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A General Setting for Dedekind's Axiomatization of the Positive Integers

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A General Setting for Dedekind's Axiomatization of the Positive Integers
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  A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATIONOF THE POSITIVE INTEGERS GEORGE WEAVER Abstract.  A Dedekind algebra is an ordered pair ( B ,  h ) where  B  is a non-empty set and  h  is a similarity transformation on B. Among the Dedekindalgebras is the sequence of the positive integers. From a contemporary per-spective, Dedekind established that the second-order theory of the sequenceof the positive integers is categorical and finitely axiomatizable. The purposehere is to show that this seemingly isolated result is a consequence of moregeneral results in the model theory of second-order languages. Each Dedekindalgebra can be decomposed into a family of disjoint, countable subalgebrascalled the configurations of the algebra. There are  ℵ 0  isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal valuedfunction on  ω  called its configuration signature. The configuration signaturecounts the number of configurations in each isomorphism type that occurs inthe decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of anycountably infinite Dedekind algebra is categorical, and there are countablyinfinite Dedekind algebras whose second-order theories are not finitely axiom-atizable. It is shown that there is a condition on configuration signatures nec-essary and sufficient for the second-order theory of a Dedekind algebra to befinitely axiomatizable. It follows that the second-order theory of the sequenceof the positive integers is categorical and finitely axiomatizable. 1.  Introduction In a letter to Hans Keferstein (see Dedekind [1890]) Dedekind posed  the semantic completeness problem   for the theory of the sequence of the positive integers:What are the mutually independent fundamental properties of thesequence  N  , that is, those properties that are not derivable fromeach other and from which all others follow.Dedekind offered  the finitary categoricity solution   to this problem: he exhibited afinite set of properties of the sequence of the positive integers and claimed(1) that no one of them was derivable from the others; and(2) that all models of these properties are isomorphic.From a modern perspective, the reasoning from the second claim to semantic com-pleteness is weak (see Dedekind [1887]  § 134, pages 95-96). In essence, Dedekindargued that isomorphisms preserve the satisfaction of properties. Therefore, giventhe second claim, any property satisfied by the sequence of positive integers is sat-isfied by all models of the finite set Dedekind presented. Hence, every propertysatisfied by the sequence of the positive integers is a consequence of this set. With Date  : 5-21-09.1991  Mathematics Subject Classification.  03B15, 03C85. Key words and phrases.  second-order logic, categoricity, Dedekind, positive integers. 1  2 GEORGE WEAVER some modification in terminology, this general line of reasoning was repeated inthe early decades of the 20 th century (Corcoran [1980], Scanlan [1991]). However,these inferences are not evident. There is no explanation of the conditions underwhich a property is satisfied or not satisfied. The specification of the collection of properties satisfied by the sequence, while suggestive, is imprecise. For example,consider the following from Dedekind [1887]: among the properties of the sequenceof positive integers isany theorem in which we leave entirely out of consideration thespecial character of the elements  n  and discuss only such notionsas arise from the arrangement  φ .This imprecision concerning the properties not only weakens the reasoning but italso supports the contention that the semantic completeness problem does not ad-mit of a mathematical solution. For example, Kershner and Wilcox [1950] (pp.361-362) maintained that the claim to the effect that isomorphic systems satisfythe same properties is ”too broad and vague to be amenable of rigorous proof”. Fur-ther, the inference from this claim to semantic completeness appears to presupposesomething like Tarski’s  no counter interpretation   definition of logical consequence.Providing a mathematical solution to the semantic completeness problem in-volves finding a precise way to specify the collection of properties (as well as