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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the system of hyperreal numbers is a way of treating
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

infinite
and
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s R that contains numbers greater than anything of the form :1 + 1 + \cdots + 1 (for any finite number of terms). Such numbers are infinite, and their
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
s are
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. The term "hyper-real" was introduced by
Edwin Hewitt Edwin Hewitt (January 20, 1920, Everett, Washington Everett is the county seat A county seat is an administrative center, seat of government, or capital city of a county or Parish (administrative division), civil parish. The term is used in ...

Edwin Hewitt
in 1948. The hyperreal numbers satisfy the
transfer principle In model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...
, a rigorous version of
Leibniz's
Leibniz's
heuristic
law of continuity The law of continuity is a heuristic principle introduced by Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. (; or ; – 14 November 1716) was a prominent German p ...
. The transfer principle states that true first-order statements about R are also valid in *R. For example, the
commutative law In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of addition, , holds for the hyperreals just as it does for the reals; since R is a
real closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, so is *R. Since \sin()=0 for all
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s ''n'', one also has \sin()=0 for all
hyperintegerIn nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus ...
s ''H''. The transfer principle for
ultrapower The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ...
s is a consequence of
Łoś' theorem The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ...
of 1955. Concerns about the
soundness In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

soundness
of arguments involving infinitesimals date back to ancient Greek mathematics, with
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
replacing such proofs with ones using other techniques such as the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
. In the 1960s,
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
is called
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
. One immediate application is the definition of the basic concepts of analysis such as the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
and
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

integral
in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of ''f''(''x'') becomes f'(x) = \operatorname\left( \frac \right) for an infinitesimal \Delta x, where st(·) denotes the
standard part function In nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus ...

standard part function
, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable
infinite sum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.


The transfer principle

The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number ''x'', ''x'' + 0 = ''x''" still applies. The same is true for quantification over several numbers, e.g., "for any numbers ''x'' and ''y'', ''xy'' = ''yx''." This ability to carry over statements from the reals to the hyperreals is called the
transfer principle In model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...
. However, statements of the form "for any ''set'' of numbers ''S'' ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
. The transfer principle, however, does not mean that R and *R have identical behavior. For instance, in *R there exists an element ''ω'' such that : 1<\omega, \quad 1+1<\omega, \quad 1+1+1<\omega, \quad 1+1+1+1<\omega, \ldots. but there is no such number in R. (In other words, *R is not
Archimedean Archimedean means of or pertaining to or named in honor of the Greece, Greek mathematics, mathematician Archimedes and may refer to: In mathematics: *Absolute value (algebra), Archimedean absolute value *Archimedean circle *Archimedean constant *Ar ...

Archimedean
.) This is possible because the nonexistence of ''ω'' cannot be expressed as a first-order statement.


Use in analysis


Calculus with algebraic functions

Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like ''dx'', and as the symbol ∞, used, for example, in limits of integration of
improper integrals In mathematical analysis, an improper integral is the limit (mathematics), limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both end ...
. As an example of the transfer principle, the statement that for any nonzero number ''x'', ''2x'' ≠ ''x'', is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals. Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that division by zero is undefined. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite. For any finite hyperreal number ''x'', its standard part, st(''x''), is defined as the unique real number that differs from it only infinitesimally. The derivative of a function ''y''(''x'') is defined not as ''dy/dx'' but as the standard part of the corresponding difference quotient. For example, to find the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
''f′''(''x'') of the
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
''f''(''x'') = ''x''2, let ''dx'' be a non-zero infinitesimal. Then, : The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity.
Dual number In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
s are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the ''dx''2 term. In the hyperreal system, ''dx''2 ≠ 0, since ''dx'' is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity ''dx''2 is infinitesimally small compared to ''dx''; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.


Integration

One way of defining a definite integral in the hyperreal system is as the standard part of an infinite sum on a hyperfinite lattice defined as , where ''dx'' is infinitesimal, ''n'' is an infinite
hypernaturalIn nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus ...
, and the lower and upper bounds of integration are ''a'' and ''b'' = ''a'' + ''n'' ''dx.''


