Characterizing integers among rational numbers
... (i) If p ∈ / ∆a,b , then Ha,b ⊗ Qp ' M2 (Qp ), and any monic quadratic polynomial is a characteristic polynomial. (ii) Now suppose that p ∈ ∆a,b . Then Ha,b ⊗ Qp is the ramified quaternion algebra over Qp , and x2 − sx + 1 is a reduced characteristic polynomial if and only if it is a power of a mon ...
... (i) If p ∈ / ∆a,b , then Ha,b ⊗ Qp ' M2 (Qp ), and any monic quadratic polynomial is a characteristic polynomial. (ii) Now suppose that p ∈ ∆a,b . Then Ha,b ⊗ Qp is the ramified quaternion algebra over Qp , and x2 − sx + 1 is a reduced characteristic polynomial if and only if it is a power of a mon ...
Relative normalization
... proof-language of T is complex: it contains proof-variables, proof-terms, as well as the terms of the theory T (that appear in proof-terms). Moreover, we need to express usual syntactic operations, such as α-conversion, substitution, etc. For that let us consider a language of trees L generated by a ...
... proof-language of T is complex: it contains proof-variables, proof-terms, as well as the terms of the theory T (that appear in proof-terms). Moreover, we need to express usual syntactic operations, such as α-conversion, substitution, etc. For that let us consider a language of trees L generated by a ...
Elements of Finite Model Theory
... structures via the use of pseudo-finite structures (a structure is pseudo-finite, if every first-order sentence it satisfies is true in some finite structure), the breadth of such application seems limited. The Chapter then introduces a fundamental technique for establishing the inexpressibility of ...
... structures via the use of pseudo-finite structures (a structure is pseudo-finite, if every first-order sentence it satisfies is true in some finite structure), the breadth of such application seems limited. The Chapter then introduces a fundamental technique for establishing the inexpressibility of ...
The absolute proof!
... step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem. Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are abso ...
... step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem. Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are abso ...
Infinite natural numbers: an unwanted phenomenon, or a useful
... clusters are densely ordered. Schematically, the order structure of M is ω + (ω ? + ω) · ξ, where ξ is a linear dense order without endpoints and the multiplication symbol · denotes the operation of replacing each element of ξ by the structure ω ? + ω, see Fig. 1. If the model M is countable then it ...
... clusters are densely ordered. Schematically, the order structure of M is ω + (ω ? + ω) · ξ, where ξ is a linear dense order without endpoints and the multiplication symbol · denotes the operation of replacing each element of ξ by the structure ω ? + ω, see Fig. 1. If the model M is countable then it ...
College Geometry University of Memphis MATH 3581 Mathematical
... Theorem: A propositional statement in conditional (“If – Then”) form, for which a proof is given or known. Note: In mathematics and the physical sciences, we do not refer to a statement as a “theorem” unless we know it to be true. Conjecture: A statement given in the same form as a theorem, but for ...
... Theorem: A propositional statement in conditional (“If – Then”) form, for which a proof is given or known. Note: In mathematics and the physical sciences, we do not refer to a statement as a “theorem” unless we know it to be true. Conjecture: A statement given in the same form as a theorem, but for ...
Slides
... Definition (Proof-step Constraint): let A1…Ak be the Antecedents and p the Proposition of step. Then: Boolean encoding ...
... Definition (Proof-step Constraint): let A1…Ak be the Antecedents and p the Proposition of step. Then: Boolean encoding ...
Friendly Logics, Fall 2015, Homework 1
... Problem 1alt Let T be a nonempty set. Fix a 2 T and e : T ! T . Prove that there exists a function f : N ! T such that (1) f (0) = a and (2) 8n 2 N f (n + 1) = e(f (n)). (It follows immediately by induction that f is uniquely determined by properties (1) and (2). But how do you know that such an f e ...
