Decision Procedures 1: Survey of decision procedures
... • Presburger arithmetic: arithmetic equations and inequalities with addition but not multiplication, interpreted over Z or N. • Tarski arithmetic: arithmetic equations and inequalities with addition and multiplication, interpreted over R (or any real-closed ...
... • Presburger arithmetic: arithmetic equations and inequalities with addition but not multiplication, interpreted over Z or N. • Tarski arithmetic: arithmetic equations and inequalities with addition and multiplication, interpreted over R (or any real-closed ...
Monadic Predicate Logic is Decidable
... • For given sentences ϕ and ψ, does ψ follow from ϕ? (“Does ϕ have ψ as a logical consequence?”) – More precisely: Is it true that for all models M, ...
... • For given sentences ϕ and ψ, does ψ follow from ϕ? (“Does ϕ have ψ as a logical consequence?”) – More precisely: Is it true that for all models M, ...
com.1 The Compactness Theorem
... detail). By compactness, Γ ∪ ∆ is satisfiable. Any model S of Γ ∪ ∆ contains an infinitesimal, namely cS . Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that th ...
... detail). By compactness, Γ ∪ ∆ is satisfiable. Any model S of Γ ∪ ∆ contains an infinitesimal, namely cS . Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that th ...
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... reasoning used in science and law. In this course the topic is called first-order logic, the title of the required textbook as well. Most logic courses teach students how to express ideas in first-order logic, and this logic is also used to study the semantics of natural language and to investigate ...
... reasoning used in science and law. In this course the topic is called first-order logic, the title of the required textbook as well. Most logic courses teach students how to express ideas in first-order logic, and this logic is also used to study the semantics of natural language and to investigate ...
Full text - The Fibonacci Quarterly
... defines in each field Fp the set Fib[p]? By providing a negative answer to the above question, the present note establishes a worth noting, albeit negative, property of the family of modular Fibonacci sets. Our main result is the following: T h e o r e m 1: There is no formula 6{x) written in the fi ...
... defines in each field Fp the set Fib[p]? By providing a negative answer to the above question, the present note establishes a worth noting, albeit negative, property of the family of modular Fibonacci sets. Our main result is the following: T h e o r e m 1: There is no formula 6{x) written in the fi ...
Book Review: Lorenz J. Halbeisen: “Combinatorial Set Theory.”
... disjoint) family is a collection A ⊂ P∞ (N) such that any two distinct elements of A are almost disjoint and for every A ∈ P∞ (N) there is some B ∈ A such that A ∩ B is infinite. By a diagonal argument, no mad family can be countable. Also, there is always a mad family of cardinality 2ℵ0 , the size ...
... disjoint) family is a collection A ⊂ P∞ (N) such that any two distinct elements of A are almost disjoint and for every A ∈ P∞ (N) there is some B ∈ A such that A ∩ B is infinite. By a diagonal argument, no mad family can be countable. Also, there is always a mad family of cardinality 2ℵ0 , the size ...
Answers - stevewatson.info
... ie. every model of Th() satisfies but an interpretation is a model of Th() iff it is a model of by the construction of Th(), so every model of satisfies . ie. but then Th(). QED. ...
... ie. every model of Th() satisfies but an interpretation is a model of Th() iff it is a model of by the construction of Th(), so every model of satisfies . ie. but then Th(). QED. ...
Lecture 7. Model theory. Consistency, independence, completeness
... Three notions of completeness. We have now seen three notions of completeness: (i) a logic may be complete: everything which should be a theorem in the semantic sense, i.e. every sentence which is valid is indeed a theorem in the syntactic sense, i.e. is derivable, provable. (ii) Given a logic and a ...
... Three notions of completeness. We have now seen three notions of completeness: (i) a logic may be complete: everything which should be a theorem in the semantic sense, i.e. every sentence which is valid is indeed a theorem in the syntactic sense, i.e. is derivable, provable. (ii) Given a logic and a ...
An Upper Bound on the nth Prime - Mathematical Association of
... In 1845, J. Bertrand conjectured that for any integer n > 3, there exists at least one prime p between n and 2n − 2 [1]. In 1852, P. Tchebychev offered the first demonstration of this now-famous theorem. Today, Bertrand’s Postulate is often stated as, “for any positive integer n ≥ 1, there exists a ...
... In 1845, J. Bertrand conjectured that for any integer n > 3, there exists at least one prime p between n and 2n − 2 [1]. In 1852, P. Tchebychev offered the first demonstration of this now-famous theorem. Today, Bertrand’s Postulate is often stated as, “for any positive integer n ≥ 1, there exists a ...
Completeness Theorem for Continuous Functions and Product
... short, is considered as a minimal subsystem of ZF necessary for a good notion of computation. KP arises from ZF by omitting the Power Set Axiom and restricting Separation and Collection to ∆0 -formulas. An admissible set is a transitive set A such that (A, ∈) is a model of KP. The smallest example o ...
... short, is considered as a minimal subsystem of ZF necessary for a good notion of computation. KP arises from ZF by omitting the Power Set Axiom and restricting Separation and Collection to ∆0 -formulas. An admissible set is a transitive set A such that (A, ∈) is a model of KP. The smallest example o ...
HISTORY OF LOGIC
... • Set theory is the branch of mathematics that studies sets, which are collections of objects • The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s ...
