UUniversity of Tabriz F ACULTY OF M ATHEMATICAL S CIENCES U NIVERSITY OF T ABRIZ , IRAN
Decidability of the Multiplicativeand Order Theory of Numbers
Author:
Ziba A
SSADI
Supervisor:
Saeed S
ALEHI
A Thesis Submitted in Partial Fulfillment of the Requirementsfor the degree of D OCTOR OF P HILOSOPHY (Ph.D.) in Pure Mathematics ( Mathematical Logic )January 2019 a r X i v : . [ m a t h . L O ] S e p i Dedicated To My Supervisor ,Professor S
A E E D S A L E H I , to whom I owe much more thanwhat I can ever express . ii Contents
Acknowledgements vAbstract viIntroduction 11 Some Preliminaries 4 (cid:104) R ; < (cid:105) and (cid:104) Q ; < (cid:105) . . . . . . . . 82.1.2 Finite Axiomatizability of (cid:104) Z ; < (cid:105) . . . . . . . . . . . . . . 92.1.3 Finite Axiomatizability of (cid:104) N ; < (cid:105) . . . . . . . . . . . . . . 12 (cid:104) R ; < , + (cid:105) and (cid:104) Q ; < , + (cid:105) . . . . . 183.2.2 Non-finite Axiomatizability of (cid:104) R ; < , + (cid:105) and (cid:104) Q ; < , + (cid:105) . . 193.3 The Chinese Remainders . . . . . . . . . . . . . . . . . . . . . . . 193.3.1 The Bézout’s Theorem . . . . . . . . . . . . . . . . . . . . 203.3.2 The Chinese Remainder Theorem . . . . . . . . . . . . . . 213.3.3 The Generalized Chinese Remainder Theorem . . . . . . . 223.4 Integer Numbers with Order and Addition . . . . . . . . . . . . . . 263.4.1 Quantifier Elimination of (cid:104) Z ; < , + (cid:105) . . . . . . . . . . . . . 263.4.2 Non-finite Axiomatizability of (cid:104) Z ; < , + (cid:105) . . . . . . . . . . 293.5 Natural Numbers with Order and Addition . . . . . . . . . . . . . . 303.5.1 Axiomatization of (cid:104) N ; < , + (cid:105) . . . . . . . . . . . . . . . . . 303.5.2 Decidability of (cid:104) N ; < , + (cid:105) . . . . . . . . . . . . . . . . . . 31 (cid:104) N ; < , ×(cid:105) . . . . . . . . . . . . . 324.2 Integer numbers with order and multiplication . . . . . . . . . . . . 334.2.1 Non-Axiomatizability of (cid:104) Z ; < , ×(cid:105) . . . . . . . . . . . . . 334.3 Real numbers with order and multiplication . . . . . . . . . . . . . 344.3.1 Axiomatization and Quantifier Elimination of (cid:104) R ; < , ×(cid:105) . . 35v 4.3.2 Non-finite Axiomatizability of (cid:104) R ; < , ×(cid:105) . . . . . . . . . . 374.4 Rational numbers with order and multiplication . . . . . . . . . . . 384.4.1 Quantifier Elimination of (cid:104) Q ; < , ×(cid:105) . . . . . . . . . . . . . 384.4.2 Non-finite Axiomatizability of (cid:104) Q ; < , ×(cid:105) . . . . . . . . . . 44 Bibliography 49Index 52
Acknowledgements
In the Name of the Creator of Science, Mathematics and Logic
First and foremost, I would like to express my most grateful thanks to my supervisor,to whom this thesis is dedicated wholeheartedly, for teaching me a lot and taking myhands in the hard moments of wandering in the wonderland of science and research.I also thank my advisor Professor Jafarsadegh Eivazloo for studying this thesisand for teaching me.I thank Professors Mohammad Bagheri and Mohammad Shahriari and Jaber Karim-poor for refereeing the thesis and for their fruitful comments and suggestions.Last but not the least, I am grateful to my parents for their unending love and tomy brother and sister for being there when I needed them most.
Ziba A ssadi , i Abstract
Ziba A
SSADI
Decidability of the Multiplicativeand Order Theory of Numbers
The ordered structures of natural, integer, rational and real numbers are studied inthis thesis. The theories of these numbers in the language of order are decidable andfinitely axiomatizable. Also, their theories in the language of order and addition aredecidable and infinitely axiomatizable. For the language of order and multiplication,it is known that the theories of N and Z are not decidable (and so not axiomatizableby any computably enumerable set of sentences). By Tarski’s theorem, the multi-plicative ordered structure of R is decidable also. In this thesis we prove this resultdirectly by quantifier elimination and present an explicit infinite axiomatization. Thestructure of Q in the language of order and multiplication seems to be missing in theliterature. We show the decidability of its theory by the technique of quantifier elimi-nation and after presenting an infinite axiomatization for this structure, we prove thatit is not finitely axiomatizable. Keywords : Decidability, Undecidability, Completeness, Incompleteness, First-OrderTheory, Quantifier Elimination, Ordered Structures.
Introduction
Entscheidungsproblem , one of the fundamental problems of (mathematical) logic,asks for a single-input Boolean-output algorithm that takes a formula ϕ as input andoutputs ‘yes’ if ϕ is logically valid and outputs ‘no’ otherwise. Now, we know thatthis problem is not (computably) solvable. One reason for this is the existence of anessentially undecidable and finitely axiomatizable theory, see e.g. [20]; for anotherproof see [3, Theorem 11.2]. However, by Gödel’s completeness theorem, the setof logically valid formulas is computably enumerable, i.e., there exists an input-freealgorithms that (after running) lists all the valid formulas (and nothing else). Forthe structures, since their theories are complete, the story is different: the theory of astructure is either decidable or that structure is not axiomatizable (by any computablyenumerable set of sentences; see e.g. [7, Corollaries 25G and 26I] or [12, Theorem15.2]). Axiomatizability or decidability of theories of natural, integer, rational, realand complex numbers in different languages have long been considered by logiciansand mathematicians. For example, the additive theory of natural numbers (cid:104) N ; + (cid:105) was shown to be decidable by Presburger in 1929 (and by Skolem in 1930; see [19]).The multiplicative theory of the natural numbers (cid:104) N ; ×(cid:105) was announced to be de-cidable by Skolem in 1930. Then it was expected that the theory of addition andmultiplication of natural numbers would be decidable too; confirming Hilbert’s Pro-gram. But the world was shocked in 1931 by Gödel’s incompleteness theorem whichimplies that the theory of (cid:104) N ; + , ×(cid:105) is undecidable (see the subsection 4.1 below).In this thesis we study the theories of the sets N , Z , Q and R in the languages { < } , { < , + } and { < , ×} ; see the table below. N Z Q R { < } Thm. 2.1.22 Thm. 2.1.16 Thm. 2.1.11 Thm. 2.1.11 { < , + } Thm. 3.5.3 Thm. 3.4.3 Thm. 3.2.1 Thm. 3.2.1 { < , ×} Prop. 4.1.1 Prop. 4.2.2 Cor. 4.4.10 Thm. 4.3.3 { + , ×} [7] Prop. 4.2.2 Prop. 4.4.12 Subsec. 4.3Let us note that order is definable in the language { + , ×} in these sets: in N by x < y ⇐⇒ ∃ z ( z + z (cid:54) = z ∧ x + z = y ) , and in Z by Lagrange’s four square theorem x < y is equivalent with ∃ t , u , v , w ( x (cid:54) = y ∧ x + t · t + u · u + v · v + w · w = y ) . The four squaretheorem holds in Q too: for any p / q ∈ Q + we have pq > pq = a + b + c + d forsome integers a , b , c , d ; therefore, p / q = pq / q = ( a / q ) +( b / q ) +( c / q ) +( d / q ) holds. Thus, the same formula defines the order ( x < y ) in Q as well. Finally, in R the relation x < y is equivalent with the formula ∃ z ( z + z (cid:54) = z ∧ x + z · z = y ) .The decidability of N , Z , Q , R in the languages { < } and { < , + } is alreadyknown. It is also known that the theories of N and Z in the language { < , ×} areundecidable, because the addition operation is definable in the multiplicative orderedstructure of natural numbers by Tarski-Robinson’s identity. Whence, the theory of (cid:104) N ; × , < (cid:105) is undecidable. This also holds for the domain of the integer numbers,since the addition operation is definable in (cid:104) Z ; × , < (cid:105) which implies the undecid-ability of the theory of (cid:104) Z ; × , < (cid:105) . The theory of R in the language { < , ×} is de-cidable by Tarski-Seidenberg’s theorem which states the decidability of the theoryof (cid:104) R ; < , + , ×(cid:105) by showing that (cid:104) R ; < , + , ×(cid:105) is aximatizable by the theory of realclosed ordered fields. Indeed, no heavy algebraic tools are needed for axiomatizingthe multiplicative order theory of the real numbers, (cid:104) R ; × , < (cid:105) . The proof of Tarski’stheorem appears in a few number of logic books; see e.g. [1] and [10]. Interestingly,the algebraic-geometric proof is more beautiful and more clever; see e.g. [4] and [5].Although this theorem of Tarski implies the decidability of (cid:104) R ; × , < (cid:105) , it does notpresent an explicit axiomatization for this structure. Here, we prove this directlyby presenting an explicit axiomatization. Finally, the structure (cid:104) Q ; < , ×(cid:105) is studiedin this thesis (seemingly, for the first time). We show, by the method of quantifierelimination, that the theory of this structure is decidable. Here, the (super-)structure (cid:104) Q ; + , ×(cid:105) is not usable since it is undecidable (proved by Robinson [16]; see also [19,Theorem 8.30]). On the other hand its (sub-)structure (cid:104) Q ; ×(cid:105) is decidable (provedin [13] by Mostowski; see also [17]). So, the three structures (cid:104) Q ; + , ×(cid:105) and (cid:104) Q ; < , ×(cid:105) and (cid:104) Q ; ×(cid:105) are different from each other; the order relation < is not definable in (cid:104) Q ; ×(cid:105) and the addition operation + is not definable in (cid:104) Q ; < , ×(cid:105) (by our results). Chapter 1
Some Preliminaries
Definition 1.1.1 (Ordered Structure) An ordered structure is a triple (cid:104) A ; < , L (cid:105) inwhich A is a non-empty set and < is a binary relation on A which satisfies the fol-lowing axioms: ( O ) ∀ x , y ( x < y → y (cid:54) < x ) ,( O ) ∀ x , y , z ( x < y < z → x < z ) , and( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ;and L is a first-order language. ⊗⊕ Here, L could be empty, or any language, for example { + } or {×} or { + , ×} . Definition 1.2.1 (Dense Linear Order)
A linear order relation < is called dense ifit satisfies ( O ) ∀ x , y ( x < y → ∃ z [ x < z < y ]) . ⊗⊕ .3. The Main Lemma of Quantifier Elimination Definition 1.2.2 (Orders Without Endpoints)
An order relation < is called with-out endpoints if it satisfies ( O ) ∀ x ∃ y ( x < y ) , and( O ) ∀ x ∃ y ( y < x ) . ⊗⊕ Definition 1.2.3 (Discrete Order) A discrete order has the property that any ele-ment has an immediate successor (i.e., there is no other element in between them). Ifthe successor of x is denoted by s ( x ) , then a discrete order satisfies( O ) ∀ x , y ( x < y ↔ s ( x ) < y ∨ s ( x ) = y ) . ⊗⊕ Convention 1.2.4
The successor of an integer x is s ( x ) = x + (cid:126) Definition 1.3.1 (Disjunctive Normal Form)
The disjunctive normal form of a for-mula is another formula such that (i) is equivalent to the original formula, and (ii) isthe disjunction of some formulas each of wich is the conjunction of some atomic ornegated-atomic formulas. ⊗⊕ Remark 1.3.2
Every quantifier-free formula can be written equivalently in disjunc-tive normal form by elimination of connectives other than {∨ , ∧ , ¬} using DeMor-gan’s laws and the double negation rule, and distributing ∧ over ∨ , if any. (cid:126) The following lemma which is known as “
The Main Lemma of Quantifier Elimi-nation ”, has been proved in e.g. [7, Theorem 31F], [9, Lemma 2.4.30], [10, Theorem1, Chapter 4], [11, Lemma 3.1.5] and [19, Lemma 4.1, Chapter III].