... Problem 1alt Let T be a nonempty set. Fix a 2 T and e : T ! T . Prove that there exists a function f : N ! T such that (1) f (0) = a and (2) 8n 2 N f (n + 1) = e(f (n)). (It follows immediately by induction that f is uniquely determined by properties (1) and (2). But how do you know that such an f e ...
Friedman`s Translation
... Theorem 1.5. If ` A is derivable in classical predicate logic and if no free variable of R occurs in the derivation, then ` A¬R is derivable in intuitionistic predicate logic. In order to obtain Theorem 1.5 for arithmetic, it remains to show that HA∗ proves the ¬R -translation of all its axioms. The ...
... Theorem 1.5. If ` A is derivable in classical predicate logic and if no free variable of R occurs in the derivation, then ` A¬R is derivable in intuitionistic predicate logic. In order to obtain Theorem 1.5 for arithmetic, it remains to show that HA∗ proves the ¬R -translation of all its axioms. The ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... the two-sorted language with one sort for numbers and one sort for strings as the preferred language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded ari ...
... the two-sorted language with one sort for numbers and one sort for strings as the preferred language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded ari ...
Lecture 16 Notes
... We will look briefly at the incompleteness result since that has received a good deal of attention as a contrast to the Gödel completeness result for classical FOL. The folklore has it that Gödel’s result cannot be constructive.1 We will not explore Kripke models and the important result of Vim Ve ...
... We will look briefly at the incompleteness result since that has received a good deal of attention as a contrast to the Gödel completeness result for classical FOL. The folklore has it that Gödel’s result cannot be constructive.1 We will not explore Kripke models and the important result of Vim Ve ...
First-Order Default Logic 1 Introduction
... We propose a model theory for full first-order default logic that allows both closed and non-closed default theories. Beginning with first-order languages without logical equality, we note how Henkin’s proof of the completeness theorem for first-order logic yields complete algebras; that is, algebra ...
... We propose a model theory for full first-order default logic that allows both closed and non-closed default theories. Beginning with first-order languages without logical equality, we note how Henkin’s proof of the completeness theorem for first-order logic yields complete algebras; that is, algebra ...
4. Overview of Meaning Proto
... • Generally, the use theorists assume that there are many different uses of language, and that the tradi6onal theories don’t well capture them all. (E.g.: performa6ve language is a kind of use tha ...
... • Generally, the use theorists assume that there are many different uses of language, and that the tradi6onal theories don’t well capture them all. (E.g.: performa6ve language is a kind of use tha ...
Practice Problem Set 1
... (b) We now wish to find a directed graph G such that φ ∧ ψ evaluates to True over MG . Indicate with justification whether G can be a finite graph (i.e., graph with finite number of vertices) and yet cause φ ∧ ψ to evaluate to True over MG . 4. [From Quiz1, Autumn 2011] Let Σ be a relational signatu ...
... (b) We now wish to find a directed graph G such that φ ∧ ψ evaluates to True over MG . Indicate with justification whether G can be a finite graph (i.e., graph with finite number of vertices) and yet cause φ ∧ ψ to evaluate to True over MG . 4. [From Quiz1, Autumn 2011] Let Σ be a relational signatu ...
Examples of Ground Resolution Proofs 1 Ground Resolution
... James Worrell In this lecture we show how to use the Ground Resolution Theorem, proved in the last lecture, to do some deduction in first-order logic. ...
... James Worrell In this lecture we show how to use the Ground Resolution Theorem, proved in the last lecture, to do some deduction in first-order logic. ...
a simple derivation of jacobi`s four-square formula
... briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange was greatly assisted by Euler, who derived an identity which was crucial in Lagrange ...
... briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange was greatly assisted by Euler, who derived an identity which was crucial in Lagrange ...
Weak Theories and Essential Incompleteness
... theory (which is recursively axiomatizable and of which Peano arithmetic is an extension) is essentially incomplete. A theory T is decidable if the set of all its theorems, i.e. the set of all sentences provable in T, is recursive. If T is not decidable then it is undecidable. Trivially, a decidable ...
