Transfer principle
Encyclopedia
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, stating that any sentence in the first-order language of fields
true for the complex number
s is also true for any algebraically closed field of characteristic 0.
". Here infinitesimals are expected to have the "same" properties as appreciable numbers. Similar tendencies are found in Cauchy.
In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number
system. Its most common use is in Abraham Robinson
's non-standard 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 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 set
s rather than to all sets.
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 non-standard mathematics is being used.
In terms of model theory, the transfer principle states that a map from a standard model to a non-standard 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).
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 1/n for some positive integer n") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than 1/n for some positive integer n" is false; however
the correct interpretation is "every positive hyperreal is larger than 1/n for some positive hyperinteger
n". In other words, the hyperreals appear to be archimedean to an internal observer living in the non-standard 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
.
where [x] is the integer part function. By a typical application of the transfer principle, every hyperreal x satisfies the inequality
,
where *[.] is the natural extension of the integer part function. If x is infinite, then the hyperinteger
*[x] is infinite, as well.
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:
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.
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
.
, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it, that eliminates the need for an ultrafilter.)
In its most general form, transfer is a bounded elementary embedding between structures.
*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.,
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, 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, stating that any sentence in the first-order language of fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
true for the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s is also true for any algebraically closed field of characteristic 0.
History
An incipient form of a transfer principle was described by Leibniz under the name of "the Law of ContinuityLaw of Continuity
The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite"...
". Here infinitesimals are expected to have the "same" properties as appreciable numbers. Similar tendencies are found in Cauchy.
In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number
Hyperreal number
The system of hyperreal numbers represents a rigorous method of treating the 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 form1 + 1 + \cdots + 1. \, Such a number is...
system. Its most common use is in Abraham Robinson
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
's non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
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.
Transfer principle for the hyperreals
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 hyperreals. 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 set
Internal set
In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the...
s rather than to all sets.
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 non-standard mathematics is being used.
In terms of model theory, the transfer principle states that a map from a standard model to a non-standard 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).
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
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than 1/n for some positive integer n") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than 1/n for some positive integer n" is false; however
the correct interpretation is "every positive hyperreal is larger than 1/n for some positive hyperinteger
Hyperinteger
In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...
n". In other words, the hyperreals appear to be archimedean to an internal observer living in the non-standard 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
Howard Jerome Keisler
H. Jerome Keisler is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.His Ph.D...
book Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of Abraham Robinson's non-standard analysis...
.
Example
Every real x satisfies the inequalitywhere [x] is the integer part function. By a typical application of the transfer principle, every hyperreal x satisfies the inequality
,where *[.] is the natural extension of the integer part function. If x is infinite, then the hyperinteger
Hyperinteger
In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...
*[x] is infinite, as well.
Generalizations of the concept of number
Historically, the concept of numberNumber
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
has been repeatedly generalized. The addition of 0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
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 calculusInfinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...
. As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries. Keisler
Howard Jerome Keisler
H. Jerome Keisler is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.His Ph.D...
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
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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.
Differences between R and *R
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
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, but by virtue of their order they carry an order topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Constructions of the hyperreals
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 ultrafilterUltrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it, that eliminates the need for an ultrafilter.)
In its most general form, transfer is a bounded elementary embedding between structures.
Statement
The ordered fieldOrdered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
*R of nonstandard real numbers properly includes the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
, 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
-
Three examples
- 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
-
-
-
- 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
-
- 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. 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 subset of *R is one that can be placed in internal one-to-one correspondence with {1, ..., n} for some n ∈ *N.
-