Chapter 1. Some Preliminaries
Lemma 1.3.3 (The Main Lemma of Quantifier Elimination)
A theory (or a struc-ture) admits quantifier elimination if and only if every formula of the form ∃ x ( (cid:86)(cid:86) i α i ) is equivalent with a quantifier-free formula, where each α i is an atomic formula orthe negation of an atomic formula. Proof.
The “only if” part is obvious. We prove the “if” part by induction on thecomplexity of ϕ . The statement holds for quantifier-free formulas. So it suffices tocheck quantifiers: ∀ and ∃ . By the equivalence ∀ x ϕ ≡ ¬∃¬ ϕ , the universal quantifieris reducible to the existential quantifier. Therefore, the quantifier elimination of theformula ∃ x ϕ suffices, where ϕ is quantifier-free. Now, by Convention 1.3.2, everyquantifier-free formula can be written in the conjunctive normal form. So we have: ∃ x ϕ ≡ ∃ x (cid:95)(cid:95) j ( (cid:94)(cid:94) i α i , j ) ≡ (cid:95)(cid:95) j ( ∃ x ( (cid:94)(cid:94) i α i , j )) By the assumption, each formula ∃ x ( (cid:86)(cid:86) i α i , j ) is equivalent with a quantifier-free for-mula. So, the formula ∃ x ϕ is also equivalent with a quantifier-free formula. (cid:2)(cid:1) Remark 1.3.4
In the presence of a linear order relation ( < ) by the two equivalences ( s (cid:54) = t ) ↔ ( s < t ∨ t < s ) and ( s (cid:54) < t ) ↔ ( t < s ∨ t = s ) , which follow from theaxioms { O , O , O } (of Definition 1.1.1), we do not need to consider the negatedatomic formulas (when there is no relation symbol other than < , = ). (cid:126) Chapter 2
Ordered Structures of Numbers
Definition 2.1.1 (Theory) A theory is a set of sentences which is closed under thelogical deduction. ⊗⊕ Definition 2.1.2 (Complete theory)
A theory T is said to be complete if for everysentence σ either σ ∈ T or ( ¬ σ ) ∈ T . ⊗⊕ Remark 2.1.3
Since the theory of a structure is a set of sentences which are satisfiedwithin that structure, this theory is complete. (cid:126)
Definition 2.1.4 (Decidable set)
A set A of expressions is decidable if and only ifthere exists an effective procedure that, given an expression α , will decide whetheror not α ∈ A . ⊗⊕ Definition 2.1.5 (Effectively enumerable set)
A set A of expressions is effectivelyenumerable if and only if there exists an effective procedure that lists, in some order,the members of A . ⊗⊕ Definition 2.1.6 (Axiomatizability)
The theory of a structure A = (cid:104) A ; L (cid:105) is ax-iomatizable if and only if there exists a decidable set of L − sentences such that theset of its logical consequences is equal to the theory of A . ⊗⊕• The structure A is finitely axiomatizable if the above set of sentences is finite. Chapter 2. Ordered Structures of Numbers
Proposition 2.1.7
For a finite or countable language:(1) An axiomatizable theory is effectively enumerable.(2) A complete axiomatizable theory is decidable.
Proof.
These results have been proved in e.g. [7, Corollaries 25F and 25G]. (cid:2)(cid:1)
Remark 2.1.8
By Remark 2.1.3 and Proposition 2.1.7 the theory of an axiomatiz-able structure is decidable. ⊗⊕ Definition 2.1.9 (The theory of) A structure A = (cid:104) A ; L (cid:105) admits quantifier elimina-tion if and only if every formula in the language L is equivalent to a quantifier-freeformula in the same language with the same free variables. ⊗⊕• Since every atom can be proved or disproved, so can the quantifier-free sentences.Whence, the
Quantifier Elimination Algorithm is in fact a
Decision Algorithm . • Here, we have presented axiomatizations for structures and have eliminated thequantifiers of their theories. Whence, axiomatizability and decidability of the struc-tures are proved this way. (cid:104) R ; < (cid:105) and (cid:104) Q ; < (cid:105) Convention 2.1.10
The axioms of The Finite Theory of Dense Linear Orders With-out Endpoints are as follows:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( O ) ∀ x , y ( x < y → ∃ z [ x < z < y ]) ( O ) ∀ x ∃ y ( x < y ) ( O ) ∀ x ∃ y ( y < x ) (cid:126) .1. Axiomatizability and Quantifier Elimination • Here, we present a syntactic (proof-theoretic) proof.
Theorem 2.1.11
The finite theory of dense linear orders without endpoints (with theaxioms { O , O , O , O , O , O } ) completely axiomatizes the order theory of the real andrational numbers and, moreover, the structures (cid:104) R ; < (cid:105) and (cid:104) Q ; < (cid:105) admit quantifierelimination, and so their theories are decidable. Proof.
By Remark 1.3.4, all the atomic formulas are either of the form u < v or u = v for some variables u and v . If both of the variables are equal then u < u is equivalentwith ⊥ by O and u = u is equivalent with (cid:62) . So, by Lemma 1.3.3, it suffices toeliminate the quantifier of the formulas of the form ∃ x ( (cid:94)(cid:94) i <(cid:96) y i < x ∧ (cid:94)(cid:94) j < m x < z j ∧ (cid:94)(cid:94) k < n x = u k ) (2.1)where y i ’s, z j ’s and u k ’s are variables.Now, if n (cid:54) = (cid:94)(cid:94) i <(cid:96) y i < u ∧ (cid:94)(cid:94) j < m u < z j ∧ (cid:94)(cid:94) k < n u = u k .So, let us suppose that n =
0. Then if (cid:96) = m =
0, the formula (2.1) is equivalentwith the quantifier-free formula (cid:62) , by the axioms O and O (with O and O ) respec-tively, and if (cid:96) , m (cid:54) =
0, it is equivalent with the quantifier-free formula (cid:86)(cid:86) i <(cid:96) , j < m y i < z j by the axiom O (with O and O ). (cid:2)(cid:1) Corollary 2.1.12
In fact, for any set A such that Q ⊆ A ⊆ R , the structure (cid:104) A ; < (cid:105) can be completely axiomatized by the finite set of axioms { O , O , O , O , O , O } . (cid:2)(cid:1) (cid:104) Z ; < (cid:105) Proposition 2.1.13
The theory of the structure (cid:104) Z ; < (cid:105) does not admit quantifierelimination.0 Chapter 2. Ordered Structures of Numbers
Proof.
We show that the formula ∃ x ( y < x < z ) is not equivalent with any quantifier-free formula in the language { < } (note that it is not equivalent with y < z ): allthe atomic formulas with the free variables y and z are y < z , z < y , y = y ( ≡ (cid:62) ) , z = z ( ≡ (cid:62) ) , y < y ( ≡ ⊥ ) and z < z ( ≡ ⊥ ) . None of the propositional compositions ofthese formulas can be equivalent to the formula ∃ x ( y < x < z ) . (cid:2)(cid:1) Remark 2.1.14
If we add the successor operation s to the language, we will have: ∃ x ( y < x < z ) ⇐⇒ s ( y ) < z ,and we will show that the process of quantifier elimination will go through in thislanguage [Theorem 2.1.16]. (cid:126) Convention 2.1.15
The axioms of The Finite Theory of Discrete Linear Orders With-out Endpoints are as follows:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( O ) ∀ x , y ( x < y ↔ s ( x ) < y ∨ s ( x ) = y ) ( O ) ∀ x ∃ y ( s ( y ) = x ) (cid:126) • The following has been proved earlier; see [15, Theorem 2.12].
Theorem 2.1.16
The finite theory of discrete linear orders without endpoints, con-sisting of the axioms { O , O , O , O , O } , completely axiomatizes the order theory ofthe integer numbers and, moreover, the structure (cid:104) Z ; < , s (cid:105) admits quantifier elimina-tion, and so its theory is decidable. Proof.
We note that all the terms in the language { < , s } are of the form s n ( y ) forsome variable y and n ∈ N . So, by Remark 1.3.4, all the atomic formulas are of .1. Axiomatizability and Quantifier Elimination s n ( u ) = s m ( v ) or s n ( u ) < s m ( v ) , for some variables u , v . If a variable x appears in the both sides of an atomic formula, then we have either s n ( x ) = s m ( x ) or s n ( x ) < s m ( x ) . The formula s n ( x ) = s m ( x ) is equivalent with (cid:62) when n = m andwith ⊥ otherwise; also s n ( x ) < s m ( x ) is equivalent with (cid:62) when n < m and with ⊥ otherwise. So, it suffices to consider the atomic formulas of the form t < s n ( x ) or s n ( x ) < t or s n ( x ) = t , for some x -free term t and n ∈ N + . Now, by Lemma 1.3.3,we eliminate the quantifier of the following formulas ∃ x ( (cid:94)(cid:94) i <(cid:96) t i < s p i ( x ) ∧ (cid:94)(cid:94) j < m s q j ( x ) < s j ∧ (cid:94)(cid:94) k < n s r k ( x ) = u k ) . (2.2)The axiom O proves [ a < b ] ↔ [ s ( a ) < s ( b )] and [ a = b ] ↔ [ s ( a ) = s ( b )] ; so wecan assume that p i ’s and q j ’s and r k ’s in the formula (2.2) are equal to each other, sayto α . Then, by O , the formula (2.2) is equivalent with ∃ y ( (cid:94)(cid:94) i <(cid:96) t (cid:48) i < y ∧ (cid:94)(cid:94) j < m y < s (cid:48) j ∧ (cid:94)(cid:94) k < n y = u (cid:48) k ) , (2.3)for some (possibly new) terms t (cid:48) i , s (cid:48) j , u (cid:48) k (and y = s α ( x ) ).Now, if n (cid:54) =
0, then the formula (2.3) is equivalent with the quantifier-free formula (cid:94)(cid:94) i <(cid:96) t (cid:48) i < u (cid:48) ∧ (cid:94)(cid:94) j < m u (cid:48) < s (cid:48) j ∧ (cid:94)(cid:94) k < n u (cid:48) = u (cid:48) k .Let us then assume that n =
0. The formula ∃ x ( (cid:94)(cid:94) i <(cid:96) t i < x ∧ (cid:94)(cid:94) j < m x < s j ) (2.4)is equivalent with the quantifier-free formula (cid:94)(cid:94) i , j s ( t i ) < s j by the axiom O . (cid:2)(cid:1) Chapter 2. Ordered Structures of Numbers (cid:104) N ; < (cid:105) Proposition 2.1.17
The theory of the structure (cid:104) N ; < (cid:105) does not admit quantifierelimination. Proof.
