Mathematical Logic Fall 2004 Professor R. Moosa Contents
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
Mathematical Structures for Reachability Sets and Relations Summary
... sets by formulæ from a decidable logical formalism. Presburger arithmetic would be a serious candidate for such a logic, but we know that it is not expressive enough. Indeed, Hopcroft and Pansiot (1979) proved that the reachability sets of any VASS of dimension 2 are effectively definable in Presbur ...
... sets by formulæ from a decidable logical formalism. Presburger arithmetic would be a serious candidate for such a logic, but we know that it is not expressive enough. Indeed, Hopcroft and Pansiot (1979) proved that the reachability sets of any VASS of dimension 2 are effectively definable in Presbur ...
Lecture01 - Mathematics
... theory is parallel to that of propositional logic. Sets are foundational mathematical objects. In some sense practically every mathematical object (numbers, functions, surfaces, etc.) is a set. iii) Relations: In mathematics it is frequently useful to describe or define some relationship between mat ...
... theory is parallel to that of propositional logic. Sets are foundational mathematical objects. In some sense practically every mathematical object (numbers, functions, surfaces, etc.) is a set. iii) Relations: In mathematics it is frequently useful to describe or define some relationship between mat ...
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... situation on pattern of example 2, has been called in [9] the exceptions-first principle. It is a very general and formal (it does not depends on the semantics of the theory) principle. It applies to all contexts independently on the subject matter, might it be temporal projection, causal relations, ...
... situation on pattern of example 2, has been called in [9] the exceptions-first principle. It is a very general and formal (it does not depends on the semantics of the theory) principle. It applies to all contexts independently on the subject matter, might it be temporal projection, causal relations, ...
CptS 440 / 540 Artificial Intelligence
... • Fuzzy Logic is a multivalued logic that allows intermediate values to be defined between conventional evaluations like yes/no, true/false, black/white, etc. • Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor of computer science at the University of California in Berkeley. • The conce ...
... • Fuzzy Logic is a multivalued logic that allows intermediate values to be defined between conventional evaluations like yes/no, true/false, black/white, etc. • Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor of computer science at the University of California in Berkeley. • The conce ...
ON LOVELY PAIRS OF GEOMETRIC STRUCTURES 1. Introduction
... 2. Lovely pairs of geometric structures We begin by translating to the setting of geometric structures, the definitions used by Vassiliev in [24]. Let T be a complete theory in a language L such that for any model M |= T , the algebraic closure satisfies the Exchange Property and that eliminates th ...
... 2. Lovely pairs of geometric structures We begin by translating to the setting of geometric structures, the definitions used by Vassiliev in [24]. Let T be a complete theory in a language L such that for any model M |= T , the algebraic closure satisfies the Exchange Property and that eliminates th ...
Bounded Functional Interpretation
... of the analysis, not by the interpretation itself. At the same time, a version of the (intuitionistically acceptable) FAN theorem is interpreted by the b.f.i.: ∀g ≤1 f ∃nA(g, n) → ∃k∀g ≤1 f ∃n ≤ kA(g, n), where A is any formula, provided that we read the relation ≤1 intensionally (more on this below ...
... of the analysis, not by the interpretation itself. At the same time, a version of the (intuitionistically acceptable) FAN theorem is interpreted by the b.f.i.: ∀g ≤1 f ∃nA(g, n) → ∃k∀g ≤1 f ∃n ≤ kA(g, n), where A is any formula, provided that we read the relation ≤1 intensionally (more on this below ...
Ordered Groups: A Case Study In Reverse Mathematics 1 Introduction
... Mathematics ∗ Reed Solomon August 28, 2003 ...
... Mathematics ∗ Reed Solomon August 28, 2003 ...
Examples of equivalence relations
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
... Let X be a set and let x, y, and z be elements of X. An equivalence relation, ~, on X is a relation on X such that: Reflexive Property: x is equivalent to x for all x in X. Symmetric Property: if x is equivalent to y, then y is equivalent to x. Transitive Property: if x is equivalent to y and y is e ...
An Introduction to Equivalence Relations and Partitions 1 2 3 Set X
... • equality (=) relation between elements of any set. Suppose A, B and C are three equal sets of natural numbers. Let a, b, c be the elements of the set A, B, C. Reflexive Property: a is equivalent to a for all a in A. Symmetric Property: if a is equivalent to b, then b is equivalent to a. Transitive ...
... • equality (=) relation between elements of any set. Suppose A, B and C are three equal sets of natural numbers. Let a, b, c be the elements of the set A, B, C. Reflexive Property: a is equivalent to a for all a in A. Symmetric Property: if a is equivalent to b, then b is equivalent to a. Transitive ...
p. 1 Math 490 Notes 4 We continue our examination of well
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
Topological Completeness of First-Order Modal Logic
... among formulas of L; that is, ϕ ≈α ψ iff ϕ and ψ share the same variable structure possibly with relabeling of bound variables. Also write ≈f for sharing the same variable structure possibly with relabeling of free variables. More precisely, ϕ ≈f ψ iff ϕ - ψ and ψ - ϕ for the transitive closure - of ...
... among formulas of L; that is, ϕ ≈α ψ iff ϕ and ψ share the same variable structure possibly with relabeling of bound variables. Also write ≈f for sharing the same variable structure possibly with relabeling of free variables. More precisely, ϕ ≈f ψ iff ϕ - ψ and ψ - ϕ for the transitive closure - of ...
lecture6.1
... She chooses two (large) prime numbers, p and q and computes n=pq and (n) . [“large” =512 bits +] She chooses a number e such that e is relatively prime to (n) and computes d, the inverse of ...
... She chooses two (large) prime numbers, p and q and computes n=pq and (n) . [“large” =512 bits +] She chooses a number e such that e is relatively prime to (n) and computes d, the inverse of ...
Axiomatic Set Teory P.D.Welch.
... It is thus consistent with ZF that V = L is true! If V = L is true, then there are many consequences for mathematics: the study of L is now highly developed and many consequences for analysis, algebra,... have been shown to hold in L whose proof either remains elusive, or else is downright unprovabl ...
... It is thus consistent with ZF that V = L is true! If V = L is true, then there are many consequences for mathematics: the study of L is now highly developed and many consequences for analysis, algebra,... have been shown to hold in L whose proof either remains elusive, or else is downright unprovabl ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.