AA xiomatic ( and N on – A xiomatic ) M athematics S AEED S ALEHI
15 August 2020
Table 1: A
XIOMATIZABILITY
N Z Q R C { < } ✓ ✓ ✓ ✓ – { + } ✓ ✓ ✓ ✓ ✓ { < , + } ✓ ✓ ✓ ✓ – { + , × } xx xx xx ✓ ✓ { × } ✓ ✓ ✓ ✓ ✓ { < , × } xx xx ✓ ✓ – exp xx – – ? xx A X IOMAT IZIN G mathematical structures andtheories, or postulating them as Russell put it, is anobjective of Mathematical Logic. Some axiomaticsystems are nowadays mere definitions, such as theaxioms of
Group Theory ; but some systems are much deeper,such as the axioms of
Complete Ordered Fields with whichReal Analysis starts. Groups abound in mathematical sciences,while by Dedekind’s theorem there exists only one completeordered field, up to isomorphism. Cayley’s theorem in AbstractAlgebra implies that the axioms of group theory completelyaxiomatize the class of permutation sets that are closed undercomposition and inversion; cf. e.g. [7].In this article, we survey some old and new results on thefirst-order axiomatizability of various mathematical structures(Table 1). The (non-)axiomatizability of many structures inTable 1 are known from almost a century ago; for example,the axiomatizability of ⟨ C ; + , ×⟩ follows from Tarski’s theorem(1936), and the non-axiomatizability of ⟨ N ; + , ×⟩ follows fromG¨odel’s theorem (1931). The question of the axiomatizabilityof e.g. ⟨ Q ; < , ×⟩ seemed to be missing in the literature, whichwas shown to be axiomaitzable in [1] for the first time; Tarski’sresult implies the axiomatizability of ⟨ C ; ×⟩ , but one explicitaxiomatization for it was presented in [20] for the first time.We will also review identities over + , × , exp that hold in the setof positive real numbers (Table 2). The identities on Table 2,except the last row which contains three dots, do completelyaxiomatize the identities that hold in the set of positive realnumbers ( R + ) over the indicated operations. Whether all theidentities in the table completely axiomatize the identities inthe structure ⟨ R + ; , + , × , exp ⟩ is the well-known Tarski’s High-School Problem , which has an interesting history. Table 2: A
XIOMS FOR I DENTITIES (over R + ) { + } x + ( y + z ) = ( x + y ) + z , x + y = y + x { × } x ⋅ ( y ⋅ z )=( x ⋅ y ) ⋅ z , x ⋅ y = y ⋅ x , x ⋅ = x { + , × } x ⋅ ( y + z ) = ( x ⋅ y ) + ( x ⋅ z ){ exp } ( x y ) z = ( x z ) y , x = x , x = { × , exp } x ( y ⋅ z ) = ( x y ) z , ( x ⋅ y ) z = x z ⋅ y z { + , × , exp } x ( y + z ) = x y ⋅ x z , ⋯ The method of “postulating” what we want has many advantages;they are the same as the advantages of theft over honest toil. –B ERTRAND R USSELL (1919,
Introduction to Mathematical Philosophy ) § B OOLEAN A LGEBRAS & P
ROPOSITIONAL L OGIC . Arguably, Modern Logic starts with Boole’s
Investigation of theLaws of Thought (1854); Boole’s axiomatic system is called“propositional logic” nowadays. It axiomatizes some of thebasic properties of the conjunction ( ∧ ), disjunction ( ∨ ), andnegation ( ¬ ) connectives. The Boolean expressions (or Booleanterms, or propositional formulas) are constructed from a fixedinfinite set of atoms, say { p , p , p , ⋯} , by means of thoseconnectives. Let us note that implication ( → ) is definable bydisjunction and negation as ( a → b ) ≡ ( ¬ a ) ∨ b , where ≡ denotes(logical) equivalence. Boole’s axiomatization is in fact nothingbut a definition of Boolean Algebras:Associativity a ∧ ( b ∧ c ) ≡ ( a ∧ b ) ∧ c , a ∨ ( b ∨ c ) ≡ ( a ∨ b ) ∨ c Commutativity a ∧ b ≡ b ∧ a , a ∨ b ≡ b ∨ a Distributivity a ∧ ( b ∨ c ) ≡ ( a ∧ b ) ∨ ( a ∧ c ) , a ∨ ( b ∧ c ) ≡ ( a ∨ b ) ∧ ( a ∨ c ) Idempotence a ∧ a ≡ a , a ∨ a ≡ a Truth and Falsum a ∨ ( ¬ a ) ≡⊺ , a ∧⊺≡ a , a ∧ ( ¬ a ) ≡(cid:150) , a ∨(cid:150)≡ a de Morgan’s Laws ¬ ( a ∧ b ) ≡ ( ¬ a ) ∨ ( ¬ b ) , ¬ ( a ∨ b ) ≡ ( ¬ a ) ∧ ( ¬ b ) Many more identities can be deduced (proved) from theabove axioms, such as the following:1
XAMPLE (i) It immediately follows from the axioms that a ≡ a ∧⊺≡ a ∧ ( p ∨¬ p ) ≡ ( a ∧ p )∨( a ∧¬ p ) .(ii) The absorbing properties of truth and falsum, i.e., a ∨⊺ ≡ ⊺ and a ∧(cid:150)≡(cid:150) follow also from the axioms. We show the former: a ∨⊺≡ a ∨ ( a ∨¬ a ) ≡ ( a ∨ a ) ∨ ( ¬ a ) ≡ a ∨ ( ¬ a ) ≡⊺ .(iii) One can also prove the absorption laws : a ∧ ( a ∨ b ) ≡ a and a ∨ ( a ∧ b ) ≡ a . Let us show the latter by using (ii) above: a ∨ ( a ∧ b ) ≡ ( a ∧⊺ ) ∨ ( a ∧ b ) ≡ a ∧ ( ⊺∨ b ) ≡ a ∧ ( b ∨⊺ ) ≡ a ∧⊺≡ a .(iv) The double negation law ¬¬ a ≡ a can be proved as follows: ¬¬ a ≡ ( ¬¬ a ) ∧ ⊺ ≡ ( ¬¬ a ) ∧ ( a ∨ ¬ a ) ≡ ( ¬¬ a ∧ a ) ∨ ( ¬¬ a ∧ ¬ a ) ≡ ( ¬¬ a ∧ a ) ∨ ( (cid:150) ) ≡ ( a ∧¬¬ a ) ∨ ( a ∧¬ a ) ≡ a ∧ ( ¬¬ a ∨¬ a ) ≡ a ∧⊺≡ a . ◇ We show that all the valid laws, according to the truth-tablesemantics, are provable from the axioms; thus it is a completeaxiomatic system (for Boolean equivalences): T HEOREM If a ≡ b is valid according to thetruth-table semantics, then it is provable from the axioms. A proof can proceed by normalizing the Boolean terms, orpropositional formulas. A (propositional) formula a is said tobe in disjunctive normal form (DNF) when it is a disjunction ofsome formulas each of which is a conjunction of some atomsor negated atoms; i.e., a = ⋁⋁ i c i where each c i is ⋀⋀ j (cid:96) ( i,j ) forsome atoms or negated-atoms (cid:96) ( i,j ) .If p is an atom, then ( p ) and ( ¬ p ) are both DNF; if q isanother atom, then the four formulas ( p ) ∨ ( q ) , ( p ) ∨ ( ¬ p ∧ q ) , ( p ∧¬ q ) ∨ ( q ) , and ( p ∧ q ) ∨ ( p ∧¬ q ) ∨ ( ¬ p ∧ q ) are equivalent DNF’s.Every propositional formula can be seen to be equivalent to aDNF formula, and this can be proved by the above axioms:firstly implication ( → ) does not appear in our formulas; andsecondly by the double negation law, proved in Example 1(iv),and de Morgan’s laws, negations ( ¬ ) can be pushed as far aspossible inside the sub-formulas, so that they appear at mostbehind atoms. Finally, by distributing all the conjunctions overdisjunctions, if any, an equivalent DNF formula is obtained;and this equivalence is provable from the above axioms.Now the proof goes as follows: assume that all the atomsthat appear in a and b belong to the set { p , ⋯ , p k } ; a and b are provably equivalent to some DNF formulas, such as e.g. a ≡ ⋁⋁ i c i and b ≡ ⋁⋁ j d j where c i ’s and d j ’s are conjunctions ofsome atoms or negated atoms. By Example 1(i) we can assumethat all the atoms p , ⋯ , p k appear exactly once in each c i and d j . By this assumption, we show that each c i is equal to some d j , and vice versa. Thus, a and b are provably equivalent. Fora fixed c i consider the evaluation that maps an atom to ⊺ if itappears positively in c i , and maps it to (cid:150) if it appears negativelyin c i . Under that evaluation, c i , and so a , is mapped to ⊺ ; thus b should be mapped to ⊺ too. So, some d j should be mapped to ⊺ under that evaluation; and this is possible only when d j = c i .The completeness of Propositional Logic with respect totruth-table semantics follows from Theorem 2. For example,the validity of the formula [( p → q ) → p ] → p , Peirce’s Law, canbe proved by first translating a → b to ¬ a ∨ b , and then showingthe equivalence ( ¬ [ ¬ ( ¬ p ∨ q ) ∨ p ] ∨ p ) ≡⊺ by the above axioms.We will come back to mathematical Identities at the end ofthe paper; before that let us study the axiomatizabilitiy of somemathematical structures. § A
XIOMATIZABILITY & Q
UANTIFIER E LIMINATION . A first-order structure consists of a non-empty set D , which iscalled domain (universe), together with a first-order language L , consisting of some constant, relation or function symbolsthat are interpreted over the domain. The abstract definitionof a structure A = ⟨ D ; L ⟩ from Model Theory is not neededhere (see e.g. [17] for more details). In the first-order setting,the quantifiers ( ∀ , ∃ ) range over the elements of the domain inquestion (which are taken to be number sets N , Z , Q , R , and C , here). So, subsets of the domain cannot be quantified; thus,the statement “for every nonempty and bounded subset there isa supremum for it” is not first-order, while “every element hasan inverse” is so.One reason for studying mathematical structures and theoriesin the setting of first-order logic is that despite of the fact thatthis logic is too weak to represent some fundamental properties(such as begin well-ordered or completeness of ordered sets)it has some other nice properties such as the compactness and semantic completeness (proved by G¨odel 1930).On the other hand, second-order logic may seem to be a moreexpressive framework for studying mathematical theories andstructures (in which one can express the properties of beginwell-ordered and completeness of ordered sets). But it has itsown foundational problems; the same problems that set theoryhas with incompleteness and truth (proved by G¨odel 1931). Infact, as Quine put it, the second-order logic is “set theory insheep’s clothing” (this is actually the title of the fourth sectionof the fifth chapter of Quine’s Philosophy of Logic , 1986).So, we have chosen first-order logic as the framework of ourstudy; though, the study could be undertaken in the frameworkof second-order logic as well. Let us recall that a sentence isa formula without any free variables, i.e., all of its variablesare quantified; and a theory is a set of sentences. We saw inthe previous section that propositional logic is axiomatizable;so a way of axiomatizing a structures is reducing its first-ordertheory to propositional logic which is usually done through theprocess of Quantifier Elimination. D EFINITION QE ) A theory T issaid to admit quantifier elimination ( QE ) when there exists analgorithm that for a given formula ϕ (⃗ x ) as input, with theshown free variables, outputs a quantifier-free formula θ (⃗ x ) with exactly the same free variables ( ⃗ x ) such that T proves thesentence ∀ ⃗ x [ ϕ (⃗ x ) ↔ θ (⃗ x )] . ◇ So, if T admits QE , then every first-order sentence over itslanguage is equivalent in T to an algorithmically calculablequantifier-free sentence. Quantifier elimination is usually doneby the means of the following fundamental lemma which isproved also in [6, Theorem 31F], [13, Theorem 4.1], and [23,Lemma III.4.1]. L EMMA
A theory T admits QE if and only if there exists an algorithmthat for every given formula of the form ∃ x γ ( x ) , where γ ( x ) is a conjunction of some atoms or negated atoms, outputs aquantifier-free formula θ such that the free variables of θ areall the free variables of γ ( x ) other than x , and the universalclosure of [ ∃ x γ ( x ) ↔ θ ] is provable in T . ϕ be an arbitrary formula. We show that it is T –equivalent toa quantifier-free formula with the same free variables (as of ϕ )and that quantifier-free formula can be found algorithmically.Take one of the innermost quantifiers of ϕ ; such as ∀ x θ ( x ) or ∃ x θ ( x ) where θ is a quantifier-free formula. In the formercase consider ¬∃ x ¬ θ ( x ) ; so without loss of generality we canassume that the quantifier is existential. We saw (in the Proof ofTheorem 2) that every propositional formula is equivalent to aDNF formula. So, ∃ x θ ( x ) ≡ ∃ x ⋁⋁ i γ i ( x ) ≡ ⋁⋁ i ∃ x γ i ( x ) , whereeach γ i ( x ) is a conjunctions of some atomic or negated atomicformulas. By the assumption, the existing algorithm can finda T –equivalent quantifier-free formula for each ∃ x γ i ( x ) ; thusthat algorithm can find a T –equivalent formula for ϕ with oneless quantifier (than ϕ ). So, by an inductive argument one canshow the existence of an algorithm that outputs a quantifier-freeformula with the same free variables (as of ϕ ) that is moreover T –equivalent to ϕ .Quantifier Elimination is applicable for axiomatizing thecomplete first-order theory of a structure A when we have acandidate theory T in a way that (i) all the axioms of T are truein A , (ii) T admits QE , and (iii) T decides (i.e., either proves orrefutes) every atomic sentence. Then, T is a complete theory,in the sense that it either proves or refutes every sentence overthe language of T , and so it completely axiomatizes A . Thus, T proves every sentence that is true in A , and refutes everysentence that is not true in A . In the following, we will studysome axiomatizations of number systems ( N , Z , Q , R , C ) overthe first-order languages that may contain < , + , × , or exp .For a structure that is known to be axiomatizable (wedo not have a clear criterion for axiomatizability or non-axiomatizability of a given structure), we introduce a theorythat is true in that structure and decides every atomic sentence,and show that it admits QE . Thus, the proposed theory doescompletely axiomatize the structure. § N UMBER S YSTEMS (Order & Addition).