We show that the formula ∃ x ( s ( x ) = y ) is not equivalent with any quantifier-free formula. All the atomic formulas with the free variable y are either of the form y < y or y = y . The equivalences ( y < y ) ≡ ⊥ and ( y = y ) ≡ (cid:62) show that none ofthe propositional compositions of them can be equivalent to ∃ x ( s ( x ) = y ) , becauseits truth depends on y (it is equivalent with ⊥ for y = (cid:62) otherwise). (cid:2)(cid:1) Remark 2.1.18
By adding the constant to the language { < } we will have: ∃ x ( x < y ) ⇐⇒ < y .Still quantifier elimination is not possible [Proposition 2.1.19, below]. (cid:126) Proposition 2.1.19
The theory of the structure (cid:104) N ; < , (cid:105) does not admit quantifierelimination. Proof.
It suffices to show that the formula ∃ x ( y < x < z ) is not equivalent with anyquantifier-free formula. All the atomic formulas with the free variables y and z are y = z =
0, 0 < y , 0 < z , y = y ( ≡ (cid:62) ) , z = z ( ≡ (cid:62) ) , y < y ( ≡ ⊥ ) , z < z ( ≡ ⊥ ) , y = z , z = y , z < y and y < z . None of the propositional compositions of these formulas canbe equivalent with the formula ∃ x ( y < x < z ) . (cid:2)(cid:1) Remark 2.1.20
If we add the successor operation s to the language { < } we willhave: ∃ x ( y < x < z ) ⇐⇒ s ( y ) < z ,and now we show that the quantifier elimination is still not possible in the language { < , s } [Proposition 2.1.21, below]. (cid:126) .1. Axiomatizability and Quantifier Elimination Proposition 2.1.21
The theory of the structure (cid:104) N ; < , s (cid:105) does not admit quantifierelimination. Proof.
We show that the formula ∃ x ( s ( x ) = y ) is not equivalent with any quantifier-free formula. All the atomic formulas with the free variable y are either of the form s n ( y ) < s m ( y ) or s n ( y ) = s m ( y ) which do not depend on y and are equivalent to either (cid:62) or ⊥ . So, the formula ∃ x ( s ( x ) = y ) (which is equivalent with ⊥ for y = (cid:62) otherwise) is not equivalent with any quantifier-free { < , s } -formula. (cid:2)(cid:1) In the following we will show the quantifier elimination of the theory of the structure (cid:104) N ; < , s , (cid:105) . This theorem has been proved in [7, Theorem 32A]. Theorem 2.1.22
The following axioms completely axiomatize the order theory ofthe ordered natural numbers:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( O ) ∀ x , y ( x < y ↔ s ( x ) < y ∨ s ( x ) = y ) ( O ◦ ) ∀ x ∃ y ( x (cid:54) = → s ( y ) = x ) ( O ) ∀ x ( x (cid:54) < ) and, moreover, the structure (cid:104) N ; < , s , (cid:105) admits quantifier elimination, and so itstheory is decidable. Proof.
All the atomic formulas of the free variable u in the language { < , s , } areof the form s n ( u ) = s m ( u ) or s n ( u ) < s m ( u ) or s n ( ) = s m ( u ) or s n ( ) < s m ( u ) or s n ( u ) < s m ( ) . The formula s n ( u ) = s m ( u ) is equivalent with (cid:62) when n = m andwith ⊥ otherwise; also s n ( u ) < s m ( u ) is equivalent with (cid:62) when n < m and with ⊥ otherwise. So, it suffices to consider the atomic formulas of the form t < s n ( x ) or s n ( x ) < t or s n ( x ) = t for some x -free term t and n ∈ N + . Now, by Lemma 1.3.3 andthe presence of < , which eliminates the negation already, we eliminate the quantifier4 Chapter 2. Ordered Structures of Numbers of the following formulas ∃ x ( (cid:94)(cid:94) i <(cid:96) t i < s p i ( x ) ∧ (cid:94)(cid:94) j < m s q j ( x ) < s j ∧ (cid:94)(cid:94) k < n s r k ( x ) = u k ) . (2.5)By the provable formulas s ( x ) < s ( y ) ⇔ x < y and s ( x ) = s ( y ) ⇔ x = y ,the formula (2.5), for N = max { p i , q j , r k } , is equivalent with ∃ x (cid:16) (cid:94)(cid:94) i <(cid:96) s N − p i ( t i ) < s N ( x ) ∧ (cid:94)(cid:94) j < m s N ( x ) < s N − q j ( s j ) ∧ (cid:94)(cid:94) k < n s N ( x ) = s N − r k ( u k ) (cid:17) . (2.6)Now for y = s N ( x ) , t (cid:48) i = s N − p i ( t i ) , s (cid:48) j = s N − q j ( s j ) and u (cid:48) k = s N − r k ( u k ) the for-mula (2.6) is equivalent with ∃ y ( (cid:94)(cid:94) i <(cid:96) t (cid:48) i < y ∧ (cid:94)(cid:94) j < m y < s (cid:48) j ∧ (cid:94)(cid:94) k < n y = u (cid:48) k ∧ s N ( ) (cid:54) y ) .So, it suffices to eliminate the quantifiers of the following formulas: ∃ y ( (cid:94)(cid:94) i <(cid:96) t i < y ∧ (cid:94)(cid:94) j < m y < s j ∧ (cid:94)(cid:94) k < n y = u k ) . (2.7)If n (cid:54) =
0, then the formula (2.7) is equivalent with the following quantifier-free for-mula: (cid:94)(cid:94) i <(cid:96) t i < u ∧ (cid:94)(cid:94) j < m u < s j ∧ (cid:94)(cid:94) k < n u = u k .And, if n =
0, then we eliminate the quantifier of: ∃ y ( (cid:94)(cid:94) i <(cid:96) t i < y ∧ (cid:94)(cid:94) j < m y < s j ) . (2.8) .1. Axiomatizability and Quantifier Elimination (cid:96) =
0, then the formula (2.8) is equivalent with the following quantifier-freeformula: (cid:94)(cid:94) j < m < s j .If m =
0, then the formula (2.8) is equivalent with (cid:62) .Finally, if (cid:96) (cid:54) = (cid:54) = m , then the formula (2.8) is equivalent with the following quantifier-free formula: (cid:94)(cid:94) i , j s ( t i ) < s j . (cid:2)(cid:1) Chapter 3
Additive Ordered Structures
In this chapter, we study the structures of the sets N , Z , Q , R over the language { + , < } . Definition 3.1.1 (Group) A group is a structure (cid:104) G ; ∗ , e , ι (cid:105) , where ∗ is a binary op-eration on G , e is a constant (a special element of G ) and ι is a unary operation on G ,which satisfy the following axioms: ∀ x , y , z [ x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ z ] ; ∀ x ( x ∗ e = x ) ; ∀ x ( x ∗ ι ( x ) = e ) . ⊗⊕• A group is called non-trivial when ∃ x ( x (cid:54) = e ) . Definition 3.1.2 (Abelian group)
A group is called abelian when it satisfies thecommutativity axiom: ∀ x , y ( x ∗ y = y ∗ x ) . ⊗⊕ .1. Some Group Theory Definition 3.1.3 (Divisible group)
A group is called divisible when for any n ∈ N + we have ∀ x ∃ y [ x = y ∗ · · · ∗ y (cid:124) (cid:123)(cid:122) (cid:125) n-times ] . ⊗⊕ Definition 3.1.4 (Ordered group) An ordered group is a group equipped with anorder relation < (which satisfies O , O , O ) such that also the axiom ∀ x , y , z ( x < y → x ∗ z < y ∗ z ∧ z ∗ x < z ∗ y ) is satisfied in it. ⊗⊕ Remark 3.1.5
The axioms of The Theory of Non-trivial Ordered Divisible AbelianGroups in the language L = { < , + , − , 0 } are as follows:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( A ) ∀ x , y , z ( x + ( y + z ) = ( x + y ) + z ) ( A ) ∀ x ( x + = x ) ( A ) ∀ x ( x + ( − x ) = ) ( A ) ∀ x , y ( x + y = y + x ) ( A ) ∀ x , y , z ( x < y → x + z < y + z ) ( A ) ∃ y ( y (cid:54) = ) ( A ) ∀ x ∃ y ( x = n (cid:5) y ) n ∈ N + (cid:126) Chapter 3. Additive Ordered Structures (cid:104) R ; < , + (cid:105) and (cid:104) Q ; < , + (cid:105) Theorem 3.2.1
The infinite theory of non-trivial ordered divisible abelian groupscompletely axiomatizes the order and additive theory of the real and rational num-bers and, moreover, the structures (cid:104) R ; < , + , − , (cid:105) and (cid:104) Q ; < , + , − , (cid:105) admit quantifierelimination, and so their theories are decidable [11, Corollary 3.1.17]. Proof.
