Compactness theorem

The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.

Robinson's principle:[5][6] If a first-order sentence holds in every field of characteristic zero, then there exists a constant ${\displaystyle p}$ such that the sentence holds for every field of characteristic larger than ${\displaystyle p.}$ This can be seen as follows: suppose ${\displaystyle \varphi }$ is a sentence that holds in every field of characteristic zero. Then its negation ${\displaystyle \lnot \varphi ,}$ together with the field axioms and the infinite sequence of sentences

$${\displaystyle 1+1\neq 0,\;\;1+1+1\neq 0,\;\ldots }$$

is not satisfiable (because there is no field of characteristic 0 in which ${\displaystyle \lnot \varphi }$ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset ${\displaystyle A}$ of these sentences that is not satisfiable. ${\displaystyle A}$ must contain ${\displaystyle \lnot \varphi }$ because otherwise it would be satisfiable. Because adding more sentences to ${\displaystyle A}$ does not change unsatisfiability, we can assume that ${\displaystyle A}$ contains the field axioms and, for some ${\displaystyle k,}$ the first ${\displaystyle k}$ sentences of the form ${\displaystyle 1+1+\cdots +1\neq 0.}$ Let ${\displaystyle B}$ contain all the sentences of ${\displaystyle A}$ except ${\displaystyle \lnot \varphi .}$ Then any field with a characteristic greater than ${\displaystyle k}$ is a model of ${\displaystyle B,}$ and ${\displaystyle \lnot \varphi }$ together with ${\displaystyle B}$ is not satisfiable. This means that ${\displaystyle \varphi }$ must hold in every model of ${\displaystyle B,}$ which means precisely that ${\displaystyle \varphi }$ holds in every field of characteristic greater than ${\displaystyle k.}$ This completes the proof.

The Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence ${\displaystyle \varphi }$ in the language of rings is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes ${\displaystyle p}$ for which ${\displaystyle \varphi }$ is true in some algebraically closed field of characteristic ${\displaystyle p,}$ in which case ${\displaystyle \varphi }$ is true in all algebraically closed fields of sufficiently large non-0 characteristic ${\displaystyle p.}$[5] One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials ${\displaystyle \mathbb {C} ^{n}\to \mathbb {C} ^{n}}$ are surjective[5] (indeed, it can even be shown that its inverse will also be a polynomial).[7] In fact, the surjectivity conclusion remains true for any injective polynomial ${\displaystyle F^{n}\to F^{n}}$ where ${\displaystyle F}$ is a finite field or the algebraic closure of such a field.[7]

A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'. To achieve this, let ${\displaystyle T}$ be the initial theory and let ${\displaystyle \kappa }$ be any cardinal number. Add to the language of ${\displaystyle T}$ one constant symbol for every element of ${\displaystyle \kappa .}$ Then add to ${\displaystyle T}$ a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of ${\displaystyle \kappa ^{2}}$ sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of ${\displaystyle T,}$ or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least ${\displaystyle \kappa }$.

A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let ${\displaystyle \Sigma }$ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ${\displaystyle \varepsilon }$ to the language and adjoining to ${\displaystyle \Sigma }$ the axiom ${\displaystyle \varepsilon >0}$ and the axioms ${\displaystyle \varepsilon <{\tfrac {1}{n}}}$ for all positive integers ${\displaystyle n.}$ Clearly, the standard real numbers ${\displaystyle \mathbb {R} }$ are a model for every finite subset of these axioms, because the real numbers satisfy everything in ${\displaystyle \Sigma }$ and, by suitable choice of ${\displaystyle \varepsilon ,}$ can be made to satisfy any finite subset of the axioms about ${\displaystyle \varepsilon .}$ By the compactness theorem, there is a model ${\displaystyle {}^{*}\mathbb {R} }$ that satisfies ${\displaystyle \Sigma }$ and also contains an infinitesimal element ${\displaystyle \varepsilon .}$

A similar argument, this time adjoining the axioms ${\displaystyle \omega >0,\;\omega >1,\ldots ,}$ etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization ${\displaystyle \Sigma }$ of the reals.[8]

It can be shown that the hyperreal numbers ${\displaystyle {}^{*}\mathbb {R} }$ satisfy the transfer principle:[9] a first-order sentence is true of ${\displaystyle \mathbb {R} }$ if and only if it is true of ${\displaystyle {}^{*}\mathbb {R} .}$