Let us first study the order relation ( < ) in number systems. Werecall that an order is a binary relation that is anti-symmetric,transitive, and linear (see the axioms A < , T < , L < in Theorem 5).The order is dense in Q and R (see D < in Theorem 5) and hasno endpoints (see U < and B < in Theorem 5). This is all the first-order theory of order can say in Q and R , since it is a completetheory. However, the structure ⟨ Q ; <⟩ is very different from thestructure ⟨ R ; < ⟩ , since the latter is complete (every nonemptyand bounded subset has a supremum) while the former is not. T HEOREM ⟨ R ; <⟩ and ⟨ Q ; <⟩ ) The (finite) theory of dense linear orders without endpoints(with the following axioms) completely axiomatizes both of thestructures ⟨ R ; < ⟩ and ⟨ Q ; < ⟩ . ( A < ) ∀ x, y ( x < y → y ≮ x )( T < ) ∀ x, y ( x < y < z → x < z )( L < ) ∀ x, y ( x < y ∨ x = y ∨ y < x )( D < ) ∀ x, y ( x < y → ∃ w [ x < w < y ])( U < ) ∀ x ∃ u ( x < u )( B < ) ∀ x ∃ v ( v < x ) For a proof, note that the axioms are true in ⟨ R ; < ⟩ and ⟨ Q ; < ⟩ ; so, it suffices to show that the above theory admits QE .For that we use Lemma 4 and show the equivalence of everyformula of the form ∃ x ⋀⋀ i γ i ( x ) to a quantifier-free formula,where each γ i is an atom or negated atom. The equivalences ¬ ( a < b ) ↔ ( a = b ) ∨ ( b < a ) and ¬ ( a = b ) ↔ ( a < b ) ∨ ( b < a ) ,which are provable in the theory, allow us to neglect negatedatomic formulas. Thus, we need to eliminate the quantifier ofthe formulas of the form ∃ x ( ⋀⋀ i u i < x ∧ ⋀⋀ j x < v j ∧ ⋀⋀ k x = w k ) only—note that x = x is equivalent to ⊺ , and x < x to (cid:150) . But thatformula is equivalent to ⋀⋀ i u i < w ∧ ⋀⋀ j w < v j ∧ ⋀⋀ k w = w k ,if the conjunction ⋀⋀ k x = w k is non-empty, and to ⋀⋀ i,j u i < v j ,if it is empty (non-existent) and none of the other conjunctionsare empty; if any of ⋀⋀ i u i < x or ⋀⋀ j x < v j is also empty, thenthe original formula is equivalent to ⊺ .The order relation behaves very differently on Z and N , sincehere it is a discrete order, in the sense that every element hasan immediate successor. Let us denote the successor function x ↦ ( x + ) by s ; and let ( x ⩽ y ) abbreviate ( x < y ) ∨ ( x = y ) . Fora proof of the following theorem, first proved by A. Robinsonand E. Zakon (1960, Theorem 2.12), see e.g. [1, Theorem 2]. T HEOREM ⟨ Z ; < ⟩ ) The (finitelyaxiomatized) theory of discrete linear orders without endpointscompletely axiomatizes the structure ⟨ Z ; < , s ⟩ ; this theoryconsists of the axioms A < , T < , L < (Theorem 5) along with ( S < ) ∀ x, y ( x < y ↔ s ( x ) ⩽ y )( P < ) ∀ x ∃ w ( s ( w ) = x ) The following has been proved in e.g. [6, Theorem 32A]. T HEOREM ⟨ N ; < ⟩ ) The (finitelyaxiomatizable) theory of discrete linear orders with the leastelement and without the last element completely axiomatizesthe structure ⟨ N ; , s , < ⟩ ; this theory consists of the axioms A < , T < , L < (Theorem 5) together with S < (Theorem 6) and ( Z < ) ∀ x ( ⩽ x )( P < ) ∀ x ∃ w ( < x → s ( w ) = x ) Let us note that ∀ x [ x < s ( x )] is provable from S < ; and soone can show that ∀ x, y [ x < y ↔ s ( x ) < s ( y )] follows from S < , T < and L < . Therefore, Peano’s axioms ∀ x ( s ( x ) ≠ ) and ∀ x, y ( s ( x ) = s ( y ) → x = y ) are provable from the axiom system A < , T < , L < , Z < , and S < .We now study the addition operation ( + ) in number systems.The most obvious properties of addition are associativity andcommutativity (see A + and C + in Theorem 8). Of course, in allof our number systems there is an additive unit element (zero ), and in all but one (the natural numbers) every element has anadditive inverse (the minus element). In C , R , and Q additionis torsion-free and divisible (see T + and D + in Theorem 8); itis hard to find any other property of + in C , R , Q that does notfollow from the above-mentioned properties.For axiomatizing the structures ⟨ C ; +⟩ , ⟨ R ; +⟩ , and ⟨ Q ; +⟩ weadd the constant symbol and the unary function symbol − tothe language; needless to say, n x abbreviates the expression x + ⋯ + x ( n times) for n ∈ N .3 HEOREM ⟨ C ; +⟩ , ⟨ R ; +⟩ , ⟨ Q ; +⟩ ) The first-order theory of non-trivial, divisible, torsion-free, andcommutative groups (with the following infinite set of axioms)completely axiomatizes the structures ⟨ Q ; , − , +⟩ , ⟨ R ; , − , +⟩ ,and ⟨ C ; , − , +⟩ . ( A + ) ∀ x, y, z ( x + ( y + z ) = ( x + y ) + z )( C + ) ∀ x, y ( x + y = y + x )( U + ) ∀ x ( x + = x )( I + ) ∀ x ( x + ( − x ) = )( N + ) ∃ u ( u ≠ )( T + ) { ∀ x ( n x = → x = )} n > ( D + ) { ∀ x ∃ v ( x = n v )} n > We show that the theory admits QE by using Lemma 4. Everyatomic formula in the language { , − , +} that contains x can beequivalently written in the form n x = t for some n ∈ N + andsome x –free term t . By a = b ←→ k a = k b , which is provablefrom the above axioms, it suffices to eliminate the quantifierof ∃ x ( ⋀⋀ i q x = t i ∧ ⋀⋀ j q x ≠ s j ) , which by D + (for n = q ) isequivalent to ∃ y ( ⋀⋀ i y = t i ∧ ⋀⋀ j y ≠ s j ) . Now, if the conjunct ⋀⋀ i y = t i is nonempty, then this is equivalent to ⋀⋀ i t = t i ∧⋀⋀ j t ≠ s j , and if ⋀⋀ i y = t i is empty, then it is equivalent to ⊺ ,since by N + there are infinitely many members (for any u ≠ we have n u ≠ m u for every n ≠ m ).The axiomatization of ⟨ Z ; +⟩ illustrates a case that one mightneed to substantially enrich the language of the structure tohave QE . As an example, the formula ∃ v ( x = v + v ) , statingthat x is even, is not equivalent to any quantifier-free formulain ⟨ Z ; , − , +⟩ . However, if we add the binary relation symbol ≡ of congruence modulo to the language, then that formulawill be equivalent to x ≡ .The quantifier elimination of the theory of the structure ⟨ Z ; , , { ≡ n } n > , − , +⟩ can be shown by using a generalizedform of the Chinese Remainder Theorem in Number Theory.The Chinese remainder theorem says that a given system ofcongruence equations { x ≡ n i r i } i < N has a solution (in Z ) if n i and n j are coprime for every i < j < N . The generalizedChinese remainder theorem says that the system { x ≡ n i r i } i < N of congruence equations has a solution if and only if for every i < j < N we have r i ≡ d i,j r j , where d i,j is the greatest commondivisor of n i and n j . Since such systems either have no solutionor have infinitely many solutions, then we can state this moregeneral theorem as follows. P ROPOSITION If n i > for every i < N , then for every { r i } i < N and { s j } j < M , ∃ x ( ⋀⋀ i < N x ≡ n i r i ∧ ⋀⋀ j < M x ≠ s j ) ⇐⇒ ⋀⋀ i < N r i ≡ d i,j r j , where d i,j is the greatest common divisor of n i and n j . For three different proofs of Proposition 9, which is a kindof QE by itself, see [20, Propositions 4.5 and 4.1] and [1,Proposition 2] which are due to Ore (1951), Mahler (1958) andFraenkel (1963) respectively.We add the congruence relations ≡ n modulo every natural n > , along with the constant , to the language; let i abbreviate + ⋯ + ( i times) for every i ∈ N . T HEOREM
