 L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, 1960.
 Robert Goldblatt, Lectures on the hyperreals : an introduction to nonstandard analysis, Springer, 1998.
 Abraham Robinson, Nonstandard Analysis, Princeton University Press 1966. The classic introduction to nonstandard analysis.
External links 