lec26-first-order
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
The Non-Euclidean Revolution Material Axiomatic Systems and the
... Material Axiomatic Systems and the Turtle Club Example Recall that a material axiomatic system consists of four parts: the primitive (or undefined) terms, the defined terms, the axioms (or assumptions) used as the starting point for deduction, and the theorems (or statements requiring proof). A proo ...
... Material Axiomatic Systems and the Turtle Club Example Recall that a material axiomatic system consists of four parts: the primitive (or undefined) terms, the defined terms, the axioms (or assumptions) used as the starting point for deduction, and the theorems (or statements requiring proof). A proo ...
The Axiom of Choice
... were seeking. Does it seem somehow unsatisfying that we magically showed that there has to be a maximal subset satisfying P , without giving any indication of what it might be? Again, this is a typical example of a proof using the axiom of choice (in Zorn’s lemma form). Zorn’s lemma can also be used ...
... were seeking. Does it seem somehow unsatisfying that we magically showed that there has to be a maximal subset satisfying P , without giving any indication of what it might be? Again, this is a typical example of a proof using the axiom of choice (in Zorn’s lemma form). Zorn’s lemma can also be used ...
Godel`s Incompleteness Theorem
... • We tried a very small set of 6 axioms, called the Peano axioms, designed for a small subset of mathematics: natural number arithmetic. • We found that we could indeed prove several (non-trivial) theorems about arithmetic from the Peano Axioms. Cool! • We also found that some arithmetical truths co ...
... • We tried a very small set of 6 axioms, called the Peano axioms, designed for a small subset of mathematics: natural number arithmetic. • We found that we could indeed prove several (non-trivial) theorems about arithmetic from the Peano Axioms. Cool! • We also found that some arithmetical truths co ...
PDF
... of the following forms: B → C, ¬B, or ∀xB, where B, C are wff’s. If A were B → C or ¬B, by induction, since B and C were in Γ, A is in Γ as a result. If A were ∀xB, then A is quasi-atomic, and therefore in Γ by the definition of Γ. Unique readability follows from the unique readability of wff’s of p ...
... of the following forms: B → C, ¬B, or ∀xB, where B, C are wff’s. If A were B → C or ¬B, by induction, since B and C were in Γ, A is in Γ as a result. If A were ∀xB, then A is quasi-atomic, and therefore in Γ by the definition of Γ. Unique readability follows from the unique readability of wff’s of p ...
First order theories
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
First order theories - Decision Procedures
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
... Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula? ...
Set theory, by Thomas Jech, Academic Press, New York, 1978, xii +
... result of mathematical significance was GödeFs proof, around 1938, that the GCH is consistent with ZFC. Much later, around 1963, Cohen showed that the negation of GCH (and in fact of CH), is also consistent with ZFC. It was perhaps Cohen's proof which had the greater influence on mathematics; Gödel' ...
... result of mathematical significance was GödeFs proof, around 1938, that the GCH is consistent with ZFC. Much later, around 1963, Cohen showed that the negation of GCH (and in fact of CH), is also consistent with ZFC. It was perhaps Cohen's proof which had the greater influence on mathematics; Gödel' ...
on Computability
... Godel's Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the sys ...
... Godel's Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the sys ...
Arithmetic as a theory modulo
... In deduction modulo, the notions of language, term and proposition are those of predicate logic. But, a theory is formed with a set of axioms Γ and a congruence ≡ defined on propositions. Such a congruence may be defined by a rewrite system on terms and on propositions (as propositions contain binde ...
... In deduction modulo, the notions of language, term and proposition are those of predicate logic. But, a theory is formed with a set of axioms Γ and a congruence ≡ defined on propositions. Such a congruence may be defined by a rewrite system on terms and on propositions (as propositions contain binde ...
PHIL012 Class Notes
... • “a=b” means that “a” and “b” are names that refer to the same objects, which can denote numbers or sets. • “a=b” also means that whatever claims are made of a must also be true of b (and vice versa) if “a=b” is true. ...
... • “a=b” means that “a” and “b” are names that refer to the same objects, which can denote numbers or sets. • “a=b” also means that whatever claims are made of a must also be true of b (and vice versa) if “a=b” is true. ...
Assignment 6
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
Introduction to Theoretical Computer Science, lesson 3
... Formula A is satisfiable in interpretation I, if there exists valuation v of variables that |=I A[v]. Formula A is true in interpretation I, |=I A, if for all possible valuations v holds that |=I A[v]. Model of a formula A is an interpretation I, in which A is true (that means for all valuations of ...