Firstly, let us note that O , O and O can be proved from the presented axioms:if a < b then by A there exists some c such that c + c = a + b ; one can easily showthat a < c < b holds. Thus O is proved; for O note that for any < a we have a < a + a by A . A dual argument can prove the axiom O . Also, the equivalences(i) [ a < b ] ↔ [ n (cid:5) a < n (cid:5) b ] and(ii) [ a = b ] ↔ [ n (cid:5) a = n (cid:5) b ] can be proved from the axioms: (i) follows from A (with O , O , O ) and (ii) followsfrom ∀ x ( n (cid:5) x = → x = ) which is derived from A (with O , O , O ).Secondly, every term containing x is equal to n (cid:5) x + t for some x -free term t and n ∈ Z −{ } . So, every atomic formula containing x is equivalent with n (cid:5) x (cid:3) t where (cid:3) ∈ { = , < , > } . Whence, by Remark 1.3.3, it suffices to prove the equivalence of theformula ∃ x ( (cid:94)(cid:94) i <(cid:96) t i < p i (cid:5) x ∧ (cid:94)(cid:94) j < m q j (cid:5) x < s j ∧ (cid:94)(cid:94) k < n r k (cid:5) x = u k ) (3.1)with a quantifier-free formula. By the equivalences (i) and (ii) above, we can assumethat p i ’s and q j ’s and r k ’s in the formula (3.1) are equal to each other, say to α . Thenby A , the formula (3.1) is equivalent with ∃ y ( (cid:94)(cid:94) i <(cid:96) t (cid:48) i < y ∧ (cid:94)(cid:94) j < m y < s (cid:48) j ∧ (cid:94)(cid:94) k < n y = u (cid:48) k ) (3.2) .3. The Chinese Remainders t (cid:48) i , s (cid:48) j , u (cid:48) k (and y = α (cid:5) x ).Now, if n (cid:54) = (cid:94)(cid:94) i <(cid:96) t (cid:48) i < u ∧ (cid:94)(cid:94) j < m u < s (cid:48) j ∧ (cid:94)(cid:94) k < n u = u (cid:48) k .So, let us suppose that n =
0. Then if (cid:96) = m =
0, the formula (3.2) is equivalentwith the quantifier-free formula (cid:62) , by the axioms O and O (with O and O ) respec-tively, and if (cid:96) , m (cid:54) =
0, it is equivalent with the quantifier-free formula (cid:86)(cid:86) i <(cid:96) , j < m t (cid:48) i < s (cid:48) j by the axiom O (with O and O ). (Compare with the proof of Theorem 2.1.11) (cid:2)(cid:1) (cid:104) R ; < , + (cid:105) and (cid:104) Q ; < , + (cid:105) Proposition 3.2.2
The structures (cid:104) R ; < , + (cid:105) and (cid:104) Q ; < , + (cid:105) are not finitely axiomati-zable. Proof.
It suffices to note that for a given natural number N , the set Q / N ! = { m / ( N ! ) k | m ∈ Z , k ∈ N } of rational numbers, where N ! = × × · · · × N , is closed under addition and sosatisfies the axioms O , O , O , A , A , A , A , A , A and the finite number of theinstances of the axiom A (for n = · · · , N ) but does not satisfy the instance of A for n = p , where p is a prime number larger than N ! . (cid:2)(cid:1) For eliminating the quantifiers of the formulas of the structure (cid:104) Z ; < , + (cid:105) , we addthe (binary) congruence relations {≡ n } n (cid:62) (modulo standard natural numbers) to thelanguage; let us note that a ≡ n b is equivalent with ∃ x ( a + n (cid:5) x = b ) . About thesecongruence relations the following Generalized Chinese Remainder Theorem will beuseful later.0 Chapter 3. Additive Ordered Structures
The Chinese Remainder Theorem has been an important tool in astronomical cal-culations and in religious observance (what day does Easter fall on?); it has been asource for mathematical puzzles. It has been abstracted in algebra to a theorem onthe isomorphism of one homomorphic image of a ring of a given type to a productof two homomorphic images of the ring; it has been applied by computer scientiststo obtain multiple precision, and, somewhere along the way, it has been used in logicas a means of coding finite sequences [19].
Lemma 3.3.1 [Bézout’s Identity] Given integers a and b , not both of wich are zero,and for d which is the greatest common divisor of a and b , there exist integers x and y such that d = ax + by . Proof.
Consider the set S of all the positive linear combinations of a and b : S = { au + bv | u , v ∈ Z , au + bv > } .Notice first that S is not empty. For example, if a (cid:54) =
0, then the integer | a | = au + b .0lies in S , where we choose u = u = − a is positive or negative.By virtue of the Well-Ordering Principle, S must contain a smallest element d . Thus,from the very definition of S , there exist integers x and y for which d = ax + by holds.We claim that d is the greatest common divisor of a and b .By the Division Algorithm, we can obtain integers q and r such that a = qd + r ,where 0 ≤ r < d . Then r can be written in the form r = a − qd = a − q ( ax + by )= a ( − qx ) + b ( − qy ) If r were positive, then this representation would imply that r is a member of S ,contradicting the fact that d is the least integer in S (recall that r < d ). Therefore, .3. The Chinese Remainders r =
0, and so a = qd , or equivalently d | a . By similar reasoning, d | b , the effect ofwhich is to make d a common divisor of a and b .Now if c is an arbitrary positive common divisor of the integers a and b , then weconclude that c | ( ax + by ) ; that is, c | d and c = | c | ≤ | d | = d , so that d is greater thanevery positive common divisor of a and b . Piecing the bits of information together,we see that d is the greatest common divisor of a and b . (cid:2)(cid:1) Proposition 3.3.2 [Chinese Remainder] For integers n , n , · · · , n k (cid:62) t , t , · · · , t k , there exists some integer x such that x ≡ n i t i for i = · · · , k . Proof.
We take m = n n · · · n k . Since the integers n , n , · · · , n k (cid:62) ( n , mn ) = ( n , mn ) = ( n k , mn k ) = c , c , · · · , c k and d , d , · · · , d k such that: c n + d mn = c n + d mn = c k n k + d k mn k = x = k ∑ i = d i t i mn i Chapter 3. Additive Ordered Structures satisfies the conclusion of the theorem.For j = · · · , k we have: x = d j t j mn j + ∑ i (cid:54) = j d i t i mn i by (3.4) = t j ( − c j n j ) + ∑ i (cid:54) = j d i t i mn i = t j + n j ( − t j c j + ∑ i (cid:54) = j d i t i mn j n i ) So, x ≡ n j t j holds for j = · · · , k . (cid:2)(cid:1) Lemma 3.3.3
For integers n , n , · · · , n k + we have: n k + ∧ ( n ∨ n ∨ · · · ∨ n k ) = ( n k + ∧ n ) ∨ ( n k + ∧ n ) ∨ · · · ∨ ( n k + ∧ n k ) ,where n i ∨ n j = max { n i , n j } and n i ∧ n j = min { n i , n j } . Proof.
First we take: β = ( n k + ∧ n ) ∨ ( n k + ∧ n ) ∨ · · · ∨ ( n k + ∧ n k ) and α = n k + ∧ ( n ∨ n ∨ · · · ∨ n k ) .Without loss of generality, we can assume that n (cid:62) n (cid:62) · · · (cid:62) n k . There are threecases to be considered: ( a ) n k + (cid:62) n ; for which we have α = n = β . ( b ) n j (cid:62) n k + (cid:62) n j + for some 0 (cid:54) j < k ; for which we have α = n k + = β . .3. The Chinese Remainders ( c ) n k (cid:62) n k + ; for which we also have α = n k + = β . (cid:2)(cid:1) Lemma 3.3.4
For integers n , n , · · · , n k , let n be the least common multiplier of n , · · · , n k and d i , j be the greatest common divisor of n i and n j for i (cid:54) = j . Then thegreatest common divisor of integers n and n k + is the least common multiplier of d k + , · · · , d k , k + . Proof.
Suppose that ρ , ρ , ρ , · · · is the sequence of all prime numbers (
2, 3, 5, · · · ) .If n j = ∏ i ρ m i ( j ) i for j =
0, 1, · · · , k +
1, then [ n , n , n , · · · , n k ] = ∏ i ρ m i ( ) ∨ m i ( ) ∨···∨ m i ( k ) i and d j , k + = ( n j , n k + ) = ∏ i ρ m i ( j ) ∧ m i ( k + ) i .So, by Lemma 3.3.3: ( n k + , [ n , n , n , · · · , n k ]) = ∏ i ρ m i ( k + ) ∧ ( m i ( ) ∨ m i ( ) ∨···∨ m i ( k )) i = ∏ i ρ ( m i ( k + ) ∧ m i ( )) ∨ ( m i ( k + ) ∧ m i ( )) ∨···∨ ( m i ( k + ) ∧ m i ( k )) i = [( n , n k + ) , ( n , n k + ) , · · · , ( n k , n k + )]= [ d k + , d k + , · · · , d k , k + ] . (cid:2)(cid:1) Proposition 3.3.5 (The Generalized Chinese Remainder)
For integers t , t , · · · , t k and n , n , · · · , n k (cid:62)
2, we have: ∃ x ( (cid:94) k (cid:94) i = x ≡ n i t i ) ⇐⇒ (cid:94)(cid:94) (cid:54) i < j (cid:54) k t i ≡ d i , j t j where d i , j is the greatest common divisor of n i and n j for i (cid:54) = j ; see [8].4 Chapter 3. Additive Ordered Structures
Proof.
The ‘only if’ part is easy: For integers t , t , · · · , t k and n , n , · · · , n k (cid:62) x such that x ≡ n i t i holds for i = · · · , k . By d i , j | n j and d i , j | n i for i (cid:54) = j , we have: x ≡ d i , j t j and x ≡ d i , j t i .And so, t i ≡ d i , j t j .We prove the ‘if’ part by induction on k . For k = k = a , a such that a n + a n = d . (3.5)Also, by the assumption there exists some c such that t − t = cd . (3.6)Now, if we take x to be a ( n / d ) t + a ( n / d ) t , then by (3.5) and (3.6) wehave x = t − a n c and x = t + a n c .And so we have: x ≡ n t and x ≡ n t .For the induction step ( k +
1) we note that by the assumption, t i ≡ d i , j t j holds for each0 (cid:54) i < j (cid:54) k +
1, and suppose that the following relations hold for some integer x (the induction hypothesis): x ≡ n t x ≡ n t ... x ≡ n k t k (3.7) .3. The Chinese Remainders n be the least common multiplier of n , · · · , n k ; then the greatest common divisor m of n and n k + is the least common multiplier of d k + , · · · , d k , k + by Lemma 3.3.4.Now, by (3.7) we have: x ≡ d k + t x ≡ d k + t ... x ≡ d k , k + t k (3.8)and by the assumption we have: t ≡ d k + t k + t ≡ d k + t k + ... t k ≡ d k , k + t k + (3.9)so by (3.8) and (3.9) x ≡ d k + t k + x ≡ d k + t k + ... x ≡ d k , k + t k + (3.10)thus x ≡ m t k + holds by (3.10) and so, for some c we have: x − t k + = mc . (3.11)By Lemma 3.3.1, there are a , b such that an + bn k + = m . (3.12)6 Chapter 3. Additive Ordered Structures
Now, by (3.11) and (3.12) for y = x − anc , we have: y = t k + + bn k + c ≡ n k + t k + .And also y ≡ n i x ≡ n i t i holds for each 0 (cid:54) i (cid:54) k . (cid:2)(cid:1) (cid:104) Z ; < , + (cid:105) Theorem 3.4.3 has been proved, in various formats, in e.g. the following references:[3, Chapter 24], [7, Theorem 32E], [9, Corollary 2.5.18], [10, Secion III, Chapter 4],[11, Corollary 3.1.21], [12, Theorem 13.10] and [19, Section 4, Chapter III]. • Here, we present a slightly different proof.