10 (An Axiomatization for ⟨ Z ; +⟩ ) The theorywhose axioms are A + , C + , U + , I + , and T + (Theorem 8) withthe following axioms completely axiomatizes the structure ⟨ Z ; , , { ≡ n } n > , − , +⟩ . ( E + ) { ∀ x, y [ x ≡ n y ↔∃ u ( x = y + n u )]} n > ( E + ) { ∀ x [ ⋁⋁ i < n ( x ≡ n i )]} n > ( E ′ + ) { ⋀⋀ < i < n ( i ≢ n )} n > For showing that the theory admits QE by Lemma 4, we notethat every atomic formula of x in { , , − , + } ∪ { ≡ n ∣ n > } isequivalent to either m x = t or m x ≡ n t for some m, n ∈ N + andsome x –free term t . By the provable equivalence ( a ≢ n b ) ↔⋁⋁ < i < n ( a ≡ n b + i ) it suffices to show that the formula ∃ x ( ⋀⋀ i q i x ≡ n i r i ∧ ⋀⋀ j q j x ≠ s j ∧ ⋀⋀ k q k x = t k ) is equivalent to a quantifier-free formula. From the provableequivalences ( a = b ) ↔ ( k a = k b ) and ( a ≡ n b ) ↔ ( k a ≡ kn k b ) we can assume that all the q i ’s, q j ’s and q k ’s are equal, to say q . Then, the above formula is equivalent to ∃ y ( y ≡ q ∧ ⋀⋀ i y ≡ n i r i ∧ ⋀⋀ j y ≠ s j ∧ ⋀⋀ k y = t k ) . We can assume that the conjunct ⋀⋀ k y = t k is empty (see theproofs of Theorems 5,8); now the result immediately followsfrom Proposition 9 (which is provable from the axioms).As for N , even the language { , , − , + } ∪ { ≡ n ∣ n > } is notsufficiently rich for QE , as the formula ∃ v ( x + v = y ) is notequivalent to a quantifier-free formula (it is equivalent to x ⩽ y )in N . Here, QE is possible when we add the order relation tothe language. T HEOREM
11 (An Axiomatization for ⟨ N ; < , +⟩ ) The theorywith the axioms A < , T < , L < (Theorem 5), S < (Theorem 6), Z < , P < (Theorem 7), A + , C + , U + (Theorem 8), E + , E + (Theorem 10)with the following axioms completely axiomatizes the structure ⟨ N ; , , < , { ≡ n } n > , +⟩ . ( M + ) ∀ x, y ( x < y → ∃ v [ x + v = y ])( O + ) ∀ x, y, z ( x < y → x + z < y + z ) A proof of Theorem 11 can be found in [6, Theorem 32E](without presenting an explicit axiomatization; though one cansee that the proof goes through with our suggested axioms). Fora proof of the following theorem see e.g. [1, Theorem 5]; otherproofs can be found in [13, § 4.III] and [23, §§ III.4.2]. T HEOREM
12 (An Axiomatization for ⟨ Z ; < , +⟩ ) The theorywith the axioms A < , T < , L < (Theorem 5), S < , P < (Theorem 6), A + , C + , U + , I + (Theorem 8), E + (Theorem 10) and finally O + (Theorem 11), with s ( x ) set to x + , completely axiomatizes thestructure ⟨ Z ; , , < , { ≡ n } n > , − , +⟩ . The following theorem (stating that the order and additionstructure of rational and real numbers can be axiomatized bythe theory of non-trivial divisible commutative ordered groups)can be proved by combining the techniques of the proofs ofTheorems 5 and 8 (cf. [1, Theorem 4]).4
HEOREM
13 (Axiomatizing ⟨ Q ; < , +⟩ and ⟨ R ; < , +⟩ ) Thetheory with the axioms A < , T < , L < (Theorem 5), A + , C + , U + , I + , N + , D + (Theorem 8), and finally O + (Theorem 11) completelyaxiomatizes the structures ⟨ Q ; , < , − , +⟩ and ⟨ R ; , < , − , +⟩ . Let us note that the axioms D < , U < , B < (in Theorem 5) and T + (in Theorem 8) are provable from the axiom system presentedin Theorem 13 just the way that are proved in classical analysis. § N UMBER S YSTEMS (Addition & Multiplication).D
EFINITION
14 (Field) A field is a structure over { , , + , − , × , − } that satisfies A + , C + , U + , I + (Theorem 8), andthe following axioms: ( A × ) ∀ x, y, z ( x ⋅ ( y ⋅ z ) = ( x ⋅ y ) ⋅ z )( C × ) ∀ x, y ( x ⋅ y = y ⋅ x )( U × ) ∀ x ( x ⋅ = x )( I × ) ∀ x ( x ≠ → x ⋅ x − = )( D × ) ∀ x, y, z [ x ⋅ ( y + z ) = ( x ⋅ y ) + ( x ⋅ z )] A field has characteristic zero if it moreover satisfies ( C ) { n ≠ } n > where, as we recall, n abbreviate + ⋯ + ( n times). ◇ The field ⟨ C ; + , ×⟩ is well known to be algebraically closedsince it satisfies the Fundamental Theorem of Algebra, i.e., ithas a root for every non-trivial polynomial (with coefficients in C ). It can be even said that it was created for having all the rootsof the polynomials (with real or complex coefficients). This isall one can say about the complex field in the first-order setting,since the theory of algebraically closed fields of characteristiczero is complete, and so it axiomatizes ⟨ C ; + , ×⟩ ; see also [15].The following result was proved by Tarski (1936); see e.g. [13,§ 4.IV] for a proof. T HEOREM
15 (An Axiomatization for ⟨ C ; + , ×⟩ ) The theoryof algebraically closed fields of characteristic zero, with theaxioms A + , C + , U + , I + , A × , C × , U × , I × , D × , C (Definition 14)along with the following axioms, completely axiomatizes thestructure ⟨ C ; , , − , + , × , − ⟩ . ( FTA C ) { ∀ ⟨ a i ⟩ i < n ∃ x ( x n +∑ i < n a i x i = )} n > For super-careful readers let us note that (i) every non-trivialpolynomial can be taken to be a monic by dividing it with theleading (non-zero) coefficient; (ii) the multiplicative inversion( x ↦ x − ) is not really a total function, since it is not definedon zero, but one can make the convention − = without anydanger; (iii) and finally, x i abbreviates the algebraic expression x × ⋯ × x ( i times) of course.For studying the structure ⟨ R ; + , ×⟩ we first note that orderis definable in it: u ⩽ v ⇐⇒ ∃ x ( u + x = v ) ; and ⟨ R ; < , + , ×⟩ is an ordered field, see e.g. [17]. An ordered field satisfies theorder axioms A < , T < , L < (Theorem 5), the axioms of fields (inDefinition 14), O + (Theorem 11), and O × (Theorem 16 below).Of course this is not all one can say about ⟨ R ; < , + , ×⟩ . Onthe other hand, not much can one say about it in the first-orderframework; only that every positive real number has a squareroot, and every polynomial of even degree can be factorized into some quadratic polynomials (see FTA R in Theorem 16).This last statement is indeed equivalent to (a real version of)the fundamental theorem of algebra. As some examples, let usnote quadratic factorizations of the following quartics: x + = ( x + √ x + )( x − √ x + ) , x − x + = ( x + √ x + )( x − √ x + ) , and x − x + = ( x + √ r x + √ rr √ r − )( x − √ r x + r √ r − √ r ) ,where r is the unique positive real number that satisfies thecubic equation r − r − = . For a proof of the following resultof Tarski (1936) see e.g. [21, Appendix] which is a modifiedversion of the proof presented in [13, § 4.V]. T HEOREM
16 (An Axiomatization for ⟨ R ; < , + , ×⟩ ) Theoryof real closed ordered fields which is axiomatized by A < , T < , L < (Theorem 5), the axioms of fields (Definition 14), and O + (Theorem 11) along with the following axioms completelyaxiomatizes the structure ⟨ R ; , , < , − , + , × , − ⟩ . ( O × ) ∀ x, y, z ( < z ∧ x < y → x ⋅ z < y ⋅ z )( S × ) ∀ x ( < x →∃ u [ x = u ])( FTA R ) { ∀ ⟨ a i ⟩ i < n ∃ ⟨ b j , c j ⟩ j < n ∀ x [( x n +∑ i < n a i x i ) =∏ j < n ( x + b j x + c j )]} n > We note that by S × (and the axioms of ordered fields)the high-school equivalence for the existence of the roots ofquadratic polynomials can be proved: ∃ x ( x + bx + c = ) ↔∃ x [( x + b ) = b − c ] ↔ b ⩾ c .It can be easily seen that FTA C (in Theorem 15) is equivalentto the statement that every monic is equal to a product of somelinear polynomials: FTA C ≡≡ { ∀ ⟨ a i ⟩ i < n ∃ ⟨ b j ⟩ j < n ∀ x [( x n +∑ i < n a i x i ) =∏ j < n ( x + b j )]} n > which resembles FTA R (in Theorem 16). Let us note a coupleof consequences of FTA R (from [21]): P ROPOSITION
17 (
FTA R (cid:212)⇒ RCF + IVT ) If every even-degree polynomial can be factorized into some quadraticpolynomials in an ordered field in which every positive elementhas a square root, then every odd-degree polynomial has a rootand the polynomial intermediate value theorem holds in it.
