Transfer Principle

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Concept in Model Theory

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Overview

In the intricate world of model theory, a transfer principle is essentially a rule, a guiding axiom that allows us to extrapolate truths from one structure to another. It’s the idea that if a statement holds true for a particular mathematical system—a “structure”—then it must also hold true for a different, often related, structure. Think of it as a bridge, allowing us to cross from a known land of mathematical facts to an unknown one, with a guarantee that the fundamental laws of logic remain intact.

One of the earliest and most significant illustrations of this principle is the Lefschetz principle. This principle, in essence, declares that any statement formulated in the precise language of first-order logic concerning fields that is demonstrably true for the complex numbers will also be true for any algebraically closed field operating within characteristic 0. It’s a powerful assertion, suggesting a deep underlying unity in the algebraic behavior of certain number systems.

History

The seeds of the transfer principle can be traced back to the philosophical musings of Gottfried Wilhelm Leibniz. He spoke of a "Law of Continuity," a concept that hinted at the idea that infinitesimally small quantities should behave in the same manner as their larger, more tangible counterparts. This wasn't a rigorous mathematical statement, of course, but rather a philosophical intuition that later formalizations would capture. It was a precursor to the idea that the properties of the finite could be reliably extended to the infinite, or vice versa, through a carefully defined logical framework.

This underlying principle can also be seen as a more formal, rigorous articulation of the principle of permanence. This principle, in its various forms, suggests that mathematical structures and their properties should persist across extensions or generalizations. Thinkers like Augustin-Louis Cauchy also toyed with similar ideas. He utilized infinitesimals in his foundational work, employing them to define both the continuity of functions in his Cours d'Analyse and a concept that foreshadowed the Dirac delta function. These were attempts to grapple with the infinitely small and the infinitely large, to build a calculus that could handle change and limits with a degree of intuitive rigor, even if the formal underpinnings were still being developed.

The formalization of these ideas took a significant leap forward in 1955. Jerzy Łoś provided a crucial proof of the transfer principle for any system of hyperreal numbers. This work laid the groundwork for Abraham Robinson's development of nonstandard analysis. In Robinson's framework, the transfer principle is the cornerstone, asserting that any statement expressible in a specific formal language that holds true for the real numbers will also hold true for the hyperreal numbers. It’s a testament to the power of formal logic to bridge the gap between the standard and the nonstandard realms of mathematics.

Transfer Principle for the Hyperreals

The transfer principle, when applied to the realm of nonstandard analysis, establishes a profound logical connection between the familiar world of the real numbers, denoted as R , and its extended counterpart, the hyperreal numbers, typically symbolized as * R . This hyperreal number system is meticulously constructed to include not only the standard real numbers but also quantities that are infinitesimally small (infinitesimals) and infinitely large. This provides a mathematically sound foundation for the intuitive but historically problematic notions of infinitesimals that were first explored by Leibniz.

The core idea behind the transfer principle is elegantly simple, yet remarkably powerful: to translate statements about the real numbers, expressed in a carefully chosen formal language, into statements about the hyperreal numbers. The magic lies in the fact that the logical structure of the statements remains the same, and crucially, the truth value of these statements is preserved. Abraham Robinson, a pioneer in this field, noted that this translation is possible because, at the level of set theory, the propositions in this formal language are interpreted as applying only to internal sets, rather than to all possible sets. This restriction to internal sets is what allows the principle to function without succumbing to paradoxes. As Robinson himself articulated, the sentences of the language are interpreted in * R in the sense developed by Leon Henkin.

This fundamental theorem, which guarantees that each proposition valid over R is also valid over * R , is precisely what we call the transfer principle. It’s not a single, monolithic statement, but rather a principle that manifests in various forms depending on the specific model of nonstandard mathematics being employed.

From a model theory perspective, the transfer principle can be understood as stating that a map from a standard model (like R ) to a nonstandard model (like * R ) is an elementary embedding. This means the map preserves the truth values of all statements expressible in the formal language. In some contexts, it might be a bounded elementary embedding, which preserves truth values only for statements involving bounded quantifiers. This nuance is critical; not every statement about R translates directly to * R without careful consideration of the quantifiers involved.