... Formula A is satisfiable in interpretation I, if there exists valuation v of variables that |=I A[v]. Formula A is true in interpretation I, |=I A, if for all possible valuations v holds that |=I A[v]. Model of a formula A is an interpretation I, in which A is true (that means for all valuations of ...
The theorem, it`s meaning and the central concepts
... “This sentence cannot be proven” 4) Analysis of the formula U that shows, that neither U or ~U can be proven in the system N The idea in the proof is to construct a sentence U that is similar to “The liar’s paradox”: “This sentence is a lie”. But unlike the paradox, the sentence U is NOT a true para ...
... “This sentence cannot be proven” 4) Analysis of the formula U that shows, that neither U or ~U can be proven in the system N The idea in the proof is to construct a sentence U that is similar to “The liar’s paradox”: “This sentence is a lie”. But unlike the paradox, the sentence U is NOT a true para ...
KNOWLEDGE
... some kind which ‘exists’ in the world. One objection is that it is the meaning of statements or beliefs which count, and this is what a proposition (p) is. Propositions rather than beliefs carry truth or falsity. I should say “p is true and I believe it” rather than “I believe p”. ...
... some kind which ‘exists’ in the world. One objection is that it is the meaning of statements or beliefs which count, and this is what a proposition (p) is. Propositions rather than beliefs carry truth or falsity. I should say “p is true and I believe it” rather than “I believe p”. ...
PDF
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
Propositional logic
... Definition: a set of wffs S logically implies a wff a, S |= a, provided that for each assignment s such that s(b) = T for each bŒS, s(a) = T (if S = ∅, write |= a and a is a tautology). ...
... Definition: a set of wffs S logically implies a wff a, S |= a, provided that for each assignment s such that s(b) = T for each bŒS, s(a) = T (if S = ∅, write |= a and a is a tautology). ...
PDF
... 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 ...
Lecture 10. Model theory. Consistency, independence
... A set of axioms ∆ is semantically complete with respect to a model M, or weakly semantically complete, if every sentence which holds in M is derivable from ∆. Three notions of completeness. We have now seen three notions of completeness: (i) a logic may be complete: everything which should be a theo ...
... A set of axioms ∆ is semantically complete with respect to a model M, or weakly semantically complete, if every sentence which holds in M is derivable from ∆. Three notions of completeness. We have now seen three notions of completeness: (i) a logic may be complete: everything which should be a theo ...
On the paradoxes of set theory
... Russell announced his famous “antinomy” and this time there was no attempt to avoid the difficulty by subterfuge or other wise, and the problem of “what to do?” had to be met. This was a development of great importance which occur red at the end of the nineteenth and at the beginning of the twentiet ...
... Russell announced his famous “antinomy” and this time there was no attempt to avoid the difficulty by subterfuge or other wise, and the problem of “what to do?” had to be met. This was a development of great importance which occur red at the end of the nineteenth and at the beginning of the twentiet ...
Hierarchical Introspective Logics
... unprovable, yet true, proposition in a generic sort of formal system is accompanied also by a proof, under reasonable hypotheses, of the truth of that "unprovable" proposition. How is this achieved? Well, of course, the proof does NOT occur WITHIN the formal system for which the Goedel proposition w ...
... unprovable, yet true, proposition in a generic sort of formal system is accompanied also by a proof, under reasonable hypotheses, of the truth of that "unprovable" proposition. How is this achieved? Well, of course, the proof does NOT occur WITHIN the formal system for which the Goedel proposition w ...
Why the Sets of NF do not form a Cartesian-closed Category
... knowledge—and quite possibly at least five times since my guess would be that Dana Scott and Solomon Feferman discovered proofs in addition to the three proofs known to me; I know Randall Holmes did, and when I met Edmund Robinson in about 1980 the first question he asked me—on learning that I studi ...
... knowledge—and quite possibly at least five times since my guess would be that Dana Scott and Solomon Feferman discovered proofs in addition to the three proofs known to me; I know Randall Holmes did, and when I met Edmund Robinson in about 1980 the first question he asked me—on learning that I studi ...
KRIPKE-PLATEK SET THEORY AND THE ANTI
... [4]). Instead of the Foundation Axiom these set theories adopt the so-called AntiFoundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical ec ...
... [4]). Instead of the Foundation Axiom these set theories adopt the so-called AntiFoundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical ec ...