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Inmodel theory, atransfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples wasthe Lefschetz principle, which states that any sentence in thefirst-order language offields that is true for thecomplex numbers is also true for anyalgebraically closed field ofcharacteristic 0.

An incipient form of a transfer principle was described byLeibniz under the name of "theLaw of Continuity".^[1] Hereinfinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of theprinciple of permanence. Similar tendencies are found inCauchy, who used infinitesimals to define both thecontinuity of functions (inCours d'Analyse) and a form of theDirac delta function.^[1]: 903

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

The transfer principle concerns the logical relation between the properties of the real numbersR, and the properties of a larger field denoted *R called thehyperreal 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 overR in a suitable language ofmathematical 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 tointernal sets rather than to all sets. AsRobinson put it,the sentences of [the theory] are interpreted in *R inHenkin's sense.^[2]

The theorem to the effect that each proposition valid overR, 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 anelementary embedding (an embedding preserving thetruth values of all statements in a language), or sometimes abounded elementary embedding (similar, but only for statements withbounded 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-Archimedeanordered field and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than${\displaystyle 1/n}$ for some positive integer${\displaystyle n}$") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than${\displaystyle 1/n}$ for some positive integer${\displaystyle n}$" is false; however the correct interpretation is "every positive hyperreal is larger than${\displaystyle 1/n}$ for some positivehyperinteger${\displaystyle n}$". In other words, the hyperreals appear to be Archimedean to an internal observer living in the nonstandard universe, but appearto be non-Archimedean to an external observer outside the universe.

A freshman-level accessible formulation of the transfer principle isKeisler's bookElementary Calculus: An Infinitesimal Approach.

Every real${\displaystyle x}$ satisfies the inequality$${\displaystyle x\ge \lfloor x\rfloor ,}$$where${\displaystyle \lfloor \cdot \rfloor }$ is theinteger part function. By a typical application of the transfer principle, every hyperreal${\displaystyle x}$ satisfies the inequality$${\displaystyle x\ge {}^{\ast }\lfloor x\rfloor ,}$$where${\displaystyle {}^{\ast }\lfloor \cdot \rfloor }$ is the natural extension of the integer part function. If${\displaystyle x}$ is infinite, then thehyperinteger${\displaystyle {}^{\ast }\lfloor x\rfloor }$ is infinite, as well.

Historically, the concept ofnumber has been repeatedly generalized. The addition of0 to the natural numbers${\displaystyle \mathbb{N}}$ was a major intellectual accomplishment in its time. The addition of negative integers to form${\displaystyle \mathbb{Z}}$ already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers${\displaystyle \mathbb{Q}}$, is more familiar to a layperson than their completion${\displaystyle \mathbb{R}}$, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by${\displaystyle \mathbb{Q}}$. 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 theinfinitesimal 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."

Theself-consistent development of the hyperreals turned out to be possible if every truefirst-order logic statement that uses basic arithmetic (thenatural 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 thatR and *R have identical behavior. For instance, in *R there exists an elementω such that

but there is no such number inR. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number likeω is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.

The hyperreals *R form anordered field containing the realsR as a subfield. Unlike the reals, the hyperreals do not form a standardmetric space, but by virtue of their order they carry an ordertopology.

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 anultrafilter, 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 boundedelementary embedding between structures.

Theordered field^*R ofnonstandard real numbers properly includes thereal fieldR. Like all ordered fields that properly includeR, this field isnon-Archimedean. It means that some membersx ≠ 0 of^*R areinfinitesimal, i.e.,

for every finitecardinal number n.

The only infinitesimal inR is 0. Some other members of^*R, the reciprocalsy of the nonzero infinitesimals, are infinite, i.e.,

for every finitecardinal number n.