spellingout the conditions under which these properties are satisfied or not satisfied). Theproperties that Dedekind exhibited are each expressible by a sentence in a second-order language. Call the set of second-order sentences that express the properties Dedekind’s axioms  . Satisfaction of formulas in second-order languages is easilydefined and it is a simple matter to show that satisfaction is preserved under iso-morphisms. Hence, isomorphism implies equivalence in second-order languages.Therefore, every second-order sentence true of the sequence of the positive integersis a consequence of the axioms. Thus, it seems natural to reformulate Dedekind’sresult within the model theory of second-order languages: there is a finite set of second-order sentences such that each member of the set is true of the sequence of the positive integers and all models of the set are isomorphic. Hence, the second-order theory of the sequence of the positive integers is finitely axiomatizable andcategorical. It is arguable that this interpretation of Dedekind’s work might benarrower than he intended because there are properties in his sense that are not ex-pressible in a second-order language. For example, there are passages in Dedekind[1887] in which it appears that some of these properties involve infinite sequencesof quantifiers (pp. 46-52). However, because Dedekind’s axioms are categorical,broader interpretations are available. Since satisfaction of formulas in  ω -order lan-guages (including those that include sentences with infinite sequences of quantifiers)is preserved under isomorphisms, the  ω -order theory of the sequence the positiveintegers is also finitely axiomatizable. More generally, if   L  is a language that in-cludes the second-order language and satisfaction for formulas in  L  is preservedunder isomorphisms, then the theory of the sequence of the positive integers in  L is also finitely axiomatizable.The second-order interpretation of Dedekind is used below. The purpose of thispaper is to place this interpretation of Dedekind in a more general context. Aclass of algebras (the Dedekind algebras) that includes the sequence of the positive  A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATION OF THE POSITIVE INTEGERS3 integers is introduced. A  Dedekind algebra  1 is an ordered pair  B =( B ,  h B ) where B  is a nonempty set ( the domain of   B )  and  h B  is an injection (or  similarity transformation  ) on  B . In essence, these algebras are the focus of the second, third,fourth and fifth sections of   The Nature and Meaning of the Numbers  . Among themis the sequence of the positive integers. This paper departs from Dedekind in threerespects: (1) in formulating the discussion in terms of second-order languages andtheir model theory; (2) in investigating conditions both necessary and sufficientfor the second-order theory of a countably infinite Dedekind algebra to be finitelyaxiomatizable and categorical, rather than investigating the second-order theoryof a particular Dedekind algebra (viz. the sequence of the positive integers); and(3) in focusing on configurations (the smallest set containing an object and closedunder both  h A  and its inverse), rather than chains (kette).The paper is divided into eleven sections. While the primary focus of the paperis countably infinite Dedekind algebras, many of the results presented for the count-able case are consequences of results that hold for all infinite Dedekind algebras.Sections two and three present some topics from universal algebra. Sectiontwo introduces the notion of the configuration of an element in the domain of a Dedekind algebra and classifies the types of these configurations. It is shownthat the configurations in a Dedekind algebra form a partition of the domain of thealgebra and that there are  ω -many types of configurations: progressions, bigressionsand, for each  n ,  n -loops. Section three introduces the notion of the configurationsignature of a Dedekind algebra and establishes that the configuration signaturecodes the structure of the algebra. The configuration signature of the Dedekindalgebra  A  is a cardinal valued function defined on  ω  that counts the number of configurations in A of the various types: the value at 0 is the number of progressions,the value at 1 is the number of bigressions and the value at  n  + 1 is the numberof n-loops . For the purposes of this preliminary discussion, think of the value of this function at  n  as the number of configurations