Convention 3.4.1
The Axioms of the Theory of Non-trivial Discretely Ordered AbelianGroups with the Division Algorithm are as follows:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( A ) ∀ x , y , z ( x + ( y + z ) = ( x + y ) + z ) ( A ) ∀ x ( x + = x ) ( A ) ∀ x ( x + ( − x ) = ) ( A ) ∀ x , y ( x + y = y + x ) ( A ) ∀ x , y , z ( x < y → x + z < y + z ) ( O ◦ ) ∀ x , y (cid:0) x < y ↔ x + (cid:54) y (cid:1) ( A ◦ ) ∀ x ∃ y (cid:0) (cid:87)(cid:87) i < n x = n (cid:5) y + ¯ i (cid:1) n ∈ N + , ¯ i = + · · · + (cid:124) (cid:123)(cid:122) (cid:125) i-times (cid:126) .4. Integer Numbers with Order and Addition Proposition 3.4.2
The theory of the structure (cid:104) Z ; < , + , − , , (cid:105) does not admit quan-tifier elimination. Proof.
It suffices to show that the formula ∃ x ( x + x = y ) is not equivalent with anyquantifier-free formula. All the terms including the free variable y in the language (cid:104) + , − , , (cid:105) are equal to m . y for some m ∈ Z , so all the atomic formulas are m . y = k , m . y > k or m . y < k , for some m , k ∈ Z . It is easy to see that all the definable sets ofthe above structure are finite or co-finite, whereas the set { y ∈ Z | ∃ x ( x + x = y ) } isneither finite nor co-finite. (cid:2)(cid:1) Theorem 3.4.3
The infinite theory of non-trivial discretely ordered abelian groupswith the division algorithm, that is O , O , O , A , A , A , A , A , O ◦ , A ◦ , completelyaxiomatizes the order and additive theory of the integer numbers and, moreover, the(theory of the) structure (cid:104) Z ; < , + , − , , , {≡ n } n (cid:62) (cid:105) admits quantifier elimination, sohas a decidable theory. Proof.
Indeed, the axiom A ◦ is equivalent with ∀ x (cid:95)(cid:95) i < n (cid:0) x ≡ n ¯ i ∧ (cid:94)(cid:94) i (cid:54) = j < n x (cid:54)≡ n ¯ j (cid:1) ,which is rather easy to verify, and so the negation signs behind the congruences canbe eliminated by ( a (cid:54)≡ n b ) ↔ (cid:95)(cid:95) < i < n ( a ≡ n b + ¯ i ) .Since every term containing the variable x is equal to n (cid:5) x + t , for some x -freeterm t and n ∈ Z −{ } , every atomic formula containing x is equivalent with n (cid:5) x (cid:3) t where (cid:3) ∈ { = , < , > , {≡ n } n (cid:62) } and t is an x -free term. Whence, by Remark 1.3.3, itsuffices to prove the equivalence of the formula ∃ x ( (cid:94)(cid:94) i < m a i (cid:5) x ≡ n i t i ∧ (cid:94)(cid:94) j < p u j < b j (cid:5) x ∧ (cid:94)(cid:94) k < q c k (cid:5) x < v k ∧ (cid:94)(cid:94) (cid:96)< r d (cid:96) (cid:5) x = w (cid:96) ) (3.13)8 Chapter 3. Additive Ordered Structures with some quantifier-free formula, where a i ’s, b j ’s, c k ’s and d (cid:96) ’s are natural num-bers and t i ’s, u j ’s, v k ’s and w (cid:96) ’s are x -free terms.By the equivalences(i) [ a < b ] ↔ [ n (cid:5) a < n (cid:5) b ] ,(ii) [ a = b ] ↔ [ n (cid:5) a = n (cid:5) b ] ,(iii) [ a ≡ m b ] ↔ [ n (cid:5) a ≡ nm n (cid:5) b ] ,which are provable from the axioms, we can assume that a i ’s, b j ’s, c k ’s and d (cid:96) ’s inthe formula (3.13) are equal to each other, say to α . Now, (3.13) is equivalent with ∃ y ( y ≡ α ∧ (cid:94)(cid:94) i < m y ≡ n i t (cid:48) i ∧ (cid:94)(cid:94) j < p u (cid:48) j < y ∧ (cid:94)(cid:94) k < q y < v (cid:48) k ∧ (cid:94)(cid:94) (cid:96)< r y = w (cid:48) (cid:96) ) , (3.14)for y = α (cid:5) x and some (possibly new) terms t (cid:48) i ’s, u (cid:48) j ’s, v (cid:48) k ’s and w (cid:48) (cid:96) ’s.If r (cid:54) =
0, then (3.14) is readily equivalent with the quantifier-free formula whichresults from substituting w (cid:48) with y . So, it suffices to eliminate the quantifier of ∃ x ( (cid:94)(cid:94) i < m x ≡ n i t i ∧ (cid:94)(cid:94) j < p u j < x ∧ (cid:94)(cid:94) k < q x < v k ) . (3.15)By the equivalence of the formula ∃ x ( θ ( x ) ∧ u < x ∧ u < x ) with the formula (cid:2) ∃ x ( θ ( x ) ∧ u < x ) ∧ u (cid:54) u (cid:3) ∨ (cid:2) ∃ x ( θ ( x ) ∧ u < x ) ∧ u (cid:54) u (cid:3) ,we can assume that p (cid:54) q (cid:54) x -congruences ∃ x ( θ ( x ) ∧ x ≡ n t ∧ x ≡ n t ) is equivalent with the following formula with just one x -congruence ∃ x ( θ ( x ) ∧ x ≡ n t ) ∧ t ≡ d t , .4. Integer Numbers with Order and Addition d is the greatest common divisor of n and n , n is their least common multi-plier, and t = a ( n / d ) t + a ( n / d ) t where a , a satisfy a n + a n = d (see theproof of Proposition 3.3.5). So, we can assume that m (cid:54) m = s ( x ) = x + just like the way formula (2.4) was equivalentwith some quantifier-free formula).So, suppose m =
1. In this case, if any of p or q is equal to 0 then (3.15) is equivalentwith (cid:62) (since any congruence can have infinitely large or infinitely small solutions).Finally, if we have p = q = = m , then the formula ∃ x ( x ≡ n t ∧ u < x ∧ x < v ) is equivalent with the formula ∃ y ( r < n (cid:5) y (cid:54) s ) for x = t + n (cid:5) y , r = u − t and s = v − t − . Now, the formula ∃ y ( r < n (cid:5) y (cid:54) s ) is equivalent with the quantifier-freeformula (cid:87)(cid:87) i < n ( s ≡ n ¯ i ∧ r + ¯ i < s ) , since there are some q and some i < n such that s = qn + i . The existence of some y such that r < n (cid:5) y (cid:54) s is then equivalent with r < nq ( = s − i ). (cid:2)(cid:1) (cid:104) Z ; < , + (cid:105) Proposition 3.4.4
The theory of (cid:104) Z ; < , + (cid:105) cannot be axiomatized finitely. Proof.
We show that O , O , O , A , A , A , A , A , O ◦ and any finite number ofthe instances of A ◦ cannot prove all the instances of A ◦ . To see this take p to be asufficiently large prime number and put N = ( p − ) ! . Let us recall that the (rational)set Q / N = { m / N k | m ∈ Z , k ∈ N } (Theorem 3.2.2) is closed under the additionoperation and x (cid:55)→ x / n for any 1 < n < p . Define the set A = ( Q / N ) × Z and putthe structure A = (cid:104) A ; < A , + A , − A , A , A (cid:105) on it by the following: ( < A ) : ( a , (cid:96) ) < A ( b , m ) ⇐⇒ ( a < b ) ∨ ( a = b ∧ (cid:96) < m ) ; (+ A ) : ( a , (cid:96) ) + A ( b , m ) = ( a + b , (cid:96) + m ) ; ( − A ) : − A ( a , (cid:96) ) = ( − a , − (cid:96) ) ; ( A ) : A = ( , ) ;0 Chapter 3. Additive Ordered Structures ( A ) : A = ( , ) .It is straightforward to see that A satisfies the axioms O , O , O , A , A , A , A , A and O ◦ ; but does not satisfy A ◦ for n = p since the equality ( , ) = p (cid:5) ( a , (cid:96) ) + ¯ i for any a ∈ Q / N , (cid:96) ∈ Z , i ∈ N (with i < p ) implies that a = / p but 1 / p (cid:54)∈ Q / N . However, A satisfies the finite number of the instances of A ◦ (for any 1 < n < p ): for any element ( a , (cid:96) ) ∈ A we have a = m / N k for some m ∈ Z , k ∈ N , and (cid:96) = nq + r for some q , r with 0 (cid:54) r < n ; now, ( a , (cid:96) ) = n (cid:5) (cid:0) m (cid:48) / N k + , q (cid:1) + A ( r ) (where m (cid:48) = m · ( N / n ) ∈ Z )and so ( a , (cid:96) ) = n (cid:5) (cid:0) m (cid:48) / N k + , q (cid:1) + A ¯ r (where ¯ r = A + A · · · + A A for r times). (cid:2)(cid:1) (cid:104) N ; < , + (cid:105) Theorem 3.5.1
The following axioms completely axiomatize the theory of the struc-ture of (cid:104) N ; < , + , , (cid:105) :( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( O ) ∀ x , y ( x < y ↔ x + < y ∨ x + = y ) ( O ◦ ) ∀ x ∃ y ( x (cid:54) = → y + = x ) ( O ) ∀ x ( x (cid:54) < ) ,( A ) ∀ x , y , z ( x + ( y + z ) = ( x + y ) + z ) ( A ) ∀ x ( x + = x ) ( A ) ∀ x , y ( x + y = y + x ) ( A ) ∀ x , y , z ( x < y → x + z < y + z ) ( A ◦ ) ∀ x ∃ y (cid:0) (cid:87)(cid:87) i < n x = n (cid:5) y + ¯ i (cid:1) n ∈ N + , ¯ i = + · · · + (cid:124) (cid:123)(cid:122) (cid:125) i-times and, moreover, the structure (cid:104) N ; < , + , , {≡ n } n (cid:62) (cid:105) admits quantifier elimination, andso its theory is decidable. .5. Natural Numbers with Order and Addition Proof.
The quantifier elimination of this structure is shown in [7, Theorem 32E]. (cid:2)(cid:1) (cid:104) N ; < , + (cid:105) Here, we use the super-structure (cid:104) Z ; < , + (cid:105) to show the decidability of the theory ofnatural numbers with order and addition. Remark 3.5.2
The set of natural numbers is definable in structure (cid:104) Z ; < , + (cid:105) by“ x ∈ N ” ⇐⇒ ∃ y ( y + y = y ∧ y (cid:54) x ) . (cid:126) Theorem 3.5.3
The theory of the structure (cid:104) N ; < , + (cid:105) is decidable. Proof.