The underlying set of the field^*R is the image ofR under a mappingA ↦ ^*A from subsetsA ofR to subsets of^*R. In every case

${\displaystyle A\subseteq {}^{\ast }A,}$

with equality if and only ifA is finite. Sets of the form^*A for some${\displaystyle A\subseteq \mathbb{R}}$ are calledstandard subsets of^*R. The standard sets belong to a much larger class of subsets of^*R calledinternal sets. Similarly each function

${\displaystyle f:A\to \mathbb{R}}$

extends to a function

${\displaystyle {}^{\ast }f:{}^{\ast }A\to {}^{\ast }\mathbb{R};}$

these are calledstandard functions, and belong to the much larger class ofinternal functions. Sets and functions that are not internal areexternal.

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

Thetransfer 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 asand,or,not,if...then..., and the quantifiers

${\displaystyle \mathrm{\forall }x\in \mathbb{R}\text{ and }\mathrm{\exists }x\in \mathbb{R}.}$

For example, one such proposition is

${\displaystyle \mathrm{\forall }x\in \mathbb{R}\mathrm{\exists }y\in \mathbb{R}x+y=0.}$

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

${\displaystyle \mathrm{\forall }x\in {}^{\ast }\mathbb{R}}$

replaces

${\displaystyle \mathrm{\forall }x\in \mathbb{R},}$

and similarly for${\displaystyle \mathrm{\exists }}$.

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

• The set

${\displaystyle [0,1{]}^{\ast }=\{x\in \mathbb{R}:0\le x\le 1{\}}^{\ast }}$

must be

${\displaystyle \{x\in {}^{\ast }\mathbb{R}:0\le x\le 1\},}$

including not only members ofR 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

${\displaystyle \mathrm{\forall }x\in \mathbb{R}(x\in [0,1]\text{ if and only if }0\le x\le 1)}$

is true inR, 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 ofN inR is simple enough for the transfer principle to apply to it) and must containn + 1 if it containsn, but must not contain anything betweenn andn + 1. Members of

${\displaystyle {}^{\ast }\mathbb{N}\setminus \mathbb{N}}$

are "infinite integers".)

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

${\displaystyle \mathrm{\forall }A\subseteq \mathbb{R}\dots \text{ or }\mathrm{\exists }A\subseteq \mathbb{R}\dots .}$

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

${\displaystyle [\mathrm{\forall }\text{ internal }A\subseteq {}^{\ast }\mathbb{R}\dots ]}$

and

${\displaystyle [\mathrm{\exists }\text{ internal }A\subseteq {}^{\ast }\mathbb{R}\dots ].}$

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

• Every nonemptyinternal 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 nonemptyinternal subset of^*N has a smallest member. Consequently the set

${\displaystyle {}^{\ast }\mathbb{N}\setminus \mathbb{N}}$

of all infinite integers is external.

• Ifn 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:

${\displaystyle \mathrm{\forall }n\in \mathbb{N}\mathrm{\exists }A\subseteq \mathbb{N}\mathrm{\forall }x\in \mathbb{N}[x\in A\text{ iff }x\le n].}$

Consequently

${\displaystyle \mathrm{\forall }n\in {}^{\ast }\mathbb{N}\mathrm{\exists }\text{ internal }A\subseteq {}^{\ast }\mathbb{N}\mathrm{\forall }x\in {}^{\ast }\mathbb{N}[x\in A\text{ iff }x\le n].}$

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

${\displaystyle \mathrm{\forall }f:A\to \mathbb{R}\dots }$

with

${\displaystyle \mathrm{\forall }\text{ internal }f:{}^{\ast }A\to {}^{\ast }\mathbb{R}\dots }$

when applying the transfer principle, and similarly with${\displaystyle \mathrm{\exists }}$ in place of${\displaystyle \mathrm{\forall }}$.

For example: Ifn is an infinite integer, then the complement of the image of any internalone-to-one functionƒ 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 (pronouncedstar-finite) subset of^*R is one that can be placed ininternal one-to-one correspondence with {1, ..., n} for somen ∈ ^*N.

• Elementary Calculus: An Infinitesimal Approach
• Principle of Permanence
• Generality of algebra

• 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,S2CID 15104702
• 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

• Mathematical principles
• Model theory
• Nonstandard analysis

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