It's crucial to understand that the transfer principle, while powerful, doesn't imply that R and * R are identical. In fact, they possess fundamentally different properties. The apparent contradictions that can arise are often due to a misapplication or misunderstanding of the principle. For instance, the real numbers R form an Archimedean ordered field, meaning that for any two positive numbers, however small one is, you can always find a sufficiently large integer multiple of it that exceeds the other. The hyperreal numbers * R , on the other hand, form a non-Archimedean ordered field. This means there exist numbers in * R that are smaller than any positive real number (infinitesimals) and numbers larger than any real number (infinitely large).

The statement "every positive real is larger than 1/n for some positive integer n" is true for R . If you naively apply the transfer principle, you might conclude that "every positive hyperreal is larger than 1/n for some positive integer n." This statement, however, is false in * R . The correct interpretation, respecting the non-Archimedean nature of * R , is that "every positive hyperreal is larger than 1/n for some positive hyperinteger n." This subtle distinction highlights how the transfer principle operates within the internal structure of the nonstandard model. To an "internal observer" living within the hyperreal universe, the field * R might appear Archimedean. However, to an "external observer" outside this universe, the non-Archimedean nature becomes apparent.

A more accessible introduction to the transfer principle can be found in Howard Jerome Keisler's textbook, Elementary Calculus: An Infinitesimal Approach. Keisler masterfully uses the transfer principle to build a calculus based on infinitesimals, making sophisticated concepts available to a broader audience.

Example

Let's illustrate this with a concrete example. Consider the inequality:

$x \geq \lfloor x \rfloor$

This statement, where $\lfloor x \rfloor$ represents the integer part or floor function, is true for every real number $x$. By the transfer principle, this same inequality must also hold for every hyperreal number $x$:

$x \geq {}^{*}\!\lfloor x \rfloor$

Here, ${}^{*}\!\lfloor x \rfloor$ is the natural extension of the floor function to the hyperreal numbers. If $x$ is an infinitely large hyperreal number, then ${}^{*}\!\lfloor x \rfloor$ will also be an infinitely large hyperinteger. This demonstrates how properties of real numbers, when expressible in the appropriate language, are preserved in the hyperreal system.

Generalizations of the Concept of Number

The evolution of the concept of number is a fascinating journey, marked by successive generalizations that have expanded the scope and power of mathematics. The inclusion of 0 into the set of natural numbers, $\mathbb{N}$, was not merely an arithmetic convenience; it was a profound conceptual leap. Subsequently, the introduction of negative integers to form the set of integers, $\mathbb{Z}$, marked a significant departure from the realm of direct, tangible experience into the more abstract domain of mathematical models.

The extension to the rational numbers, $\mathbb{Q}$, is perhaps more familiar to the layperson than their completion, the real numbers, $\mathbb{R}$. This familiarity arises partly because, in terms of practical measurement and computation, the distinction between rational and real numbers can be subtle. The notion of an irrational number, for instance, is fundamentally alien to the most powerful floating-point computer, which operates on finite approximations. The necessity for extending the number system to include irrationals stems not from empirical observation but from the internal logical and structural demands of mathematics itself.

The introduction of infinitesimals into mathematical discourse occurred at a time when the burgeoning field of the infinitesimal calculus demanded them. As mentioned earlier, the rigorous mathematical justification for this conceptual leap was significantly delayed, taking centuries to fully materialize. Howard Jerome Keisler, in his work, aptly captures this historical context:

"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 development of a self-consistent framework for the hyperreals was achieved by postulating that every true first-order logic statement involving basic arithmetic operations (addition, multiplication, comparison) and quantifying only over the real numbers, would retain its truth value when reinterpreted to quantify over hyperreal numbers. This is the essence of the transfer principle. For example, the statement:

$\forall x \in \mathbb{R} \quad \exists y \in \mathbb{R} \quad x < y$

which asserts that for every real number, there exists another real number greater than it, is a fundamental property of the ordered field of real numbers. The transfer principle ensures that this property also holds for the hyperreal numbers:

$\forall x \in {}^{\star}\mathbb{R} \quad \exists y \in {}^{\star}\mathbb{R} \quad x < y$

Similarly, the statement:

$\forall x \in \mathbb{R} \quad x < x + 1$

which expresses that adding 1 to any real number always results in a larger number, is also preserved:

$\forall x \in {}^{\star}\mathbb{R} \quad x < x + 1$

These examples, while illustrative, only scratch the surface. Many formulas in analysis involve quantification over higher-order objects like functions and sets, making the application of the transfer principle more intricate and requiring a deeper understanding of internal and external entities within the nonstandard framework.

Differences Between R and *R

Despite the power of the transfer principle in preserving logical truths, it's crucial to recognize that the real numbers (R) and the hyperreal numbers (*R) are fundamentally different structures. The transfer principle does not imply that *R behaves identically to R in all respects.

A key difference lies in the Archimedean property. While R is Archimedean, *R is not. This means that in *R, there exist elements, such as $\omega$, that are "infinitely large." Such an element $\omega$ would satisfy inequalities like:

$1 < \omega, \quad 1+1 < \omega, \quad 1+1+1 < \omega, \quad 1+1+1+1 < \omega, \ldots$

This is possible because the non-existence of such a number cannot be expressed as a first-order statement in the language for which the transfer principle holds. The reciprocals of these infinitely large numbers are the infinitesimals, quantities smaller than any positive real number but greater than zero.

While *R does not form a standard metric space, its order structure endows it with an order topology. This topological structure, combined with the presence of infinitesimals and infinite numbers, allows for a richer and more nuanced analysis than is possible with the real numbers alone.

Constructions of the Hyperreals

The hyperreal numbers can be approached in two primary ways: axiomatically or through more constructive methods. The axiomatic approach, as championed by figures like Robinson, posits two fundamental axioms:

1. The existence of at least one infinitesimal number (different from zero).
2. The validity of the transfer principle for a specific formal language.

From these axioms, the entire structure of the hyperreals can be derived.

The more constructive approaches, however, aim to build the hyperreal numbers from existing mathematical objects, typically within the framework of set theory. One prominent constructive method involves the use of ultrafilters. An ultrafilter on a set of indices (often the natural numbers) is used to construct *R as an ultraproduct of copies of R. While this method provides a concrete construction, the existence of non-principal ultrafilters (those not generated by a single element) relies on the Axiom of Choice, and they cannot be explicitly constructed in a definable way.

Vladimir Kanovei and Saharon Shelah have made significant contributions by providing a construction of a definable, countably saturated elementary extension of the structure consisting of the real numbers and all finitary relations on them. This work offers a more tangible and accessible way to work with nonstandard models of the reals, even if the underlying set-theoretic machinery remains complex.

In its most general formulation within model theory, the transfer principle is understood as a bounded elementary embedding between structures. This means it's a map that preserves the structure of the models and the truth of statements, particularly those with bounded quantifiers.

Statement

The ordered field *R, comprising the nonstandard real numbers, is a proper extension of the real field R. As is characteristic of any ordered field that properly contains R, *R is non-Archimedean. This non-Archimedean nature implies the existence of non-zero elements $x$ in *R that are infinitesimal. An infinitesimal is defined as an element whose absolute value, when added to itself a finite number of times (say, $n$ times, where $n$ is any finite cardinal number), remains less than 1:

$\underbrace {|x| + \cdots + |x|} _{n \text{ terms}} < 1$

In the standard real numbers R, the only infinitesimal is 0. However, in *R, there are other infinitesimals. Consequently, there exist elements $y$ in *R, which are the reciprocals of the non-zero infinitesimals, that are infinite. These infinite elements are larger in magnitude than any finite sum of 1s:

$\underbrace {1 + \cdots + 1} _{n \text{ terms}} < |y|$

for every finite cardinal number $n$.