Properties

The hyperreals *R form an
ordered field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, but by virtue of their order they carry an
order topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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
Vladimir Kanovei Vladimir G. Kanovei (born 1951) is a Russian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ...
and
Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical str ...

Saharon Shelah
shows that there is a definable, countably saturated (meaning ω-saturated, but not, of course, countable) elementary extension of the reals, which therefore has a good claim to the title of ''the'' hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the
continuum hypothesis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The condition of being a hyperreal field is a stronger one than that of being a
real closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.


Development

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 set-theoretic object called an
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (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 pr ...
, but the ultrafilter itself cannot be explicitly constructed.


From Leibniz to Robinson

When
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

Newton
and (more explicitly)
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...
introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as
Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithm ...

Euler
and
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

Cauchy
. Nonetheless these concepts were from the beginning seen as suspect, notably by
George Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne The Bishop of Cloyne is an episcopal title that takes its name after the small town of Cloyne in County Cork, Republic of Ireland Irela ...

George Berkeley
. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where ''dx'' is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When in the 1800s
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
was put on a firm footing through the development of the (ε, δ)-definition of limit by
Bolzano Bolzano ( or ; german: Bozen (formerly ), ; bar, Bozn; lld, Balsan or ) is the capital city A capital or capital city is the municipality holding primary status in a Department (country subdivision), department, country, Constituent state, ...

Bolzano
, Cauchy,
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Weierstrass
, and others, infinitesimals were largely abandoned, though research in non-Archimedean fields continued (Ehrlich 2006). However, in the 1960s
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
.. The classic introduction to nonstandard analysis. Robinson developed his theory nonconstructively, using
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
; however it is possible to proceed using only
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
and
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers ''per se'', aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by Hewitt (1948) by purely algebraic techniques, using an ultrapower construction.


The ultrapower construction

We are going to construct a hyperreal field via
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
s of reals. In fact we can add and multiply sequences componentwise; for example: : (a_0, a_1, a_2, \ldots) + (b_0, b_1, b_2, \ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, \ldots) and analogously for multiplication. This turns the set of such sequences into a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
, which is in fact a real
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
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 self-consistent 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+\epsilon, where \epsilon is a certain infinitesimal number. Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion: : (a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff (a_0 \leq b_0) \wedge (a_1 \leq b_1) \wedge (a_2 \leq b_2) \ldots 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 In the mathematical field of order theory, an ultrafilter on a given partially ordered set (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 pr ...
''U'' on the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s; these can be characterized as ultrafilters that do not contain any finite sets. (The good news is that
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
guarantees the existence of many such ''U''; 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 is in ''U''. This is a
total preorder The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy. In mathematics Mathematics (from Ancient Greek, Gre ...
and it turns into a
total order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
I in the commutative ring A (namely, the set of the sequences that vanish in some element of ''U''), and then to define *R as A/I; as the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
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. The maximality of I follows from the possibility of, given a sequence ''a'', constructing a sequence ''b'' inverting the non-null elements of ''a'' and not altering its null entries. If the set on which ''a'' vanishes is not in ''U'', the product ''ab'' is identified with the number 1, and any ideal containing 1 must be ''A''. In the resulting field, these ''a'' and ''b'' are inverses. The field A/''U'' is an
ultrapower The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ...
of R. Since this field contains R it has
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
at least that of the
continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models explain variation as involving gradual quantitative transitions without abrupt changes or discontinuities. In contrast, categorical theories or models explain variatio ...
. Since A has cardinality :(2^)^ = 2^ =2^, it is also no larger than 2^, and hence has the same cardinality as R. 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
; in
ZFC
ZFC
with the continuum hypothesis we can prove this field is unique up to
order isomorphismIn the mathematical field of order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (g ...
, and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields that are both countably indexed ultrapowers of the reals. For more information about this method of construction, see
ultraproduct The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ( ...
.