... • Set theory is the branch of mathematics that studies sets, which are collections of objects • The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s ...
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... theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. Basically, we need to show that every axiom is a tautology, and that the inferen ...
... theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. Basically, we need to show that every axiom is a tautology, and that the inferen ...
A Finite Model Theorem for the Propositional µ-Calculus
... 2. ≤ is well-founded, and there is no infinite set of pairwise ≤-incomparable elements. 3. Every countable sequence x0 , x1 , . . . has xi ≤ xj for some i < j. 4. Every countable sequence x0 , x1 , . . . has a countable monotone subsequence ...
... 2. ≤ is well-founded, and there is no infinite set of pairwise ≤-incomparable elements. 3. Every countable sequence x0 , x1 , . . . has xi ≤ xj for some i < j. 4. Every countable sequence x0 , x1 , . . . has a countable monotone subsequence ...
KRIPKE-PLATEK SET THEORY AND THE ANTI
... the basis of Kripke-Platek set theory without Foundation. This system is dubbed KPA. It is shown that the addition of AFA considerably increases the proof theoretic strength. Indeed, KPA has the same strength as the subsystem of second order arithmetic based on ∆12 Comprehension. 2. The anti-foundat ...
... the basis of Kripke-Platek set theory without Foundation. This system is dubbed KPA. It is shown that the addition of AFA considerably increases the proof theoretic strength. Indeed, KPA has the same strength as the subsystem of second order arithmetic based on ∆12 Comprehension. 2. The anti-foundat ...
An un-rigorous introduction to the incompleteness theorems
... • Logicism. The incompleteness theorems show that there is no set of axioms from which all the truths of arithmetic can be proven. So, if we think of logicism as the view that all mathematical truths are disguised versions of truths provable in some system of logic, it seems that Gödel has shown th ...
... • Logicism. The incompleteness theorems show that there is no set of axioms from which all the truths of arithmetic can be proven. So, if we think of logicism as the view that all mathematical truths are disguised versions of truths provable in some system of logic, it seems that Gödel has shown th ...
Predicate Calculus pt. 2
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
Exam #2 Wednesday, April 6
... Resolving on P gives (Q) and then on Q gives the empty clause, showing that the original formula is valid. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and {Q} and then Q to get the empty clause, we also see that the original formula is valid. 2. ((Q -> P) ^ Q) -> P The neg ...
... Resolving on P gives (Q) and then on Q gives the empty clause, showing that the original formula is valid. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and {Q} and then Q to get the empty clause, we also see that the original formula is valid. 2. ((Q -> P) ^ Q) -> P The neg ...
Modal Logic
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
Math 245 - Cuyamaca College
... Introduction to discrete mathematics. Includes basic logic, methods of proof, sequences, elementary number theory, basic set theory, elementary counting techniques, relations, and recurrence relations. Prerequisite “C” grade or higher or “Pass” in MATH 280 or equivalent Entrance Skills Without the f ...
... Introduction to discrete mathematics. Includes basic logic, methods of proof, sequences, elementary number theory, basic set theory, elementary counting techniques, relations, and recurrence relations. Prerequisite “C” grade or higher or “Pass” in MATH 280 or equivalent Entrance Skills Without the f ...
Homework 5
... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...
... During the second half of this course you should work on a self-chosen project related to the topic of applied logic. This could, for instance, be a literature study about an interesting or the implementation (and documentation) of a proof environment. We will discuss a few possibilities in class. P ...
Lecture 10. Model theory. Consistency, independence
... Partee, ter Meulen and Wall use the term formal system where we are now using the term relational algebra. With this natural extension to the notions of isomorphism, etc., we can say what it means for an axiom system to be categorical. An axiom system is categorical if all of its models are isomorph ...
... Partee, ter Meulen and Wall use the term formal system where we are now using the term relational algebra. With this natural extension to the notions of isomorphism, etc., we can say what it means for an axiom system to be categorical. An axiom system is categorical if all of its models are isomorph ...
lecture05
... Proof of Trakhtenbrot’s Theorem s = {<, T0(¢,¢), T1(¢, ¢), (Hq(¢, ¢))q 2 Q} The formula fM will say the following: • < is a linear order • T0(p, t) = the tape holds 0 at position p and time t • T1(p, t) = similar • Hq(p, t) = the machine is in state q at time t, and the head is over position p of t ...
... Proof of Trakhtenbrot’s Theorem s = {<, T0(¢,¢), T1(¢, ¢), (Hq(¢, ¢))q 2 Q} The formula fM will say the following: • < is a linear order • T0(p, t) = the tape holds 0 at position p and time t • T1(p, t) = similar • Hq(p, t) = the machine is in state q at time t, and the head is over position p of t ...
Keiichi Komatsu
... Theorem 1. The class number of ( m ) is odd. In our lectures, we prove the above theorem not using class field theory. After 50 years, Iwasawa proved the following theorem using class field theory: Theorem 2 [2]. Let k be an algebraic number field, K a finite Galois extension of k and the Galois g ...
... Theorem 1. The class number of ( m ) is odd. In our lectures, we prove the above theorem not using class field theory. After 50 years, Iwasawa proved the following theorem using class field theory: Theorem 2 [2]. Let k be an algebraic number field, K a finite Galois extension of k and the Galois g ...