We show that the decidability of the structure (cid:104) Z ; < , + (cid:105) implies the decid-ability of the structure (cid:104) N ; < , + (cid:105) . Relativization ψ N of a { < , + } -formula ψ resultedfrom substituting any subformula of the form ∀ x θ ( x ) by ∀ x [ “ x ∈ N ” → θ ( x )] and ∃ x θ ( x ) by ∃ x [ “ x ∈ N ” ∧ θ ( x )] by Remark 3.5.2 has the following property: (cid:104) N ; < , + (cid:105) | = ψ ⇐⇒ (cid:104) Z ; < , + (cid:105) | = ψ N .So, the theory of the structure (cid:104) N ; < , + (cid:105) is decidable (cid:2)(cid:1) Chapter 4
Multiplicative Ordered Structures
In this chapter we consider the theories of the number sets N , Z , R and Q over thelanguage { < , ×} . (cid:104) N ; < , ×(cid:105) Proposition 4.1.1
The theory of the structure (cid:104) N ; < , ×(cid:105) is undecidable. Proof.
First we notice that the addition operation is definable in (cid:104) N ; < , ×(cid:105) , since ( ) successor s is definable from < : y = s ( x ) ⇐⇒ x < y ∧ ¬∃ z ( x < z < y ) ; ( ) and addition is definable from the successor and multiplication: z = x + y ⇐⇒ (cid:2) ¬∃ u ( s ( u ) = z ) ∧ x = y = z (cid:3) ∨ (cid:2) ∃ u ( s ( u ) = z ) ∧ s ( z · x ) · s ( z · y ) = s ( z · z · s ( x · y )) (cid:3) .(The above identity was first introduced by Robinson [16]; also see e.g. [3, Chapter24] or [7, Exercise 2 on page 281].)Now by (1) and (2), the structure (cid:104) N ; < , ×(cid:105) can interpret the structure (cid:104) N ; + , ×(cid:105) whose theory is undecidable by Gödel’s Incompleteness theorem. Thus, the theory .2. Integer numbers with order and multiplication (cid:104) N ; < , ×(cid:105) is undecidable (see [3, Theorem 17.4], [7, Corollary 35A],[9, Theorem 4.1.7], [12, Chapter 15] or [19, Corollary 6.4 in Chapter III] for a proofof the undecidability of the structure (cid:104) N ; < , ×(cid:105) and some more details). (cid:2)(cid:1) Corollary 4.1.2
The structure (cid:104) N ; < , ×(cid:105) can not be axiomatized by any computablyenumerable set of sentences. (cid:2)(cid:1) (cid:104) Z ; < , ×(cid:105) The undecidability of the theory of the structure (cid:104) N ; + , ×(cid:105) also implies the undecid-ability of the theories of the structures (cid:104) Z ; + , ×(cid:105) and (cid:104) Z ; < , ×(cid:105) . Proposition 4.2.1
The theory of the structure (cid:104) Z ; + , ×(cid:105) is undecidable. Proof.
By Lagrange’s Four Square Theorem (see e.g. [12, Theorem 16.6]) N isdefinable in (cid:104) Z ; + , ×(cid:105) : u ∈ N ⇐⇒ ∃ x , y , z , t ( u = x · x + y · y + z · z + t · t ) .Whence, (cid:104) N ; + , ×(cid:105) is definable in (cid:104) Z ; + , ×(cid:105) , and so (cid:104) Z ; + , ×(cid:105) has an undecidabletheory by Gödel’s Incompleteness theorem (see e.g. [12, Theorem 16.7] or [19,Corollary 8.29 in Chapter III]). (cid:2)(cid:1) Proposition 4.2.2
The theory of the structure (cid:104) Z ; < , ×(cid:105) is undecidable. Proof.
First we notice that the following numbers and operations are definable inthe structure (cid:104) Z ; < , ×(cid:105) :– The number zero: u = ⇐⇒ ∀ x ( x · u = u ) .– The number one: u = ⇐⇒ ∀ x ( x · u = x ) .4 Chapter 4. Multiplicative Ordered Structures – The number − : u = − ⇐⇒ u · u = ∧ u (cid:54) = .– The additive inverse: y = − x ⇐⇒ y = ( − ) · x .– The successor: y = s ( x ) ⇐⇒ x < y ∧ ¬∃ z ( x < z < y ) .– The addition: z = x + y ⇐⇒ [ z = ∧ y = − x ] ∨ [ z (cid:54) = ∧ s ( z · x ) · s ( z · y ) = s ( z · z · s ( x · y ))] .There is another beautiful definition for + in terms of s and × in Z in [9, p. 187]: z = x + y ⇐⇒ [ z · s ( z ) = z ∧ s ( x · y ) = s ( x ) · s ( y )] ∨ [ z · s ( z ) (cid:54) = z ∧ s ( z · x ) · s ( z · y ) = s ( z · z · s ( x · y ))] .And so, the structure (cid:104) Z ; + , ×(cid:105) somehow includes the structure (cid:104) Z ; < , ×(cid:105) . By Propo-sition 4.2.1, the theory of the structure (cid:104) Z ; + , ×(cid:105) is undecidable. Thus the theory ofthe structure (cid:104) Z ; < , ×(cid:105) is undecidable too. (cid:2)(cid:1) Corollary 4.2.3
The structure (cid:104) Z ; < , ×(cid:105) can not be axiomatized by any computablyenumerable set of sentences. (cid:2)(cid:1) The structure (cid:104) R ; < , ×(cid:105) is decidable since by a theorem of Tarski the (theory of the)structure (cid:104) R ; < , + , ×(cid:105) can be completely axiomatized by the theory of real closed .3. Real numbers with order and multiplication ordered fields , and so has a decidable theory; see e.g. [10, Theorem 7, Chapter 4],[11, Theorem 3.3.15] or [12, Theorem 21.36]. Corollary 4.3.1
For the reason that the structure (cid:104) R ; < , ×(cid:105) is included in the struc-ture (cid:104) R ; < , + , ×(cid:105) , the theory of the structure (cid:104) R ; < , ×(cid:105) is also decidable. (cid:2)(cid:1) • Here, we prove the decidability of this theory directly (without using Tarski’s the-orem) and provide an explicit axiomatization for it. (cid:104) R ; < , ×(cid:105) First we study the structure (cid:104) R + ; < , ×(cid:105) . Proposition 4.3.2
The following infinite theory (of the non-trivial ordered divisibleabelian groups) completely axiomatizes the order and multiplicative theory of thepositive real numbers:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( M ) ∀ x , y , z ( x · ( y · z ) = ( x · y ) · z ) ( M ) ∀ x ( x · = x ) ( M ) ∀ x ( x · x − = ) ( M ) ∀ x , y ( x · y = y · x ) ( M ) ∀ x , y , z ( x < y → x · z < y · z ) ( M ) ∃ y ( y (cid:54) = ) ( M ) ∀ x ∃ y ( x = y n ) n (cid:62) (cid:104) R + ; < , × , (cid:3) − , (cid:105) admits quantifier elimination, and so its theory isdecidable. Proof.
The structure (cid:104) R + ; < , ×(cid:105) (of the positive real numbers) is (algebraically)isomorphic to the structure (cid:104) R ; < , + (cid:105) by the mapping x (cid:55)→ log ( x ) . So, Theorem 3.2.1implies the decidability of the structure (cid:104) R + ; < , ×(cid:105) . (cid:2)(cid:1) Chapter 4. Multiplicative Ordered Structures
Proposition 4.3.3
The following infinite theory completely axiomatizes the orderand multiplicative theory of the real numbers:( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( M ) ∀ x , y , z ( x · ( y · z ) = ( x · y ) · z ) ( M ◦ ) ∀ x ( x · = x ∧ x · = = − ) ( M ◦ ) ∀ x ( x (cid:54) = → x · x − = ) ( M ) ∀ x , y ( x · y = y · x ) ( M ◦ ) ∀ x , y , z ( x < y ∧ < z → x · z < y · z ) ( M • ) ∀ x , y , z ( x < y ∧ z < → y · z < x · z ) ( M ◦ ) ∃ y ( − < < < y ) ( M ◦ ) ∀ x ∃ y ( x = y n + ) n ∈ N ( M ) ∀ x ( x n = ←→ x = ∨ x = − ) n ∈ N ( M ) ∀ x ( < x ←→ ∃ y [ y (cid:54) = ∧ x = y ]) and, moreover, the structure (cid:104) R ; < , × , (cid:3) − , − , , (cid:105) admits quantifier elimination,and so its theory is decidable. Proof.
We have ( x < ) ↔ ( < − x ) by M • , M ◦ , M ◦ and M , where − x = ( − ) · x .Whence, for any formula η we have ∃ x η ( x ) ≡ ∃ x > η ( x ) ∨ η ( ) ∨ ∃ y > η ( − y ) .Also, if z is another variable in η , then η ( x , z ) is equivalent with [ < z ∧ η ( x , z )] ∨ η ( x , ) ∨ [ < − z ∧ η ( x , z )] .For the last disjunct, if we let z (cid:48) = − z , then < − z ∧ η ( x , z ) will be < z (cid:48) ∧ η ( x , − z (cid:48) ) .Thus, by introducing the constants and − (and renaming the variables if neces-sary) we can assume that all the variables of a quantifier-free formula are positive.Now, the process of eliminating the quantifier of the formula ∃ x η ( x ) , where η is the .3. Real numbers with order and multiplication and − and then reduce the desired conclusion toProposition 4.3.2. For the first part, we simplify terms so that each term is either pos-itive (all the variables are positive) or equals to or is the negation of a positive term(is − t for some positive term t ). Then by replacing = with (cid:62) and < with ⊥ ,we can assume that appears at most once in any atomic formula; also − appearsat most once since − t = − s is equivalent with t = s and − t < − s with s < t . Now,we can eliminate the constant − by replacing the atomic formulas − t = s , t = − s and t < − s by ⊥ and − t < s by (cid:62) for positive or zero terms t , s (note that − = by M ◦ ). Also the constant can be eliminated by replacing < t with (cid:62) and t < and t = (also = t ) with ⊥ for positive terms t . Thus, we get a formula whose allvariables are positive, and so we are in the realm of R + . Finally, for the second partwe have the equivalence of thus resulted formula with a quantifier-free formula byProposition 4.3.2 provided that the relativized form of the axioms O , O , O , M , M , M , M , M , M and M to R + can be proved from the axioms O , O , O , M , M ◦ , M ◦ , M , M ◦ , M • , M ◦ , M ◦ , M , and M . We need to consider M and M only, when relativized to R + , i.e., ∃ y ( < y ∧ y (cid:54) = ) and ∀ x ∃ y [ < x → < y ∧ x = y n ] . The relativization of M immediately follows from M ◦ . For the relativization of M take any a > , and any n ∈ N . Write n = k ( m + ) ; by M ◦ there exists some c such that c m + = a , and by M ◦ and M • we should have c > . Now, by using M for k times there must exist some b such that b k = c and we can assume that b > (since otherwise we can take − b instead of b ). Now, we have b k ( m + ) = c m + = a and so a = b n . (cid:2)(cid:1) (cid:104) R ; < , ×(cid:105) Proposition 4.3.4
The structure (cid:104) R + ; < , ×(cid:105) is not finitely axiomatizable. Proof.