For a proof suppose that the polynomial p ( x ) is of degree m and p ( u ) p ( v ) < holds for some u < v . Put q ( x ) to be thepolynomial p ( u ) ( + x ) m p ( u + v − u + x ) ; then q ( x ) = x m + r ( x ) for some polynomial r ( x ) with degree less than m . So, q ( x ) can be factorized to say ∏ j < m ( x + b j x + c j ) . Now we have ∏ j < m c j = q ( ) = p ( v ) p ( u ) < and so c j < for some j ; then we have b j > c j and so the quadratic x + b j x + c j = has a root, such as s . Now, r = u + v − u + s is a root of p ( x ) = that satisfies u < r < v .By a classical real analytic argument, if the intermediate valuetheorem holds for polynomials in an ordered field, then everyodd-degree polynomial has a root in it.So, the fundamental theorem of algebra is really fundamental since it can prove some basic theorems in algebra, and it is akind of fundamental theorem for the mathematical analysis ofpolynomials as well; see [21] for more details.5o far, we have observed two applications of mathematics(especially number theory and algebra) in mathematical logic:1. The (Generalized) Chinese Remainder Theorem2. The Fundamental Theorem of Algebra1. Proposition 9 was used in proving that the axiomaticsystem suggested for the additive structure of integer numbers ⟨ Z ; +⟩ has QE and so it is a complete theory (Theorem 10). Itis worth noting that G¨odel (1931, Lemma 1) also had used the(non-generalized) Chinese remainder theorem in his proof ofthe first incompleteness theorem for the coding technicalities.2. The truly fundamental theorem of elementary algebra andelementary analysis was used for axiomatizing the additive andmultiplicative structures of complex and real numbers, ⟨ C ; + , ×⟩ and ⟨ R ; + , ×⟩ , noting that order ( < ) is definable in ⟨ R ; + , ×⟩ (Theorems 15 and 16).Now, we present two applications of mathematical logic inother areas of mathematics (especially algebraic geometry):I. The Tarski-Seidenberg PrincipleII. Hilbert’s 17th ProblemI. Theorem 16, like many other theorem of QE , is proved byusing Lemma 4. Let us see how the proof can proceed: first,we note that all the atomic formulas of x over the language { , , − , + , × , − , < } are equivalent to p ( x ) = or p ( x ) > fora polynomial p . Second, negation can be eliminated (see theproof of Theorem 5), so QE over this language is equivalent to“the existence of a solution of a system of polynomial equationsand inequalities is equivalent to a system of some equations andinequalities between the coefficients of those polynomials”. Asan example, ∃ x ( ax + bx + c = ) is known to be equivalent to ( a > ∧ b ⩾ ac ) ∨ ( a = ∧ b > ) ∨ ( a = ∧ b = ∧ c = ) .This is called the Tarski-Seidenberg principle in real algebraicgeometry (see [3, §1.4]), which is exactly what the translationof Lemma 4 would be in the proof of Theorem 16 (cf. [15]).II. Hilbert’s celebrated 17th Problem asked (see e.g. [22]): Given a multivariate polynomial that takes only non-negativevalues over the reals, can it be represented as a sum of squaresof rational functions?
Let us note a couple of examples: x − x + = ( x − ) + ( x − ) + ( √ ) and ( x + y ) [ x y + x y + − x y ] = ( x − y ) + [ x y ( x + y − )] + [ xy ( x + y − )] + [ xy ( x + y − )] .A consequence of the Tarski-Seidenberg principle is theArtin-Lang Homomorphism theorem [3, Theorem 4.1.2] whichgives a positive answer to the problem; see [3, Theorem 6.1.1].Let us note that by the fundamental theorem of algebra everynon-negative polynomial of one variable can be written as asum of the squares of some polynomials; but there are non-negative polynomials of two variables that cannot be written assuch. One example (see [22]) is Motzkin (1969)’s polynomial x y + x y + − x y ; of course it is the sum of the squares ofsome rational functions (see the second example above).The next structures that we study over the language { + , × } are Q , Z , and N . Here the story becomes dramatically different.To start with, let us note that the axiomatic systems presentedfor the ordered structures ⟨ N ; < ⟩ , ⟨ Z ; < ⟩ , ⟨ Q ; < ⟩ , and ⟨ R ; < ⟩ The sum of squares for a given polynomial may not be unique, as theidentity ( x + ) = ( x ) + ( x − ) shows. were all finite (Theorems 5,6,7). Other axiomatic systems werenot finite, but were presented in a way that one can recognizewhether a given sentence is an axiom of that system or not,in the sense that a properly designed algorithm can recognizethem. In the other words, the axiomatic theories for the studiedstructures were decidable by an algorithm.To make precise the forthcoming definition, let us makethe convention that all our first-order individual variablesare ϑ , ϑ ′ , ϑ ′′ , ϑ ′′′ , ⋯ , made up from ϑ and ′ . Let us fixthe following finite set of symbols as an alphabet: A ={¬ , ∧ , ∨ , ∀ , ∃ , ( , ) , ϑ , ′ , , , < , = , + , − , × , exp } . Every formulaover the first-order language { , , < , + , − , × , exp } is a string(i.e., a finite sequence) of the elements of A . There exists analgorithm that decides (outputs yes or no ) if a given such stringas input is a well-founded formula or not. D EFINITION
18 (Decidability)
A set B of strings of symbolsfrom A is decidable when there exists an algorithm such thatfor a given string as input outputs yes if it belongs to B andoutputs no otherwise. ◇ Let us note that we have not fixed a rigorous definition for theinformal notion of algorithm in the above definition; it couldbe a recursive function or a
Turing machine (see [2]). By theChurch-Turing thesis all such formally rigorous and equivalentdefinitions do define the informal notion of algorithm; so we donot need to fix a formalization. “Axiomatizable” usually means axiomatizable by a decidable set of axioms ; though more oftenthe decidability of the axiom set is not explicitly mentioned. D EFINITION
19 (Axiomatizability)
A theory or a structure issaid to be axiomatizable when there exists a decidable set ofsentences that completely axiomatizes it. ◇ All the theories and structures that we have studied so farare axiomatizable by a decidable set of sentences. Actually, astructure is axiomatizable by a decidable set of sentences if andonly if it has a decidable theory; see e.g. [6, Corollary 26I].Decidability implies axiomatizability, since one only needsto algorithmically list all the sentences and pick the ones thathold true; thus a decidable set of axioms is obtained. If A isaxiomatizable, then for a given sentence ψ run this algorithmfor consecutive n ’s starting from n = : list all the theoremsthat are proved in n steps or less from the first n axioms (if n exceeds the number of axioms, then use all the finitely manyaxioms); if ψ or ¬ ψ appears in the list, then output yes or no accordingly. The algorithm will surely terminate (for some n )since the axiomatic system completely axiomatizes A .Now, the shocking result of G¨odel’s incompleteness theorem(1931) is that the structure ⟨ N ; + , ×⟩ is not axiomatizable. Asthe history goes, Presburger (1929) proved the axiomatizabilityof ⟨ N ; +⟩ and Skolem (1930) announced the axiomatizability of ⟨ N ; ×⟩ (see [23]); so ⟨ N ; + , ×⟩ was expected to be axiomatizable,that would confirm Hilbert’s Programme (see e.g. [5, 7]). T HEOREM
20 (Non-Axiomatizability of ⟨ N ; + , ×⟩ ) The fullfirst-order theory of ⟨ N ; + , ×⟩ is not axiomatizable by anydecidable set of sentences. Of course, there does exist an undecidable set of sentencesthat completely axiomatizes ⟨ N ; + , ×⟩ ; that is the so-called true rithmetic , the set of all the sentences that are true in N . Thenon-axiomatizability of ⟨ Z ; + , ×⟩ is inherited from ⟨ N ; + , ×⟩ since the set N is definable in ⟨ Z ; + , ×⟩ by Lagrange’s foursquare theorem (see e.g. [23, Theorem II.3.8]): let N ( x ) bethe formula ∃ u, v, w, z ( u + v + w + z = x ) . Then for every m ∈ Z we have [ m ∈ N if and only if N ( m ) is true in Z ].For every formula ϕ over { + , × } , let ϕ N result from ϕ by changing every ∀ x Θ to ∀ x [N ( x ) → Θ ] and ∃ x Θ to ∃ x [N ( x ) ∧ Θ ] ; that is relativizing all the bounded variablesto N . Now, for every sentence θ over { + , × } we have: θ is truein ⟨ N ; + , ×⟩ if and only if θ N is true in ⟨ Z ; + , ×⟩ .So, it follows that the structure ⟨ Z ; + , ×⟩ is not axiomatizableby any decidable set of sentences T , since otherwise ⟨ N ; + , ×⟩ would be axiomatizable by the decidable set of sentences T N = { θ N ∣ θ ∈ T } . For another definition of N in ⟨ Z ; + , ×⟩ seeRobinson’s paper [19] where it is proved that Z is definablein ⟨ Q ; + , ×⟩ as well (see also [18]). C OROLLARY
21 (On ⟨ Z ; + , ×⟩ and ⟨ Q ; + , ×⟩ ) The structures ⟨ Z ; + , ×⟩ and ⟨ Q ; + , ×⟩ are not axiomatizable. § N UMBER S YSTEMS (Multiplication & Exponentiation).
