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mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
has an infinite model, then for every infinite
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
''κ'' it has a model of size ''κ'', and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in
Lindström's theorem In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the '' strongest logic'' (satisfying certain conditions, e.g. closure under classical negation) h ...
to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic.


Theorem

In its general form, the Löwenheim–Skolem Theorem states that for every
signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
''σ'', every infinite ''σ''-
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
''M'' and every infinite cardinal number , there is a ''σ''-structure ''N'' such that and such that * if then ''N'' is an elementary substructure of ''M''; * if then ''N'' is an elementary extension of ''M''. The theorem is often divided into two parts corresponding to the two cases above. The part of the theorem asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the downward Löwenheim–Skolem Theorem.Nourani, C. F., ''A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos'' (
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: Apple Academic Press;
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:
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, 2014)
pp. 160–161
The part of the theorem asserting that a structure has elementary extensions of all larger cardinalities is known as the upward Löwenheim–Skolem Theorem.


Discussion

Below we elaborate on the general concept of signatures and structures.


Concepts


Signatures

A
signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
consists of a set of function symbols ''S''func, a set of relation symbols ''S''rel, and a function \operatorname: S_\cup S_ \rightarrow \mathbb_0 representing the
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. ...
of function and relation symbols. (A nullary function symbol is called a constant symbol.) In the context of first-order logic, a signature is sometimes called a language. It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains. A first-order
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature. Theories are often specified by giving a list of axioms that generate the theory, or by giving a structure and taking the theory to consist of the sentences satisfied by the structure.


Structures / Models

Given a signature ''σ'', a ''σ''-
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
''M'' is a concrete interpretation of the symbols in ''σ''. It consists of an underlying set (often also denoted by "''M''") together with an interpretation of the function and relation symbols of ''σ''. An interpretation of a constant symbol of ''σ'' in ''M'' is simply an element of ''M''. More generally, an interpretation of an ''n''-ary function symbol ''f'' is a function from ''M''''n'' to ''M''. Similarly, an interpretation of a relation symbol ''R'' is an ''n''-ary relation on ''M'', i.e. a subset of ''M''''n''. A substructure of a ''σ''-structure ''M'' is obtained by taking a subset ''N'' of ''M'' which is closed under the interpretations of all the function symbols in ''σ'' (hence includes the interpretations of all constant symbols in ''σ''), and then restricting the interpretations of the relation symbols to ''N''. An
elementary substructure In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...
is a very special case of this; in particular an elementary substructure satisfies exactly the same first-order sentences as the original structure (its elementary extension).


Consequences

The statement given in the introduction follows immediately by taking ''M'' to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. A theory is called categorical if it has only one model, up to isomorphism. This term was introduced by , and for some time thereafter mathematicians hoped they could put mathematics on a solid foundation by describing a categorical first-order theory of some version of set theory. The Löwenheim–Skolem theorem dealt a first blow to this hope, as it implies that a first-order theory which has an infinite model cannot be categorical. Later, in 1931, the hope was shattered completely by Gödel's incompleteness theorem. Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of
true arithmetic In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano a ...
, which satisfy every first-order induction axiom but have non-inductive subsets. Let N denote the natural numbers and R the reals. It follows from the theorem that the theory of (N, +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of (R, +, ×, 0, 1) (the theory of real closed fields) has a countable model. There are, of course, axiomatizations characterizing (N, +, ×, 0, 1) and (R, +, ×, 0, 1) up to isomorphism. The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first-order. For example, in the theory of the real numbers, the completeness of a linear order used to characterize R as a complete ordered field, is a non-first-order property. Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. Cantor's theorem states that some sets are uncountable. This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not
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.


Proof sketch


Downward part

For each first-order \sigma -formula \varphi(y,x_, \ldots, x_) \,, the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
implies the existence of a function :f_: M^n\to M such that, for all a_, \ldots, a_ \in M, either :M\models\varphi(f_ (a_1, \dots, a_n), a_1, \dots, a_n) or :M\models\neg\exists y\, \varphi(y, a_1, \dots, a_n) \,. Applying the axiom of choice again we get a function from the first-order formulas \varphi to such functions f_ \,. The family of functions f_ gives rise to a
preclosure operator In topology, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four ...
F on the power set of M :F(A) = \ for A \subseteq M \,. Iterating F countably many times results in a
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...
F^ \,. Taking an arbitrary subset A \subseteq M such that \left\vert A \right\vert = \kappa, and having defined N = F^(A) \,, one can see that also \left\vert N \right\vert = \kappa \,. Then N is an elementary substructure of M by the
Tarski–Vaught test In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one of ...
. The trick used in this proof is essentially due to Skolem, who introduced function symbols for the Skolem functions f_ into the language. One could also define the f_ as partial functions such that f_ is defined if and only if M \models \exists y\, \varphi(y,a_1,\dots,a_n) \,. The only important point is that F is a preclosure operator such that F(A) contains a solution for every formula with parameters in A which has a solution in M and that :\left\vert F(A) \right\vert \leq \left\vert A \right\vert + \left\vert \sigma \right\vert + \aleph_0 \,.