For the infinite axiomatizability it suffices to note that for a sufficiently large N , the set { m · ( N ! ) − k | m ∈ Z , k ∈ N } of positive real numbers is a multiplicativesubgroup and so satisfies all the axioms ( O , O , O , M , M , M , M , M , M ) and finitely8 Chapter 4. Multiplicative Ordered Structures many instances of the axiom M (for n (cid:54) N ) but not all the instances of M (for examplewhen n = p is a prime larger than N ! ). (cid:2)(cid:1) Theorem 4.3.5
The structure (cid:104) R ; < , ×(cid:105) is not finitely axiomatizable. Proof.
The set { } ∪ {− m · ( N ! ) − k , 2 m · ( N ! ) − k | m ∈ Z , k ∈ N } of real numbers, forsome N >
2, satisfies all the axioms of Theorem 4.3.3 except M ◦ ; however it satisfiesa finite number of its instances (when 2 n + (cid:54) N ) but not all the instances of M ◦ (e.g.when 2 n + N ! ). (cid:2)(cid:1) The technique of the proof of Theorem 4.3.3 enables us to consider first the multi-plicative and order structure of the positive rational numbers, that is (cid:104) Q + ; < , ×(cid:105) . (cid:104) Q ; < , ×(cid:105) Proposition 4.4.1
The theory of the structure (cid:104) Q + ; < , ×(cid:105) does not admit quantifierelimination. Proof.
We show that the formula ∃ x ( y = x n ) (for n >
1) is not equivalent with anyquantifier-free formula. All the atomic formulas of the free variable y , are y n < y m or y n = y m which do not depend on y and are equivalent with (cid:62) or ⊥ . So the formula ∃ x ( y = x n ) (which depends on y and n and can be (cid:62) or ⊥ ) is not equivalent with anyof them. (cid:2)(cid:1) Definition 4.4.2 ( ℜ ) Let ℜ n ( y ) be the formula ∃ x ( y = x n ) , stating that “ y is the n thpower of a number” (for n > ⊗⊕ Remark 4.4.3
For any r ∈ Q and any natural n > ℜ n ( r ) holds if andonly if every exponent of the unique factorization (of the numerators and denomi-nators of the reduced form) of r is divisible by n . Thus ℜ n ( r ) is an algorithmicallydecidable relation of r (and n ). (cid:126) .4. Rational numbers with order and multiplication Definition 4.4.4 ( TQ ) Let TQ be the theory axiomatized by the axioms( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( M ) ∀ x , y , z ( x · ( y · z ) = ( x · y ) · z ) ( M ) ∀ x ( x · = x ) ( M ) ∀ x ( x · x − = ) ( M ) ∀ x , y ( x · y = y · x ) ( M ) ∀ x , y , z ( x < y → x · z < y · z ) ( M ) ∃ y ( y (cid:54) = ) ( M ) ∀ x , z ∃ y ( x < z → x < y n < z ) n ∈ N , and( M ) ∀{ x j } j < q ∃ y ∀ z (cid:86)(cid:86) m j (cid:45) n ( j < q ) ( y n · x j (cid:54) = z m j ) for each n (cid:62) m j > ⊗⊕ Some explanations on the new axioms M and M are in order:The axiom M , interpreted in Q + , states that Q + is dense not only in itself but also inthe radicals of its elements (or more generally in R + : for any x , z ∈ Q + there existssome y ∈ Q + that satisfies n √ x < y < n √ z ).The axiom M , interpreted in Q + again, is actually equivalent with the fact that forany sequences x , · · · , x q ∈ Q + and m , · · · , m q ∈ N + none of which divides n (insymbols m j (cid:45) n ), there exists some y ∈ Q + such that (cid:86)(cid:86) j ¬ ℜ m j ( y n · x j ) . This axiomis not true in R + (while M is true in it) and to see that why M is true in Q + itsuffices to note that for given x , · · · , x q one can take y to be a prime number whichdoes not appear in the unique factorization (of the numerators and denominators ofthe reduced forms) of any of x j ’s. In this case y n · x j can be an m j ’s power (of arational number) only when m j divides n . The condition m j (cid:45) n is necessary, sinceotherwise (if m j | n and) if x j happens to satisfy ℜ m j ( x j ) , then no y can satisfy therelation ¬ ℜ m j ( y n · x j ) .0 Chapter 4. Multiplicative Ordered Structures • We now show that TQ completely axiomatizes the theory of the structure (cid:104) Q + ; < , × , (cid:3) − , , { ℜ n } n > (cid:105) and moreover this structure admits quantifier elimina-tion, thus the theory of the structure (cid:104) Q + ; < , ×(cid:105) is decidable. For that, we will needthe following lemmas. Lemma 4.4.5
For any x ∈ Q + and any natural n , n > ℜ n ( x ) ∧ ℜ n ( x ) ⇐⇒ ℜ n ( x ) ,where n is the least common multiplier of n and n . Proof.
Since n divides n and n , the ⇐ part is straightforward; for the ⇒ directionsuppose that x = y n = z n . By Bézout’s Identity there are some c , c ∈ Z such that c n / n + c n / n =
1; therefore, x = x c n / n · x c n / n = y c n · z c n = ( y c z c ) n ,and this completes the proof. (cid:2)(cid:1) Lemma 4.4.6
For natural numbers { n i } i < p with n i > { t i } i < p and x , (cid:94)(cid:94) i < p ℜ n i ( x · t i ) ⇐⇒ ℜ n ( x · β ) ∧ (cid:94)(cid:94) i (cid:54) = j ℜ d i , j ( t i · t − j ) ,where n is the least common multiplier of n i ’s, d i , j is the greatest common divisor of n i and n j (for each i (cid:54) = j ) and β = ∏ i < p t c i ( n / n i ) i in which c i ’s satisfy the (Bézout’s)identity ∑ i < p c i ( n / n i ) = Proof.
For t i ’s, n i ’s, c i ’s, d i , j ’s and n as given above, we show that the relation ℜ n k ( t k · β − ) holds for each fixed k < p when (cid:86)(cid:86) i (cid:54) = j ℜ d i , j ( t i · t − j ) holds. Let m k , i bethe least common multiplier of n k and n i (which is then a divisor of n ). Let us notethat d k , i / n i = n k / m k , i . Since ℜ d k , i ( t k · t − i ) , there should exists some w k , i ’s (for i (cid:54) = k ) .4. Rational numbers with order and multiplication t k · t − i = w d k , i k , i . Now, the relation ℜ n k ( t k · β − ) follows from the followingidentities: t k · β − = t ∑ i c i ( n / n i ) k · ∏ i t − c i ( n / n i ) i = ∏ i (cid:54) = k ( t k · t − i ) c i ( n / n i ) = ∏ i (cid:54) = k ( w d k , i k , i ) c i ( n / n i ) = ∏ i (cid:54) = k w c i · n k ( n / m k , i ) k , i = ( ∏ i (cid:54) = k w c i ( n / m k , i ) k , i ) n k .( ⇒ ): The relations ℜ n i ( x · t i ) and ℜ n j ( x · t j ) immediately imply that ℜ d i , j ( x · t i ) and ℜ d i , j ( x · t j ) and so ℜ d i , j ( t i · t − j ) . For showing ℜ n ( x · β ) it suffices, byLemma 4.4.5, to show that ℜ n i ( x · β ) holds for each i < p . This follows from ℜ n i ( t i · β − ) , which was proved above, and the assumption ℜ n i ( x · t i ) .( ⇐ ): From the first part of the proof we have ℜ n k ( t k · β − ) for each k < p ; nowby ℜ n ( x · β ) we have ℜ n k ( x · β ) and so ℜ n k ( x · t k ) for each k < p . (cid:2)(cid:1) • Let us note that Lemmas 4.4.5 and 4.4.6 are provable in TQ . The idea of the proofof Lemma 4.4.6 is taken from [14]. Lemma 4.4.7
The following sentences are provable in TQ , for any n > ∀ u ∃ y [ ℜ n ( y · u )] , ∀ x , u ∃ y [ x < y ∧ ℜ n ( y · u )] , ∀ z , u ∃ y [ y < z ∧ ℜ n ( y · u )] and ∀ x , z , u ∃ y [ x < z → x < y < z ∧ ℜ n ( y · u )] . Proof.
We present a proof for the last formula only. By M (of Definition 4.4.4)there exists some v such that x · u < v n < z · u . Then for y = v n · u − we will have x < y < z and ℜ n ( y · u ) . (cid:2)(cid:1) Lemma 4.4.8
The following sentences are provable in TQ , for any { m j > } j < q : ∀{ x j } j < q ∃ y [ (cid:86)(cid:86) j < q ¬ ℜ m j ( y · x j )] , ∀{ x j } j < q , u ∃ y [ u < y ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( y · x j )] ,2 Chapter 4. Multiplicative Ordered Structures ∀{ x j } j < q , v ∃ y [ y < v ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( y · x j )] and ∀{ x j } j < q , u , v ∃ y [ u < v → u < y < v ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( y · x j )] . Proof.
The first sentence is an immediate consequence of M (of Definition 4.4.4)for n =
1. We show the last sentence. There exists γ , by M , such that the relation (cid:86)(cid:86) j ¬ ℜ m j ( γ · x j ) holds. Let M = ∏ j m j ; by M there exists some δ such that theinequalities u · γ − < δ M < v · γ − holds. Now for y = γ · δ M we have u < y < v andalso (cid:86)(cid:86) j ¬ ℜ m j ( y · x j ) , since if (otherwise) we had ℜ m j ( y · x j ) , then ℜ m j ( γ · δ M · x j ) and so ℜ m j ( γ · x j ) would hold; a contradiction. (cid:2)(cid:1) Lemma 4.4.9
In the theory TQ the following formulas ∃ x [ ℜ n ( x · t ) ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( x · s j )] , ∃ x [ u < x ∧ ℜ n ( x · t ) ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( x · s j )] and ∃ x [ x < v ∧ ℜ n ( x · t ) ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( x · s j )] are equivalent with (cid:86)(cid:86) m j | n ( j < q ) ¬ ℜ m j ( t − · s j ) ;and the formula ∃ x [ u < x < v ∧ ℜ n ( x · t ) ∧ (cid:86)(cid:86) j < q ¬ ℜ m j ( x · s j )] is equivalent with (cid:86)(cid:86) m j | n ( j < q ) ¬ ℜ m j ( t − · s j ) ∧ u < v . Proof. If m j | n then ℜ n ( x · t ) implies ℜ m j ( x · t ) . Now, if ℜ m j ( t − · s j ) were true,then ℜ m j ( x · s j ) would be true too; contradicting (cid:86)(cid:86) j < q ¬ ℜ m j ( x · s j ) . Suppose nowthat the relation (cid:86)(cid:86) m j | n ¬ ℜ m j ( t − · s j ) holds. By M there exists some γ such that (cid:86)(cid:86) m j (cid:45) n ¬ ℜ m j ( γ · t − · s j ) holds. By M there exists some δ such that the inequalities u · t · γ − n < δ M · n < v · t · γ − n (if u < v ) hold, where M is the product ∏ j < q m j . For x = δ M · n · γ n · t − we have u < x < v and ℜ n ( x · t ) . We show ¬ ℜ m j ( x · s j ) for each j < q by distinguishing two cases: if m j | n then ¬ ℜ m j ( t − · s j ) implies the relation ¬ ℜ m j ( δ M · n · γ n · t − · s j ) ; if m j (cid:45) n then by ¬ ℜ m j ( γ · t − · s j ) we have the relation ¬ ℜ m j ( δ M · n · γ n · t − · s j ) . (cid:2)(cid:1) .4. Rational numbers with order and multiplication • Finally we can prove the main result which appears for the first time in thisthesis.