of type  n . It is shown that twoDedekind algebras are isomorphic iff their configuration signatures are identical.The intuition behind the next three sections is that the second-order theory of infinite Dedekind algebra  A  is categorical, if its configuration signature is  pointwise describable   in the sense that for each  n , the value of the  A ’s configuration at  n is describable by a second-order formula. There are two basic components of theintended description. Section four introduces some fundamental facts about thesecond-order theories of Dedekind algebras. It is shown that for each  n  there is asecond-order formula  φ n ( S  ) that is satisfied by a set  B  in  A  iff   B  contains exactlyone object from each configuration of type  n  in  A . Hence, if   B  satisfies  φ n ( S  ) in  A ,then the cardinality of   B  is the value of   A ’s configuration signature at  n . Sectionfive reviews the notion of a second-order charaterizable cardinal (Garland [1967])and establishes that if the cardinal  β   is second-order characterizable, then there isa formula,  β  ( S  ), that is satisfied by all and only sets of cardinality  β  . In sectionsix it is shown that the second-order theory of   A  is categorical if, for each  n , thevalue of   A ’s configuration signature at  n  is zero or a second-order characterizablecardinal. 1 The use of the term ‘Dedekind Algebra’ in the above sense was introduced in Weaver [1996].The author is not aware of an earlier use of this term in this or any other sense. This term isused in Potter [2004] (page 90) to refer to members of the isomorphism type of the sequence of the positive integers.  4 GEORGE WEAVER Since all finite and non-zero cardinals and the cardinal ℵ 0  are second-order chara-terizable, the second-order theory of every countably infinite Dedekind algebra iscategorical. Since there are 2 ℵ 0 different configuration signatures of countably infi-nite Dedekind algebras, not all of their theories are finitely axiomatizable. Sectionseven provides a condition that separates those countably infinite Dedekind alge-bras whose second-order theories are finitely axiomatizable (and categorical) fromthose whose theories are merely categorical. In fact, something stronger is estab-lished: there is a condition that separates those infinite Dedekind algebras whosesecond-order theories are finitely axiomatizable and categorical from those whosetheories are not. Consider the countable case and focus on the configuration signa-tures of these algebras, rather than the algebras themselves. For each  n , the valueof the signature at  n  is a describable cardinal. Some of the signatures are morecomplex than others. For example, some of these functions will only take finitelymany different values. Of these, some are constant valued functions, while othersare not. Roughly speaking, the second-order theory of such a Dedekind algebra iscategorical and finitely axiomatizable provided that the action of its configurationsignature is  globally describable   in the sense that there is a second-order formula φ ( S,S  ′ ) such that the sets  B ,  B ′ satisfy  φ ( S,S  ′ ) in the Dedekind algebra iff theconfiguration signature of the algebra at the cardinality of   B  is the cardinality of  B ′ . When the algebra is infinite, the notion of a describable configuration is aslightly more complex.In section eight it is shown that if   A  is any infinite Dedekind algebra whosesecond-order theory is finitely axiomatizable, then that theory is categorical. It fol-lows that the describability condition mentioned in the above paragraph separatesthose infinite Dedekind algebras whose second-order theories are finitely axiomati-zable from those whose theories are not.Section nine briefly discusses Dedekind’s concern to produce independent setsof axioms and it is observed that the sets of axioms exhibited in section seven areindependent.Section ten provides an application of Dedekind algebras in set theory: proofsof both the Cantor-Schr¨oder-Bernstein Theorem and Banach’s Theorem. Finally,section eleven outlines how the notion of the configuration of an object can beextended to a class of algebras that includes the Dedekind algebras. Configurationsignatures can be associated with each member of this class and it can be shownthat the configuration signatures of a member of the class codes the structure of themember. However, the number of types of configurations is uncountable. Hencesome results from section four fail for this extended class.2.  