The underlying set of the field *R is formed by applying a mapping $A \mapsto {}^{*}\!A$ to subsets $A$ of R. This mapping generates corresponding subsets ${}^{*}\!A$ within *R. In all cases, $A \subseteq {}^{*}\!A$, with equality holding if and only if $A$ is a finite set. Subsets of *R that are of the form ${}^{*}\!A$ for some $A \subseteq \mathbb{R}$ are termed "standard subsets" of *R. These standard subsets are a specific type within a much broader classification of subsets of *R known as "internal sets." Similarly, any function $f: A \rightarrow \mathbb{R}$ defined on a subset $A$ of R has a natural extension to a function ${}^{*}\!f: {}^{*}\!A \rightarrow {}^{*}\!\mathbb{R}$. These extensions are called "standard functions" and belong to the larger class of "internal functions." Sets and functions that are not internal are termed "external."

The significance of these internal and external distinctions becomes apparent when we consider the precise statement of the transfer principle and its implications.

The Transfer Principle:

• Statements involving functions and relations: Consider a proposition that holds true for *R and can be expressed using functions of finitely many variables (e.g., addition $x+y$), relations among finitely many variables (e.g., $x \leq y$), finite logical connectives (AND, OR, NOT, IF...THEN...), and quantifiers ranging over the real numbers ($\forall x \in \mathbb{R}$ and $\exists x \in \mathbb{R}$). For example, the proposition: $\forall x \in \mathbb{R} \quad \exists y \in \mathbb{R} \quad x+y=0$ This proposition is true in R. The transfer principle asserts that it is also true in *R, provided that the quantifiers are reinterpreted to range over hyperreal numbers: $\forall x \in {}^{\star}\mathbb{R} \quad \exists y \in {}^{\star}\mathbb{R} \quad x+y=0$

• Statements involving specific sets: If a proposition, otherwise expressible as described above, also refers to particular subsets $A \subseteq \mathbb{R}$, then the proposition holds true in R if and only if it holds true in *R with each such set $A$ replaced by its corresponding hyperreal extension ${}^{*}\!A$.

  • Example: The interval $[0, 1]$: The set ${}^{*}\![0,1] = \{x \in {}^{\star}\mathbb{R} : 0 \leq x \leq 1\}$ must include not only the real numbers between 0 and 1 (inclusive) but also hyperreal numbers between 0 and 1 that differ from the real numbers by infinitesimal amounts. This follows from the transfer principle applied to the statement: $\forall x \in \mathbb{R} \quad (x \in [0,1] \text{ if and only if } 0 \leq x \leq 1)$
  • Example: The set of natural numbers $\mathbb{N}$: The set ${}^{*}\!\mathbb{N}$ has no upper bound in ${}^{\star}\mathbb{R}$ (because the statement expressing the non-existence of an upper bound for $\mathbb{N}$ in $\mathbb{R}$ is simple enough for the transfer principle to apply). Furthermore, ${}^{*}\!\mathbb{N}$ must contain $n+1$ if it contains $n$, but it must not contain any element strictly between $n$ and $n+1$ for any standard natural number $n$. The elements of ${}^{\star}\mathbb{N} \setminus \mathbb{N}$ are precisely the "infinite integers."

• Statements involving quantifiers over sets: If a proposition, again otherwise expressible simply, involves quantifiers over subsets of R (e.g., $\forall A \subseteq \mathbb{R} \dots$ or $\exists A \subseteq \mathbb{R} \dots$), then the proposition holds true in R if and only if it holds true in *R after the standard changes (quantifiers over ${}^{\star}\mathbb{R}$ and replacement of sets $A$ by ${}^{*}\!A$) and the quantifiers over subsets are replaced by quantifiers over internal subsets of ${}^{\star}\mathbb{R}$ (i.e., $[\forall \text{ internal } A \subseteq {}^{\star}\mathbb{R} \dots]$ and $[\exists \text{ internal } A \subseteq {}^{\star}\mathbb{R} \dots]$). This distinction between internal and external sets is critical for correctly applying the transfer principle to statements involving quantification over sets.