We saw that by Tarski’s result ⟨ C ; + , ×⟩ is axiomatizable; thenits theory is decidable, and so is the theory of ⟨ C ; ×⟩ . Thus, ⟨ C ; ×⟩ is axiomatizable by a decidable set of sentences; butwhat is that axiomatic system? This question was answeredin [20, Theorem 2.2] by providing an explicit axiomatizationfor the multiplicative structure of complex numbers: T HEOREM
22 (An Axiomatization for ⟨ C ; ×⟩ ) The structure ⟨ C ; , , { ω n } n > , × , − ⟩ is axiomatizable by A × , C × , U × , I × (Theorem 16) along with the following axioms: ( Z × ) ∀ x ( x ⋅ = = − )( D × ) { ∀ x ∃ v ( x = v n )} n > ( R × ) { ∀ x [ x n = ↔⋁⋁ i < n x = ( ω n ) i ]} n > ( R × ) { ⋀⋀ i < j < n ( ω n ) i ≠ ( ω n ) j } n > where ω n is interpreted as [ cos ( π / n ) + ı ′ sin ( π / n )] for every n > ; thus ω = − , ω = (− / ) + ı ′ (√ / ) and ω = ı ′ . The same question can be asked about the real numbers: weknow that ⟨ R ; ×⟩ is decidable by Tarski’s result that ⟨ R ; + , ×⟩ is axiomatizable; but what is an explicit axiomatization for ⟨ R ; ×⟩ ? For its answer we need to add the positivity predicate ,denoted P( x ) , to the language. The following result is provedin [20, Theorem 3.3]. T HEOREM
23 (An Axiomatization for ⟨ R ; ×⟩ ) The structure ⟨ R ; , , − , P , × , − ⟩ is axiomatizable by A × , C × , U × , I × , Z × (Theorem 22) along with the following axioms: ( N × ) ∃ u ( u ≠ , , − )( D o × ) { ∀ x ∃ v ( x = v n + )} n > ( R e × ) { ∀ x ( x n = ↔ x = ∨ x =− )} n > ( P ) ∀ x (P( x ) ↔ ∃ y ≠ [ x = y ])( P × ) ∀ x, y ≠ (P( xy ) ←→ [P( x ) ↔ P( y )])( P −× ) ∀ x ≠ [ ¬ P( x ) ↔ P([ − ] x )] Let us note that the multiplicative structure of positive realnumbers ⟨ R + ; ×⟩ is a non-trivial, divisible, torsion-free, andcommutative group, since it is isomorphic to ⟨ R ; +⟩ via themapping x ↦ ln ( x ) .For axiomatizing the multiplicative structure of rationalnumbers ⟨ Q ; ×⟩ we first axiomatize the multiplicative structureof positive rational numbers ⟨ Q + ; ×⟩ noting that one can obtainan axiomatization for ⟨ Q ; ×⟩ by adding the constants , − andthe predicate P( x ) to the language and adding Z × , N × , P , P × ,and P −× (Theorem 23) to the axioms. The following is proved in[20, Theorem 4.11]: T HEOREM
24 (An Axiomatization for ⟨ Q + ; ×⟩ ) The first-order structure ⟨ Q + ; , × , − ⟩ is axiomatizable by A × , C × , U × , I × (Definition 14) along with the following axioms: ( T × ) { ∀ x ( x n = → x = )} n > ( M × ) { ∀ ⟨ x i ⟩ i < k ∃ v ∀ y ⋀⋀ i < k ( v n x i ≠ y m i )} n,k where n, k ∈ N , and no m i ∈ N divides n . The axioms M × in Theorem 24 state that for every sequence x , . . . , x k − of positive rational numbers and every sequence m , . . . , m k − of natural numbers none of which divides thenatural number n , there exists a positive rational number v suchthat for every i < k none of v n x i ’s is an m i -power of a positiverational number. To see that this holds in Q + it suffices to take v to be a prime number that does not divide the numerators anddenominators of any of x i ’s. This does not hold if some m i divides n since x i could be an m i th power; it does not hold in R + either since every positive real number has an m i th root.The next structures that we study over the language { × } are Z and N . Here too, as we saw, it suffices to study ⟨ N + ; ×⟩ first,and then for ⟨ N ; ×⟩ we need to add and the axiom Z × , and for ⟨ Z ; ×⟩ we need to add − , P and the axioms P × and P −× . Sincestudying the axioms of ⟨ N + ; ×⟩ will not be needed later, andthey are too many to be listed in the main body of the paper, andexplaining them will take much time and will distract the flowof the paper, we apologetically postpone it to the Appendix.Let us move on to the language { < , × } over which R and Q are axiomatizable, while Z and N are not. The following isproved in [1, Theorem 6]. T HEOREM
25 (An Axiomatization for ⟨ R ; < , ×⟩ ) The theorywith the axioms A < , T < , L < (Theorem 5), A × , C × , U × , I × , Z × (Theorem 22), O × (Theorem 16), D o × , R e × (Theorem 23)along with the following completely axiomatizes the structures ⟨ R ; , , − , < , × , − ⟩ . ( P < ) ∀ x ( < x → ∃ y ≠ [ x = y ])( N < ) ∃ u ( − < < < u )( O −× ) ∀ x, y, z ( z < ∧ x < y → y ⋅ z < x ⋅ z ) The axiomatizability of the structure ⟨ Q ; < , ×⟩ seemed to bemissing (or ignored) in the literature. Since ⟨ Q ; < , + , ×⟩ is notdecidable (Corollary 21), one could not immediately infer thedecidability of ⟨ Q ; < , ×⟩ . Also, + is not definable in ⟨ Q ; < , ×⟩ ,this follows from Theorem 26 below, and so Corollary 21 can-not imply its undecidability. The decidability of ⟨ Q ; < , ×⟩ wasproved, and an explicit axiomatization was provided for it, forthe first time in [1, Theorem 7]:7 HEOREM
26 (An Axiomatization for ⟨ Q ; < , ×⟩ ) The theorywith A < , T < , L < (Theorem 5), O × (Theorem 16), A × , C × , U × , I × , Z × (Theorem 22), R e × (Theorem 23), M × (Theorem 24), N < (Theorem 25), along with the following completely axiomatizesthe structure ⟨ Q ; , , − , < , × , − ⟩ . ( D ×< ) {∀ x, y ∃ v ( < x < y → x < v n < y )} n > The axioms M × in Theorem 26 state that Q + is dense in theset of its positive radicals. T HEOREM