Upward part

First, one extends the signature by adding a new constant symbol for every element of ''M''. The complete theory of ''M'' for the extended signature ''σ''' is called the ''elementary diagram'' of ''M''. In the next step one adds ''κ'' many new constant symbols to the signature and adds to the elementary diagram of ''M'' the sentences ''c'' ≠ ''c''' for any two distinct new constant symbols ''c'' and ''c'''. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least ''κ'', the downward part of this theorem guarantees the existence of a model ''N'' which has cardinality exactly ''κ''. It contains an isomorphic copy of ''M'' as an elementary substructure.


In other logics

Although the (classical) Löwenheim–Skolem theorem is tied very closely to first-order logic, variants hold for other logics. For example, every consistent theory in second-order logic has a model smaller than the first supercompact cardinal (assuming one exists). The minimum size at which a (downward) Löwenheim–Skolem–type theorem applies in a logic is known as the Löwenheim number, and can be used to characterize that logic's strength. Moreover, if we go beyond first-order logic, we must give up one of three things: countable compactness, the downward Löwenheim–Skolem Theorem, or the properties of an
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. Chang, C. C., & Keisler, H. J., ''Model Theory'', 3rd ed. ( Mineola & New York:
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, 1990)
p. 134


Historical notes

This account is based mainly on . To understand the early history of model theory one must distinguish between ''syntactical consistency'' (no contradiction can be derived using the deduction rules for first-order logic) and ''satisfiability'' (there is a model). Somewhat surprisingly, even before the completeness theorem made the distinction unnecessary, the term ''consistent'' was used sometimes in one sense and sometimes in the other. The first significant result in what later became
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
was ''Löwenheim's theorem'' in Leopold Löwenheim's publication "Über Möglichkeiten im Relativkalkül" (1915): :For every countable signature ''σ'', every ''σ''-sentence that is satisfiable is satisfiable in a countable model. Löwenheim's paper was actually concerned with the more general Peirce–Schröder ''
calculus of relatives Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
'' (
relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
with quantifiers). He also used the now antiquated notations of Ernst Schröder. For a summary of the paper in English and using modern notations see . According to the received historical view, Löwenheim's proof was faulty because it implicitly used Kőnig's lemma without proving it, although the lemma was not yet a published result at the time. In a revisionist account, considers that Löwenheim's proof was complete. gave a (correct) proof using formulas in what would later be called '' Skolem normal form'' and relying on the axiom of choice: :Every countable theory which is satisfiable in a model ''M'', is satisfiable in a countable substructure of ''M''. also proved the following weaker version without the axiom of choice: : Every countable theory which is satisfiable in a model is also satisfiable in a countable model. simplified . Finally, Anatoly Ivanovich Maltsev (Анато́лий Ива́нович Ма́льцев, 1936) proved the Löwenheim–Skolem theorem in its full generality . He cited a note by Skolem, according to which the theorem had been proved by Alfred Tarski in a seminar in 1928. Therefore, the general theorem is sometimes known as the ''Löwenheim–Skolem–Tarski theorem''. But Tarski did not remember his proof, and it remains a mystery how he could do it without the compactness theorem. It is somewhat ironic that Skolem's name is connected with the upward direction of the theorem as well as with the downward direction: :''"I follow custom in calling Corollary 6.1.4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable sets."'' – . :''"Skolem ..rejected the result as meaningless; Tarski ..very reasonably responded that Skolem's formalist viewpoint ought to reckon the downward Löwenheim-Skolem theorem meaningless just like the upward."'' – . :''"Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence."'' – .


References


Sources

The Löwenheim–Skolem theorem is treated in all introductory texts on
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
or
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
.


Historical publications

* ** () * * ** () * ** () * *


Secondary sources

* ; A more concise account appears in chapter 9 of * * * * *


External links

* * Burris, Stanley N.
Contributions of the Logicians, Part II, From Richard Dedekind to Gerhard Gentzen
* Burris, Stanley N.
Downward Löwenheim–Skolem theorem
* Simpson, Stephen G. (1998),
Model Theory
{{DEFAULTSORT:Lowenheim-Skolem Theorem Mathematical logic Metatheorems Model theory Theorems in the foundations of mathematics