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In model theory, a **transfer principle** states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0.

An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity".^{ [1] } Here infinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the principle of permanence. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function.^{ [1] }^{: 903 }

In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system. Its most common use is in Abraham Robinson's nonstandard analysis of the hyperreal numbers, where the transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.

The transfer principle concerns the logical relation between the properties of the real numbers **R**, and the properties of a larger field denoted ***R** called the hyperreal numbers. The field ***R** includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by Leibniz.

The idea is to express analysis over **R** in a suitable language of mathematical logic, and then point out that this language applies equally well to ***R**. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets. As Robinson put it, *the sentences of [the theory] are interpreted in * R in Henkin's sense.*

The theorem to the effect that each proposition valid over **R**, is also valid over ***R**, is called the transfer principle.

There are several different versions of the transfer principle, depending on what model of nonstandard mathematics is being used. In terms of model theory, the transfer principle states that a map from a standard model to a nonstandard model is an elementary embedding (an embedding preserving the truth values of all statements in a language), or sometimes a *bounded* elementary embedding (similar, but only for statements with bounded quantifiers).^{[ clarification needed ]}

The transfer principle appears to lead to contradictions if it is not handled correctly. For example, since the hyperreal numbers form a non-Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than for some positive integer ") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than for some positive integer " is false; however the correct interpretation is "every positive hyperreal is larger than for some positive hyperinteger ". In other words, the hyperreals appear to be Archimedean to an internal observer living in the nonstandard universe, but appear to be non-Archimedean to an external observer outside the universe.

A freshman-level accessible formulation of the transfer principle is Keisler's book * Elementary Calculus: An Infinitesimal Approach *.

Every real satisfies the inequality

where is the integer part function. By a typical application of the transfer principle, every hyperreal satisfies the inequality

where is the natural extension of the integer part function. If is infinite, then the hyperinteger is infinite, as well.

Historically, the concept of number has been repeatedly generalized. The addition of 0 to the natural numbers was a major intellectual accomplishment in its time. The addition of negative integers to form already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers , is more familiar to a layperson than their completion , partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by . Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer. The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence. The infinitesimals entered mathematical discourse at a time when such a notion was required by mathematical developments at the time, namely the emergence of what became known as the infinitesimal calculus. As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries. Keisler wrote:

- "In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."

The self-consistent development of the hyperreals turned out to be possible if every true first-order 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:

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.

The transfer principle however doesn't mean that **R** and ***R** have identical behavior. For instance, in ***R** there exists an element *ω* 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 *ω* 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.

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, but the ultrafilter itself cannot be explicitly constructed. Vladimir Kanovei and Shelah^{ [3] } give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it.

In its most general form, transfer is a bounded elementary embedding between structures.

The ordered field ^{*}**R** of nonstandard real numbers properly includes the real field **R**. Like all ordered fields that properly include **R**, this field is non-Archimedean. It means that some members *x* ≠ 0 of ^{*}**R** are infinitesimal, i.e.,

The only infinitesimal in *R* is 0. Some other members of ^{*}**R**, the reciprocals *y* of the nonzero infinitesimals, are infinite, i.e.,

The underlying set of the field ^{*}**R** is the image of **R** under a mapping *A* ↦ ^{*}*A* from subsets *A* of **R** to subsets of ^{*}**R**. In every case

with equality if and only if *A* is finite. Sets of the form ^{*}*A* for some are called **standard** subsets of ^{*}**R**. The standard sets belong to a much larger class of subsets of ^{*}**R** called **internal** sets. Similarly each function

extends to a function

these are called **standard functions**, and belong to the much larger class of **internal functions**. Sets and functions that are not internal are **external**.

The importance of these concepts stems from their role in the following proposition and is illustrated by the examples that follow it.

The **transfer principle:**

- Suppose a proposition that is true of
^{*}**R**can be expressed via functions of finitely many variables (e.g. (*x*,*y*) ↦*x*+*y*), relations among finitely many variables (e.g.*x*≤*y*), finitary logical connectives such as**and**,**or**,**not**,**if...then...**, and the quantifiers

- For example, one such proposition is

- Such a proposition is true in
**R**if and only if it is true in^{*}**R**when the quantifier

- replaces

- and similarly for .

- Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets . Such a proposition is true in
**R**if and only if it is true in^{*}**R**with each such "*A*" replaced by the corresponding^{*}*A*. Here are two examples:

- The set

- must be
- including not only members of
**R**between 0 and 1 inclusive, but also members of^{*}**R**between 0 and 1 that differ from those by infinitesimals. To see this, observe that the sentence - is true in
**R**, and apply the transfer principle.

- The set
^{*}**N**must have no upper bound in^{*}**R**(since the sentence expressing the non-existence of an upper bound of**N**in**R**is simple enough for the transfer principle to apply to it) and must contain*n*+ 1 if it contains*n*, but must not contain anything between*n*and*n*+ 1. Members of

- are "infinite integers".)

- Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier

- Such a proposition is true in
**R**if and only if it is true in^{*}**R**after the changes specified above and the replacement of the quantifiers with - and

The appropriate setting for the hyperreal transfer principle is the world of *internal* entities. Thus, the well-ordering property of the natural numbers by transfer yields the fact that every internal subset of has a least element. In this section internal sets are discussed in more detail.