Configurations Following Burris and Sankappanavar [1980], B =( B ,  h B ) is a  mono-unary algebra  where  B  is a non-empty set ( the domain of   B ) and  h B is a unary function on  B . A Dedekind algebra   is a mono-unary algebra in which the unary function is an injectionon the domain. Since the proper class of Dedekind algebras is a universal Horn class(quasi-variety), it is closed under subalgebras, direct, reduced and ultraproducts,subdirect products, directed unions and direct limits (see Wechler [1992]). However,the class of Dedekind algebras is not closed under homomorphic images.Given a Dedekind algebra  B ,  P  B  is the binary relation on  B  that correspondsto  h B . When ( a , b ) ∈  P  B ,  a  is said to be  the parent of   b  (in   B ) . Notice that  a  is  A GENERAL SETTING FOR DEDEKIND’S AXIOMATIZATION OF THE POSITIVE INTEGERS5 the parent of   b  in  B  iff   h − 1 B  ( b )= a . Since  h B  is an injection, parents are unique.  b is a fixed point of   h B  (i.e.  h B ( b )= b ) iff   b  is its own parent in  B . Members of   B not in  h B [ B ] (the image of   B  under  h B ) have no parents in  B . Let  A B  be thetransitive closure of   P  B  (i.e. the smallest transitive binary relation on  B  extending P  B ). When ( a , b ) ∈  A B , we say that  a  is an ancestor of   b  (in   B ) .  a  is an ancestorof   b  iff there is  n  ≥  1 such that  h n B ( a )= b .  A B ( b ) is  the set of ancestors of   b  (in   B ) .When  B  is the set of positive integers and  h B  is the successor function, then  A B is the strictly less than relation and for  b  ∈  B ,  A B ( b ) is the strict initial segmentof   b .Since  A B  is transitive, if   a  is an ancestor of   b , then  A B ( a ) ⊆  A B ( b ). If   a  hasancestors and  a   =  b  and  A B ( a ) ⊆  A B ( b ), then  a  is an ancestor of   b ; and if   a  is anancestor of   b  and  A B ( a ) is a proper subset of   A B ( b ), then  a  is an ancestor of   b .Finally, since  h B  is 1-1, if   a  and  b  are different ancestors of   c , then either  a  is anancestor of   b  or  b  is an ancestor of   a .Given the Dedekind algebra  B ,  A  ⊆  B  and  b  ∈  B ,  A  is  the configuration of   b (in   B ) iff   A  is the smallest set such that  b  ∈  A ,  A B ( b ) ⊆  A  and  A  is closed under h B . Let Σ be the collection of subsets of   B  that include  { b }∪ A B ( b ) and are closedunder  h B .  B  ∈  Σ. And, the intersection of Σ is in Σ. Hence, every member of   B has a unique configuration in  B . The set  A  is a configuration in   B  provided  A  isthe configuration of some element in  B . Each configuration is a chain (or Kette)in the sense of Dedekind [1888], but not all chains are configurations. It is shownbelow that the configurations in  B  partition  B . Lemma 2.1:  Assume that  B  is a Dedekind algebra, that  a  and  b  ∈  B  and that a  is the ancestor of   b  in  B . Then,  a  and  b  have the same configuration in  B . Proof:  Assume that  a  is an ancestor of   b  in  B . Let  A ′ be the configuration of  a  in  B  and  A  be the configuration of   b  in  B . Both  A  and  A ′ are closed under  h B .It suffices to establish the following:(1)  b  ∈  A ′ and  A B ( b ) ⊆  A ′ ; and(2)  a  ∈  A  and  A B ( a ) ⊆  A .It follows that  A  ⊆  A ′ and  A ′ ⊆  A .By assumption,  A B ( a ) ⊆  A B ( b ) and  a  ∈  A B ( b ). Since  A B ( b ) ⊆  A , condition (2)above is satisfied. Also by assumption, there is  n  ≥  1 such that  h n B ( a )= b .  a  ∈  A ′ and  A ′ is closed under  h B . Thus,  b  ∈  A ′ . Let  c  be any ancestor of   b  in  B . Either a = c ,  a  is an ancestor of   c  or  c  is an ancestor of   a . If   c = a  or  c  is an ancestor of   a ,then  c  ∈  A ′ . Suppose that  a  is an ancestor of   c . There is  n  ≥  1 such that  h n B ( a )= c . a  ∈  A ′ and  A ′ is closed under  h B . Hence,  c  ∈  A ′ ; and (1) above is established.Suppose that  A  is the configuration of   a  in  B  and that  b  ∈  A . Then, either  b = a or  a  is an ancestor of   b  or  b  is an ancestor of   a . Hence, by Lemma 2.1,  A  is also theconfiguration of   b  in  B . The following is immediate. Lemma 2.2:  Assume that  B  is a Dedekind algebra and that  A  and  A ′ areconfigurations in  B . Then  A = A ′ iff   A ∩ A ′ is non-empty.There is another charaterization of the configurations in  B . Lemma 2.3:  Assume that  B  is a Dedekind algebra,  b  ∈  B  and  A  ⊆  B . Then,the following are equivalent:
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