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. Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain 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 necessarily divide by a non-zero element. 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 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 a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field, which is called a hyperreal
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal. This construction is parallel to the construction of the reals from the rationals given by
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer Prayer is an invocation An invocation (from the Latin verb ''invocare'' "to call on, invoke, to give") may take the form of: * Supplication, prayer ...
. He started with the ring of the
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s 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, consider the zero sets of our sequences, that is, the z(a)=\, 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: #One of the sequences that vanish on two complementary sets should be declared zero #If a is declared zero, ab should be declared zero too, no matter what b is. #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
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
s ''X'' of N and to declare that a=0 if and only if z(a) belongs to ''U''. From the above conditions one can see that: #From two complementary sets one belongs to ''U'' #Any set having a subset that belongs to ''U'', also belongs to ''U''. #An intersection of any two sets belonging to ''U'' belongs to ''U''. #Finally, we do not want the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
to belong to ''U'' because then everything would belong to ''U'', as every set has the empty set as a subset. Any family of sets that satisfies (2–4) is called a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
(an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (1) also holds, U is called an
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (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 pr ...
(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. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice. Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result. 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(\)=\ where \ means "the equivalence class of the sequence \dots 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. It turns out 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 can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.


Properties of infinitesimal and infinite numbers

The finite elements F of *R form a
local ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
, and in fact a
valuation ringIn abstract algebra, a valuation ring is an integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
, 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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
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 In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such ...

standard part
of ''x'', conceptually the same as ''x'' ''to the nearest real number''. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e. x \le y implies \operatorname(x) \le \operatorname(y), but x < y does not imply \operatorname(x) < \operatorname(y). * We have, if both ''x'' and ''y'' are finite, :: \operatorname(x + y) = \operatorname(x) + \operatorname(y) :: \operatorname(x y) = \operatorname(x) \operatorname(y) * If ''x'' is finite and not infinitesimal. :: \operatorname(1/x) = 1 / \operatorname(x) * ''x'' is real if and only if :: \operatorname(x) = x The map st is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
with respect to the order topology on the finite hyperreals; in fact it is
locally constant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.


Hyperreal fields

Suppose ''X'' is a
Tychonoff space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical stru ...
, also called a T3.5 space, and C(''X'') is the algebra of continuous real-valued functions on ''X''. Suppose ''M'' is a
maximal ideal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 (terminology due to
Hewitt
Hewitt
(1948)) and ''F'' a hyperreal field. Note that no assumption is being made that the cardinality of ''F'' is greater than R; it can in fact have the same cardinality. An important special case is where the topology on ''X'' is the
discrete topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
; in this case ''X'' can be identified with a
cardinal number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
κ and C(''X'') with the real algebra Rκ of functions from κ to R. The hyperreal fields we obtain in this case are called
ultrapower The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ...
s of R and are identical to the ultrapowers constructed via free
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (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 pr ...
s in model theory.


See also

* * * * * * * - Surreal numbers are a much larger class of numbers, that contains the hyperreals as well as other classes of non-real numbers.


References


Further reading

* * Hatcher, William S. (1982) "Calculus is Algebra",
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are co ...
89: 362–370. * Hewitt, Edwin (1948
Rings of real-valued continuous functions
I. Trans. Amer. Math. Soc. 64, 45—99. * *Keisler, H. Jerome (1994) The hyperreal line. Real numbers, generalizations of the reals, and theories of continua, 207—237, Synthese Lib., 242, Kluwer Acad. Publ., Dordrecht. *


External links

* Crowell,
Calculus
'. A text using infinitesimals. * Hermoso,

'. A gentle introduction. * Keisler,

'. Includes an axiomatic treatment of the hyperreals, and is freely available under a Creative Commons license * Stroyan, ''A Brief Introduction to Infinitesimal Calculu
Lecture 1Lecture 2Lecture 3
{{DEFAULTSORT:Hyperreal Number Mathematical analysis Nonstandard analysis Field (mathematics) Real closed field Infinity Mathematics of infinitesimals