Theorem 4.4.10
The infinite theory TQ completely axiomatizes the theory of thestructure (cid:104) Q + ; < , ×(cid:105) , and moreover the structure (cid:104) Q + ; < , × , (cid:3) − , , { ℜ n } n > (cid:105) ad-mits quantifier elimination. Proof.
We are to eliminate the quantifier of the formula ∃ x ( (cid:94)(cid:94) i < p ℜ n i ( x a i · t i ) ∧ (cid:94)(cid:94) j < q ¬ ℜ m j ( x b j · s j ) ∧ (cid:94)(cid:94) k < f u k < x c k ∧ (cid:94)(cid:94) (cid:96)< g x d (cid:96) < v (cid:96) ∧ (cid:94)(cid:94) ι < h x e ι = w ι ) .(4.1)By the equivalences (i) a n < b n ↔ a < b (ii) ℜ m · n ( a n ) ↔ ℜ m ( a ) we can assume that all the a i ’s, b j ’s, c k ’s, d (cid:96) ’s and e ι ’s are equal to each other, andmoreover, equal to one (cf. the proof of Theorem 3.4.3). We can also assume that h = f , g (cid:54)
1. By Lemma 4.4.6 we can also assume that p (cid:54) q =
0, then Lemma 4.4.7 implies that the quantifier of the formula (4.1) canbe eliminated. So, we assume that q > p =
0, then the quantifier of (4.1) can be eliminated by Lemma 4.4.8.– Finally, if p = q (cid:54) = = h and f , g (cid:54) (cid:2)(cid:1) Chapter 4. Multiplicative Ordered Structures
Corollary 4.4.11
The below infinite theory completely axiomatized the theory ofthe structure (cid:104) Q ; < , ×(cid:105) :( O ) ∀ x , y ( x < y → y (cid:54) < x ) ( O ) ∀ x , y , z ( x < y < z → x < z ) ( O ) ∀ x , y ( x < y ∨ x = y ∨ y < x ) ( M ) ∀ x , y , z ( x · ( y · z ) = ( x · y ) · z ) ( M ◦ ) ∀ x ( x · = x ∧ x · = = − ) ( M ◦ ) ∀ x ( x (cid:54) = → x · x − = ) ( M ) ∀ x , y ( x · y = y · x ) ( M ◦ ) ∀ x , y , z ( x < y ∧ < z → x · z < y · z ) ( M • ) ∀ x , y , z ( x < y ∧ z < → y · z < x · z ) ( M ◦ ) ∃ y ( − < < < y ) ( M ) ∀ x ( x n = ←→ x = ∨ x = − )( M ◦ ) ∀ x , z ∃ y ( < x < z → x < y n < z ) n ∈ N ( M ) ∀{ x j } j < q ∃ y ∀ z (cid:86)(cid:86) m j (cid:45) n ( j < q ) ( y n · x j (cid:54) = z m j ) for each n (cid:62) m j > (cid:104) Q ; < , × , (cid:3) − , − , , , { ℜ n } n > (cid:105) admits quantifier elim-ination. Proof.
Quantifier elimination of the theory of (cid:104) Q ; < , × , (cid:3) − , − , , , { ℜ n } n > (cid:105) fol-lows from Theorem 4.4.10: it suffices to distinguish the signs by noting that for all x one of the three cases − x > or x = or x > holds. (cid:2)(cid:1) Proposition 4.4.12
The theory of the structure (cid:104) Q ; + , ×(cid:105) is undecidable. Proof.
Since the set of integer numbers is definable in (cid:104) Q ; + , ×(cid:105) [16], the decidabil-ity of the theory of the structure (cid:104) Q ; + , ×(cid:105) implies the decidability of the theory ofthe structure (cid:104) Z ; + , ×(cid:105) and this contradicts Proposition 4.2.1. (cid:2)(cid:1) (cid:104) Q ; < , ×(cid:105) Theorem 4.4.13
The structure (cid:104) Q + ; < , ×(cid:105) is not finitely axiomatizable. .4. Rational numbers with order and multiplication Proof.
To see that the structure (cid:104) Q + ; < , ×(cid:105) cannot be axiomatized by a finite set ofsentences we present an ordered multiplicative structure that satisfies any sufficientlylarge finite number of the axioms of TQ but does not satisfy all of its axioms. Let p be a sufficiently large prime number. The set Q / p = { m / p k | m ∈ Z , k ∈ N } is closed under addition and the operation x (cid:55)→ x / p , and the inclusions Z ⊂ Q / p ⊂ Q hold. Let ρ , ρ , ρ , · · · denote the sequence of all prime numbers (2, 3, 5, · · · ). Let ( Q / p ) ∗ be the set { ∏ i <(cid:96) ρ r i i | (cid:96) ∈ N , r i ∈ Q / p } ; this is closed under multiplicationand the operation x (cid:55)→ x / p , and we have the inclusions Q + ⊂ ( Q / p ) ∗ ⊂ R + . Thus, ( Q / p ) ∗ satisfies the axioms O , O , O , M , M , M , M , M and M of Proposition 4.3.2,and also the axiom M . However, it does not satisfy the axiom M for n = q = x = m = p because ( Q / p ) ∗ | = ∀ y ℜ p ( y ) . We show that ( Q / p ) ∗ satisfies theinstances of the axiom M when 1 < m j < p (for each j < q and arbitrary n , q ). Thus,no finite number of the instances of M can prove all of its instances (with the rest ofthe axioms of TQ ). Let x j ’s be given from ( Q / p ) ∗ ; write x j = ∏ i <(cid:96) j ρ r i , j i where wecan assume that (cid:96) j (cid:62) q . Put r j , j = u j / p v j where u j ∈ Z and v j ∈ N (for each j < q ).Define t j to be 1 when m j | u j and be m j when m j (cid:45) u j . Let y = ∏ i < q ρ ( t i / p vi + ) i ( ∈ ( Q / p ) ∗ ) .We show (cid:94)(cid:94) j < q ¬ ℜ m j ( y n · x j ) under the assumption (cid:86)(cid:86) j < q m j (cid:45) n . Take a k < q , and assume (for the sake of contra-diction) that ℜ m k ( y n · x k ) . Then ℜ m k ( ρ nt k / p vk + k · ρ u k / p vk k ) holds, and so there shouldexist some a , b such that ρ ( nt k + p u k ) / p vk + k = ρ ( m k · a ) / p b k .6 Chapter 4. Multiplicative Ordered Structures
Therefore, m k | nt k + p u k .We reach to a contradiction by distinguishing two cases:(i) if m k | u k then t k = m k | n + p u k whence m k | n , contradicting (cid:86)(cid:86) j < q m j (cid:45) n ;(ii) if m k (cid:45) u k then t k = m k and so m k | nm k + p u k whence m k | p u k which by ( m k , p ) = m k | u k , contradicting the assumption (of m k (cid:45) u k ). (cid:2)(cid:1) Chapter 5
Conclusions and Open Problems
In the following table the decidable structures are denoted by ∆ and the undecidableones by ∆ \(cid:54) : N Z Q R { < } ∆ ∆ ∆ ∆ { < , + } ∆ ∆ ∆ ∆ { < , ×} ∆ \(cid:54) ∆ \(cid:54) ∆ ∆ { + , ×} ∆ \(cid:54) ∆ \(cid:54) ∆ \(cid:54) ∆ • Decidability of the theory of the structure (cid:104) Q ; < , ×(cid:105) and also the presentationof an explicit axiomatization for the theory of the structure (cid:104) R ; < , ×(cid:105) are somenew results in this thesis. • For the theory of some other decidable structures, the old and new (syntactic)proofs were given along with some explicit axiomatizations. • It is interesting to note that8
Chapter 5. Conclusions and Open Problems – the undecidability of the theories of (cid:104) N ; < , ×(cid:105) and (cid:104) Z ; < , ×(cid:105) follow fromthe undecidability of the theories of (cid:104) N ; + , ×(cid:105) and (cid:104) Z ; + , ×(cid:105) (and thedefinability of + from < and × in N and Z ); – the decidability of the theory of the structure (cid:104) R ; < , ×(cid:105) follows from thedecidability of the theory of the structure (cid:104) R ; + , ×(cid:105) (and the definabilityof < from + and × in R ); – though, the undecidability of the additive and multiplicative structure (cid:104) Q ; + , ×(cid:105) has nothing to do with the (decidable) theory of multiplicativestructure (cid:104) Q ; < , ×(cid:105) ; as a matter of fact + is not definable in the multi-plicative structure (cid:104) Q ; < , ×(cid:105) while < is definable in (cid:104) Q ; + , ×(cid:105) . There are lots of notable sets between Q and R . For example– Q [ √ ] .– Q [ √ √ √ · · · ] .– | Ω | = the set of real numbers that are constructible by ruler and compass.– The field generated by the radicals of rational numbers (when they exist in thereal numbers).For any set A with Q ⊆ A ⊆ R , Theorem 3.2.1 axiomatizes the theory of the struc-ture (cid:104) A ; < , + (cid:105) when A is closed under the addition operation and also the operations x (cid:55)→ x / n ( n ∈ N + ) . But the theory of the structure (cid:104) A ; < , ×(cid:105) could be different,when A is closed under × (it could not be even axiomatizable, or be axiomatizableby a different set of axioms). For example, it is not yet known if the theory of thestructure (cid:104) | Ω | ; < , ×(cid:105) is decidable or not!?Investigating any of these problems could lead to some wonderful results in Math-ematical Logic and Computer Science.9 Bibliography [1] Z
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Index A Abelian Group 16Axiomatizability 7 B Bézout’s Theorem 20 C Chinese Remainder Theorem 21Complete Theory 7 D Decidable Set 7Decision Algorithm 8Dense Linear Order 4Discrete Order 5Disjunctive Normal Form 5Divisible Group 17 ndex E Effectively Enumerable Set 7Entscheidungsproblem 1 F Finitely Axiomatizable 7 G Generalized Chinese Remainder Theorem 23Group 16 L Lagrange’s Four Square Theorem 2 M Main Lemma of Quantifier Elimination 6 N Non-trivial Group 164 O Ordered Group 17Ordered Structure 4Orders Without Endpoints 5 Q Quantifier Elimination 5 S Successor 5 T Tarski-Robinson’s Identity 32Tarski-Seidenberg’s Theorem 2Theory 7