27 (Non-Axiomatizability of ⟨ N ; < , ×⟩ , ⟨ Z ; < , ×⟩ ) The full first-order theory of ⟨ N ; < , ×⟩ and ⟨ Z ; < , ×⟩ are notaxiomatizable by any decidable set of sentences. For a proof note that successor and zero are definable in bothof these structures by v = s ( u ) ⇐⇒ u < v ∧¬∃ w [ u < w < v ] and ( u = ) ⇐⇒ u × s ( u ) = u , respectively. So, + is definable in ⟨ N ; < , ×⟩ by Tarski-Robinson’s identity [19]: ( u + v = w ) ⇐⇒ [ u = v = w = ] ∨ [ w ≠ ∧ s ( wu ) s ( wv ) = s ( w s ( uv ))] . Thus, byTheorem 20, ⟨ N ; < , ×⟩ is not axiomatizable; neither is ⟨ Z ; < , ×⟩ since N is definable in it by the formula ⩽ v .The exponential function is not total in Z or Q , even whenthe base is positive: − / ∈ Z and ( / ) / ∈ Q . As for N we take exp ( x, y ) = x y with the convention that = ; and of course x = for every x > . For R and C we consider x ↦ e x forthe Napier-Euler number e in the place of exp ( x ) , since if x is negative, then the value of x y may not exist in R , such as (− ) ( / ) , and even if it exists in C it may not be unique (forexample, ( / ) could be , − , ı ′ , or − ı ′ ); indeed one can takeany positive real number for e . We also add + and × to thelanguage; so, by the real exponential field we mean ⟨ R ; + , × , e x ⟩ and by the complex exponential field we mean ⟨ C ; + , × , e x ⟩ . T HEOREM
28 (Non-Axiomatizability of ⟨ N ; exp ⟩ ) The first-order theory of ⟨ N ; exp ⟩ is not axiomatizable. Since one can define × and + in the structure ⟨ N ; exp ⟩ (see [6,Exercise 1, page 223]) by ( u × v = w ) ⇐⇒ ∀ x [ x w = ( x u ) v ] and ( u + v = w ) ⇐⇒ ∀ x [ x w = ( x u ) × ( x v )] . So, the result followsfrom Theorem 20. T HEOREM
29 (Non-Axiomatizability of ⟨ C ; + , × , e x ⟩ ) Thecomplex exponential field is not axiomatizable.
Indeed, the formula ∀ x, y ( e xy = − x = → e xy ⋅ v = ) defines Z in ⟨ C ; + , × , e x ⟩ , see e.g. [15], since ∀ x ( x = − ↔ x = ± ı ′ ) holds in C , and for every z we have [ e ± ı ′ z = if and only if z = k π for some k ∈ Z ]. The result follows from Corollary 21.One of the most exciting questions in the axiomatizabilitytheory is the question of the axiomatizability of ⟨ R ; + , × , e x ⟩ ,the real exponential field (due to Tarski) which is still open. Aninteresting instance of interaction between seemingly differentareas of mathematics (number theory and logic) is the resultof Macintyre and Wilkie [14] which states that ⟨ R ; + , × , e x ⟩ isaxiomatizable if and only if Weak Schanuel’s Conjecture is true .So, if a computer scientist or a mathematical logician shows the(non-)axiomatizability of ⟨ R ; + , × , e x ⟩ , then weak Schanuel’sconjecture is solved in computational number theory, and if a number theorist solves that problem, then we know whether ⟨ R ; + , × , e x ⟩ is axiomatizable or not. If the conjecture is true,then we have an axiomatization for ⟨ R ; + , × , e x ⟩ which is “quitecomplicated and ugly” according to Marker [15]. § I DENTITIES (O VER + , × , exp IN R + ). First-order sentences can be restricted in at least two ways: onecan consider the sentences of the form( a ) ∃ ⃗ x η (⃗ x ) where η (⃗ x ) is an equation (between two termson ⃗ x and possibly some other parameters); or( b ) ∀ ⃗ x η (⃗ x ) where η (⃗ x ) is as above.The ( a ) formulas are called diophantine equations and areclosely related to Hilbert’s 10th Problem. Since these formulasare discussed elsewhere (see e.g. [2, 5, 18]), here we discussthe formulas in ( b ), which are called identities .For a proof of the parts (i) and (ii) of the following theoremsee e.g. [11]; and for a proof of part (iii), which is due to Martin[16], see e.g. [12, Corollary 3.7]. T HEOREM
30 (Identities With A Single Operation.) (i) The identities of ⟨ R + ; +⟩ are axiomatized by ( A + ) x + ( y + z ) = ( x + y ) + z ( C + ) x + y = y + x (ii) The identities of ⟨ R + ; , ×⟩ are axiomatized by ( A × ) x ⋅ ( y ⋅ z ) = ( x ⋅ y ) ⋅ z ( C × ) x ⋅ y = y ⋅ x ( U × ) x ⋅ = x (iii) The identities of ⟨ R + ; , exp ⟩ are axiomatized by ( C ∧ ) ( x y ) z = ( x z ) y ( Z ∧ ) x = ( U ∧ ) x = x Let us note that / ∈ R + and so the identity ( U + ) x + = x is notexpressible here; and since we do not have − in our language,the identity ( I + ) x + ( − x ) = is not expressible either. The part(I) of the following theorem appears in [11]; for the part (II),which appeared in [16] first, see e.g. [12, Corollary 3.9]. T HEOREM
31 (Identities With Two Operations.) (I) The identities of ⟨ R + ; , + , ×⟩ are axiomatized by A + , C + , A × , C × , U × (Theorem 30) along with the following identity: ( D ×+ ) x ⋅ ( y + z ) = ( x ⋅ y ) + ( x ⋅ z ) (II) The identities of ⟨ R + ; , × , exp ⟩ are axiomatized by A × , C × , U × , Z ∧ , U ∧ (Theorem 30) along with the following identities: ( D ×∧ ) x ( y ⋅ z ) = ( x y ) z ( D ∧× ) ( x ⋅ y ) z = x z ⋅ y z Let us note that the axiom C ∧ (Theorem 30.iii) is provablefrom C × (Theorem 30.ii) and D ×∧ (Theorem 31.II). The axiomsin Theorem 31.I (for { + , × } ) suffices for proving many of thehigh-school identities, such as: ● The Binomial Identity: ( x + y ) n =∑ i ⩽ n ( ni ) x i y n − i , ● ( x + y + ) n =∑ ( i + j ⩽ n ) ( ni + j )( i + ji ) x i y j ,8nd the more difficult one: ( W uv ) : ( A u + B u ) v ( C v + D v ) u = ( A v + B v ) u ( C u + D u ) v , where A ( x ) = x + , B ( x ) = x + x + , C ( x ) = x + , and D ( x ) = x + x + are polynomials on x .Of course the { + , × } -identities in Theorem 31.I can prove ( W uv ) when both u and v are positive natural numbers. We nowshow that the identities of Table 2 derive ( W uv ) when at leastone of u or v is a natural number. So, we assume that say u ∈ N and note that AD = BC = x + x + x + x + x + . We have: ( A u + B u ) v ( C v + D v ) u = ( A u + B u ) v ∑ i ⩽ u ( ui ) C vi D v ( u − i ) =∑ i ⩽ u ( ui )[( A u + B u ) C i D u − i ] v =∑ i ⩽ u ( ui )([( AC ) i ( AD ) u − i ] + [( BC ) i ( BD ) u − i ]) v =∑ i ⩽ u ( ui )([( AC ) i ( BC ) u − i ] + [( AD ) i ( BD ) u − i ]) v =∑ i ⩽ u ( ui )( C u [ A i B u − i ] + D u [ A i B u − i ]) v =∑ i ⩽ u ( ui )([ C u + D u ][ A i B u − i ]) v = ( C u + D u ) v ∑ i ⩽ u ( ui )( A v ) i ( B v ) u − i = ( A v + B v ) u ( C u + D u ) v .Indeed, Wilkie’s identity ( W uv ) is true even when both u, v arevariables: since for E ( x ) = x − x + we have C = AE and D = BE , thus E uv can be factored out from both sides of ( W uv ) .Note that the positive-valued polynomial E is not expressiblein the langauge { , + , × , exp } .Tarski’s High-School Problem asked whether the identitiesof Table 2 could axiomatize all the identities of the positivecone of the real exponential field ⟨ R + ; , + , × , exp ⟩ . It was posedfirst by Doner & Tarski (1969) and was popularized in 1977by Henkin [11] as a then open problem. Wilkie [24] showedin 1981 that ( W uv ) is not derivable from Tarski’s high-schoolidentities when both u and v are variables (see also [8]).Wilkie [24] also proved that the identities of ⟨ R + ; , + , × , exp ⟩ are axiomatizable by a decidable set of identities andGurevi˘c [9] showed that it is not axiomatizable by any finiteset of identities. However, Tarski’s conjecture holds true for awide range of identities.Let us say that a term t over { , + , × , exp } is of level 1 whenfor every sub-term u v of t either u is a variable or u containsno variable; for example, x α + ( + ) β . A term t is of level 2when for every sub-term u v of t we have that u is of level 1;for example the term p ( x ) u + q ( x ) v is of level 2 when p, q arepolynomials of the variable x and u, v are variables. Let us notethat the term ( p ( x ) u + q ( x ) u ) v , which appears in ( W uv ) , is notof level 2 in general. The following theorem is proved in [10,Proposition 4.4.5]: T HEOREM
32 (Tarski’s Conjecture For Terms of Level 2) If ( r = s ) is a valid identity of the structure ⟨ R + ; , + , × , exp ⟩ where r and s are terms of level 2, then ( r = s ) can be provedfrom the identities of Table 2. So, Wilkie’s result [24] (Theorem 33 below) is a boundaryresult, since some terms in Wilkie’s identity ( W uv ) are of level 3(which are the terms with the property that for every sub-term u v of them, u is a term of level 2). T HEOREM
33 (Tarski’s Conjecture Not for Higher Levels)
The identity ( W uv ) holds in ⟨ R + ; , + , × , exp ⟩ but is not provablefrom the identities of Table 2 when u, v, x are all variables. § A PPENDIX (An Axiomatization for ⟨ N + ; ×⟩ ). An axiomatization for ⟨ N + ; , ×⟩ was presented in [4] whoseproofs are available only in French; an English exposition ofthe axioms without any proof appears in [23, § III.5]. We needthe following notation for presenting the axioms: y ⊑ x ⇐⇒ ∃ w ( y ⋅ w = x ) , P ( x ) ⇐⇒ x ≠ ∧∀ y ( y ⊑ x → y = ∨ y = x ) , R ( x, y ) ⇐⇒ P ( x ) ∧ x ⊑ y ∧ ∀ z ( P ( z ) ∧ z ≠ x → z / ⊑ y ) , and V ( x, y, z ) ⇐⇒ R ( x, z ) ∧ z ⊑ y ∧∀ w ( R ( x, w ) ∧ w ⊑ y → w ⊑ z ) ;which state, respectively, that “ y divides x ”, “ x is a prime”, “ y is a power of the prime x ”, and “ z is the largest power of theprime x that divides y ”. Here are C´egielski’s axioms ([4]): ( A × ) ∀ x, y, z ( x ⋅ ( y ⋅ z ) = ( x ⋅ y ) ⋅ z )( C × ) ∀ x, y ( x ⋅ y = y ⋅ x )( U × ) ∀ x ( x ⋅ = x )( C × ) ∀ x, y, z ( x ⋅ y = x ⋅ z → y = z )( U × ) ∀ x, y ( x ⋅ y = → x = y = )( D × ) { ∀ x, y ( x n = y n → x = y )} n > ( E × ) { ∀ x ∃ u, v ( x = u n v ∧ ∀ y, z [ x = y n z → v ⊑ z ])} n > ( P × ) ∀ x ∃ v ( P ( v ) ∧ v / ⊑ x )( R × ) ∀ u, x, y ( R ( u, x ) ∧ R ( u, y ) → x ⊑ y ∨ y ⊑ x )( V ∃ ) ∀ u, x [ P ( u ) →∃ v V ( u, x, v )]( V ⊑ ) ∀ x, y ( ∀ u, v, w [ P ( u ) ∧ V ( u, x, v ) ∧ V ( u, y, w ) → v ⊑ w ] —→ x ⊑ y )( V × ) ∀ x, y ( ∀ u, v, w [ P ( u ) ∧ V ( u, x, v ) ∧ V ( u, y, w ) —→ V ( u, x ⋅ y, v ⋅ w )])( T × ) ∀ x, y ∃ z ∀ u ( P ( u ) —→ [ u / ⊑ x → V ( u, z, )] ∧ [ u ⊑ x →∀ v { V ( u, z, v ) ↔ V ( u, y, v ) } ])( S × ) { ∀ x, y ∃ z ∀ u ( P ( u ) —→ [ u ⊑ x ⋅ y ∧∃ v, w { V ( u, x, v ) ∧ V ( u, y, w n v ) }→ V ( u, z, u )] ∧ [ ¬( u ⊑ x ⋅ y ∧∃ v, w { V ( u, x, v ) ∧ V ( u, y, w n v ) })→ V ( u, z, )])} n > By U × , C × , and U × the relation ⊑ is anti-symmetric: if a ⊑ b ⊑ a , then a = b . For every prime u and every x there existssome v , by V ∃ , such that V ( u, x, v ) . That v is unique by V ⊑ ;so let us denote it by V( u, x ) . So, if u ranges over the primes,then x = ∏ u ⊑ x V( u, x ) . Thus, V × is equivalent to V( u, xy ) = V( u, x )V( u, y ) ; and the number z in T × is ∏ u ⊑ x V( u, y ) .The axiom S × states the existence of ∏ [ u ⊑ xy, V( u,x ) ⊑ n V( u,y )] u ,where a ⊑ n b is by definition ∃ w ( aw n = b ) . Finally, we note thatthe following sentences are provable from the axioms: ( V = ) ∀ x, y ( ∀ u [ P ( u ) → V( u, x ) = V( u, y )] —→ x = y )( I × ) ∀ x ∃ w ∀ u ( P ( u ) → [ u / ⊑ x → V( u, w ) = ] ∧ [ u ⊑ x → V( u, w ) = u V( u, x )])( P ×∃ ) ∀ x ( x ≠ → ∃ u [ P ( u ) ∧ u ⊑ x ]) In fact, V = follows from V ⊑ , and I × follows from S × byputting w = xz where z is stated to exist by S × for x = y, n = .Indeed, V = is the axiom A in [4] ( V in [23]), and I × is theaxiom A in [4] ( I in [23]) which, as we saw, are redundant.For P ×∃ we note that if no prime divides α ≠ , then V( u, α ) = for every prime u ; so by V ⊑ we have α ⊑ y for every y , and thiscontradicts P × (by which there are infinitely many primes).9 eferences [1] A SSADI , Z
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Acknowledgements:
This research was partially supportedby a grant from
IPM (No. 98030022).S
AEED S ALEHI
Research Institute for Fundamental Sciences (RIFS),University of Tabriz, 29 Bahman Boulevard,P.O.Box 51666–17766, Tabriz, Iran.School of Mathematics,Institute for Research in Fundamental Sciences (
IPM ),P.O.Box 19395–5746, Tehran, Iran. [email protected]://saeedsalehi.ir/[email protected]://saeedsalehi.ir/