Three Examples

The true power and subtlety of the transfer principle for hyperreals are best understood within the context of internal entities. The principle allows us to deduce properties of internal sets and functions in the nonstandard universe based on their counterparts in the standard universe.

• Completeness of internal sets: Every non-empty internal subset of ${}^{\star}\mathbb{R}$ that is bounded above in ${}^{\star}\mathbb{R}$ possesses a least upper bound (supremum) within ${}^{\star}\mathbb{R}$. This property, a consequence of the transfer principle applied to the completeness of the real numbers, is fundamental. It also implies that the set of all infinitesimals, while non-empty and containing 0, cannot be an internal set. If it were internal, it would have a least upper bound, which would be 0, but this would contradict the existence of non-zero infinitesimals. Thus, the set of infinitesimals must be external.

• Well-ordering of internal hypernatural numbers: The well-ordering principle, which states that every non-empty subset of the natural numbers $\mathbb{N}$ has a least element, is preserved by the transfer principle for internal subsets of ${}^{\star}\mathbb{N}$. This means that every non-empty internal subset of ${}^{\star}\mathbb{N}$ has a smallest member. Consequently, the set of all infinite integers, ${}^{\star}\mathbb{N} \setminus \mathbb{N}$, which is an external set, cannot be guaranteed to have a smallest member.

• Internal sets related to finite ranges: Consider an infinite integer $n$. The set $\{1, 2, \ldots, n\}$ is a finite set of hypernatural numbers. This set, even though its upper bound $n$ is infinite, is an internal set. We can prove this by observing the following statement, which is trivially true for standard natural numbers: $\forall n \in \mathbb{N} \quad \exists A \subseteq \mathbb{N} \quad \forall x \in \mathbb{N} \quad [x \in A \text{ iff } x \leq n]$ Applying the transfer principle, this statement translates to: $\forall n \in {}^{\star}\mathbb{N} \quad \exists \text{ internal } A \subseteq {}^{\star}\mathbb{N} \quad \forall x \in {}^{\star}\mathbb{N} \quad [x \in A \text{ iff } x \leq n]$ This confirms that for any hypernatural number $n$ (finite or infinite), the set $\{x \in {}^{\star}\mathbb{N} : x \leq n\}$ is an internal set.

• Internal functions and set complements: The transfer principle also applies to internal functions. When dealing with quantifiers involving functions, such as $\forall f: A \rightarrow \mathbb{R} \dots$ or $\exists f: A \rightarrow \mathbb{R} \dots$, the transfer principle requires replacing these with quantifiers over internal functions (e.g., $\forall \text{ internal } f: {}^{*}\!A \rightarrow {}^{\star}\mathbb{R} \dots$).

  • Example: Complements of images: Let $n$ be an infinite integer. Consider the set $\{1, 2, \ldots, n\}$, which we've established is internal. Let $f$ be an internal one-to-one function from this set into the larger internal set $\{1, 2, \ldots, n, n+1, n+2, n+3\}$. By the transfer principle, the complement of the image of $f$ within $\{1, 2, \ldots, n, n+1, n+2, n+3\}$ will have exactly three members. While there might be many such functions $f$, and thus many different complements, the property of having exactly three missing elements is preserved due to the transfer principle, provided we consider only internal functions. This example highlights how the transfer principle allows us to reason about the sizes and properties of sets and functions in the nonstandard universe, particularly when dealing with finite or *-finite sets.

This last example motivates a crucial definition: A subset of ${}^{\star}\mathbb{R}$ is called *-finite (pronounced "star-finite") if it can be put into an internal one-to-one correspondence with $\{1, 2, \ldots, n\}$ for some hypernatural number $n \in {}^{\star}\mathbb{N}$. This concept is vital for understanding the finite-like behavior of certain sets within the infinite landscape of the hyperreals.