... theory (which is recursively axiomatizable and of which Peano arithmetic is an extension) is essentially incomplete. A theory T is decidable if the set of all its theorems, i.e. the set of all sentences provable in T, is recursive. If T is not decidable then it is undecidable. Trivially, a decidable ...
COMPLETENESS OF THE RANDOM GRAPH
... Definition 3.1. A graph G is a pair of sets (V, E) where E ⊆ V × V . The elements of V we call vertices, and the elments of E we call edges. Two vertices v1 , v2 ∈ V are adjacent to each other if (v1 , v2 ) ∈ E, in which case we shall write v1 ↔ v2 , and v1 6↔ v2 otherwise. Note that this is not a s ...
... Definition 3.1. A graph G is a pair of sets (V, E) where E ⊆ V × V . The elements of V we call vertices, and the elments of E we call edges. Two vertices v1 , v2 ∈ V are adjacent to each other if (v1 , v2 ) ∈ E, in which case we shall write v1 ↔ v2 , and v1 6↔ v2 otherwise. Note that this is not a s ...
Handout on Revenge
... How does the revenge paradox manifest itself in this system? We may consider a sentence, δ, which says of itself that it is not determinately true: δ ↔ ¬∆T r(pδq). From this and the above principles we may show: ¬∆∆δ Revenge results in higher order indeterminacy instead of inconsistency. If an opera ...
... How does the revenge paradox manifest itself in this system? We may consider a sentence, δ, which says of itself that it is not determinately true: δ ↔ ¬∆T r(pδq). From this and the above principles we may show: ¬∆∆δ Revenge results in higher order indeterminacy instead of inconsistency. If an opera ...
A Syntactic Characterization of Minimal Entailment
... semantics for cwaS . Certainly, it follows from the definition 3.1 that (the degree undecidability of) cwaS (Σ) is Π1 relative to Cn(Σ), or Π2 relative to Σ, and seemingly there is no good reason why it should be less. On the other hand, all asymptotically decidable problems (in particular those one ...
... semantics for cwaS . Certainly, it follows from the definition 3.1 that (the degree undecidability of) cwaS (Σ) is Π1 relative to Cn(Σ), or Π2 relative to Σ, and seemingly there is no good reason why it should be less. On the other hand, all asymptotically decidable problems (in particular those one ...
Reasoning About Recursively Defined Data
... Next, if z is not an atom, it must have projections. (4) Vz[-~atom(z) D ~x(K(z) = x)]. Vz[~atom(z) D 3 x ( L ( z ) = x)]. Finally, once an element z lies in A, all iterations of projection functions from z (as long as they are defined) must lie in A. (5) Vz[atom(z) A 3 x ( K ( z ) = x) ~ atom(K(z)) ...
... Next, if z is not an atom, it must have projections. (4) Vz[-~atom(z) D ~x(K(z) = x)]. Vz[~atom(z) D 3 x ( L ( z ) = x)]. Finally, once an element z lies in A, all iterations of projection functions from z (as long as they are defined) must lie in A. (5) Vz[atom(z) A 3 x ( K ( z ) = x) ~ atom(K(z)) ...
Lecture 9. Model theory. Consistency, independence, completeness
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
Is the principle of contradiction a consequence of ? Jean
... Moreover the two guys define model theory with the following equation universal algebra + logic = model theory Although we can enjoy their sense of humor, structures do not reduce to algebras. Model theory is a relation between syntax and semantics, syntax not in the sense of proof-theory but in the ...
... Moreover the two guys define model theory with the following equation universal algebra + logic = model theory Although we can enjoy their sense of humor, structures do not reduce to algebras. Model theory is a relation between syntax and semantics, syntax not in the sense of proof-theory but in the ...
The Decision Problem for Standard Classes
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...