- Every nonempty
*internal*subset of^{*}**R**that has an upper bound in^{*}**R**has a least upper bound in^{*}**R**. Consequently the set of all infinitesimals is external.- The well-ordering principle implies every nonempty
*internal*subset of^{*}**N**has a smallest member. Consequently the set

- The well-ordering principle implies every nonempty

- of all infinite integers is external.

- If
*n*is an infinite integer, then the set {1, ...,*n*} (which is not standard) must be internal. To prove this, first observe that the following is trivially true:

- Consequently

- As with internal sets, so with internal functions: Replace

- with
- when applying the transfer principle, and similarly with in place of .
- For example: If
*n*is an infinite integer, then the complement of the image of any internal one-to-one function*ƒ*from the infinite set {1, ...,*n*} into {1, ...,*n*,*n*+ 1,*n*+ 2,*n*+ 3} has exactly three members by the transfer principle. Because of the infiniteness of the domain, the complements of the images of one-to-one functions from the former set to the latter come in many sizes, but most of these functions are external.

- This last example motivates an important definition: A
***-finite**(pronounced**star-finite**) subset of^{*}**R**is one that can be placed in*internal*one-to-one correspondence with {1, ...,*n*} for some*n*∈^{*}**N**.

- 1 2 Keisler, H. Jerome. "Elementary Calculus: An Infinitesimal Approach". p. 902.
- ↑ Robinson, A. The metaphysics of the calculus, in Problems in the Philosophy of Mathematics, ed. Lakatos (Amsterdam: North Holland), pp. 28–46, 1967. Reprinted in the 1979 Collected Works. Page 29.
- ↑ Kanovei, Vladimir; Shelah, Saharon (2004), "A definable nonstandard model of the reals" (PDF),
*Journal of Symbolic Logic*,**69**: 159–164, arXiv: math/0311165 , doi:10.2178/jsl/1080938834

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 epsilon–delta procedures rather than infinitesimals. **Nonstandard analysis** instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.

In mathematics, a real function of real numbers is said to be **uniformly continuous** if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .

In mathematics, the **infimum** of a subset of a partially ordered set is a greatest element in that is less than or equal to each element of if such an element exists. Consequently, the term *greatest lower bound* is also commonly used.

In mathematics, the system of **hyperreal numbers** is a way of treating infinite and infinitesimal quantities. The hyperreals, or **nonstandard reals**, ***R**, are an extension of the real numbers **R** that contains numbers greater than anything of the form

In mathematical logic, the **compactness theorem** states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

In mathematics, an **infinitesimal number** is a quantity that is closer to zero than any standard real number, but that is not zero. The word *infinitesimal* comes from a 17th-century Modern Latin coinage *infinitesimus*, which originally referred to the "infinity-th" item in a sequence.

In nonstandard analysis, a branch of mathematics, **overspill** is a widely used proof technique. It is based on the fact that the set of standard natural numbers **N** is not an internal subset of the internal set ***N** of hypernatural numbers.

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.

In mathematics, **differential** refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.

In mathematics, specifically order theory, a **well-quasi-ordering** or **wqo** is a quasi-ordering such that any infinite sequence of elements from contains an increasing pair with

**Internal set theory** (**IST**) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional ZFC axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements.

In abstract algebra, a **valuation ring** is an integral domain *D* such that for every element *x* of its field of fractions *F*, at least one of *x* or *x*^{−1} belongs to *D*.

In mathematics, a subset of a preordered set is said to be **cofinal** or **frequent** in if for every it is possible to find an element in that is "larger than ".

In mathematical logic, in particular in model theory and nonstandard analysis, an **internal set** is a set that is a member of a model.

In model theory and related areas of mathematics, a **type** is an object that describes how a element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language *L* with free variables *x*_{1}, *x*_{2},…, *x*_{n} that are true of a sequence of elements of an *L*-structure . Depending on the context, types can be **complete** or **partial** and they may use a fixed set of constants, *A*, from the structure . The question of which types represent actual elements of leads to the ideas of saturated models and **omitting types**.

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 hyperreal , the unique real infinitely close to it, i.e. is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.

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 (1, 2, 3, ...) in the ultrapower construction of the hyperreals.

In mathematics, a **non-Archimedean ordered field** is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable order.

In nonstandard analysis, a discipline within classical mathematics, **microcontinuity** of an internal function *f* at a point *a* is defined as follows:

- Chang, Chen Chung; Keisler, H. Jerome (1990) [1973],
*Model Theory*, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3 - Hardy, Michael: "Scaled Boolean algebras".
*Adv. in Appl. Math.*29 (2002), no. 2, 243–292. - Kanovei, Vladimir; Shelah, Saharon (2004), "A definable nonstandard model of the reals",
*Journal of Symbolic Logic*,**69**: 159–164, arXiv: math/0311165 , doi:10.2178/jsl/1080938834 - Keisler, H. Jerome (2000). "Elementary Calculus: An Infinitesimal Approach".
- Kuhlmann, F.-V. (2001) [1994], "Transfer principle",
*Encyclopedia of Mathematics*, EMS Press - Łoś, Jerzy (1955) Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres. Mathematical interpretation of formal systems, pp. 98–113. North-Holland Publishing Co., Amsterdam.
- Robinson, Abraham (1996),
*Non-standard analysis*, Princeton University Press, ISBN 978-0-691-04490-3, MR 0205854

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