1 Introduction to Categories and Categorical Logic
... developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two examples illustrate that many important mathematical structures themselves appear as categories of particular kinds. The fact that two such different kinds ...
... developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two examples illustrate that many important mathematical structures themselves appear as categories of particular kinds. The fact that two such different kinds ...
Hilbert`s Program Then and Now
... between the basic concepts and the axioms. Of basic importance for an axiomatic treatment are, so Hilbert, investigation of the independence and, above all, of the consistency of the axioms. In his 1902 lectures on the foundations of geometry, he puts it thus: Every science takes its starting point ...
... between the basic concepts and the axioms. Of basic importance for an axiomatic treatment are, so Hilbert, investigation of the independence and, above all, of the consistency of the axioms. In his 1902 lectures on the foundations of geometry, he puts it thus: Every science takes its starting point ...
Notes on Mathematical Logic David W. Kueker
... whose sentences include formalizations of the sttements commonly used in mathematics and whose interpretatins include the usual mathematical structures. The details of this become quite intricate, which obscures the “big picture.” We therefore first consider a much simpler situation and carry out ou ...
... whose sentences include formalizations of the sttements commonly used in mathematics and whose interpretatins include the usual mathematical structures. The details of this become quite intricate, which obscures the “big picture.” We therefore first consider a much simpler situation and carry out ou ...
Approximate equivalence relations.
... If Aut(G, R) acts transitively on G, then of course |S(a)| ≥ Ok,m (1)|R(a)| for all a ∈ G. So we recover in this setting Corollary 2.19: Corollary 2.19. Fix k ∈ N, m ∈ N. Let G be a group, X a finite subset of G, 1 ∈ X = X −1 , and assume |X ·3 | ≤ k|X|. Then there exists S, 1 ∈ S = S −1 ⊂ G, such t ...
... If Aut(G, R) acts transitively on G, then of course |S(a)| ≥ Ok,m (1)|R(a)| for all a ∈ G. So we recover in this setting Corollary 2.19: Corollary 2.19. Fix k ∈ N, m ∈ N. Let G be a group, X a finite subset of G, 1 ∈ X = X −1 , and assume |X ·3 | ≤ k|X|. Then there exists S, 1 ∈ S = S −1 ⊂ G, such t ...
Belief Revision in non
... However, the notion of consistency in the object logic may differ from that of classical logic. The AGM revision relies somehow on the notion of (classical) consistency. This can be easily understood by analysing postulate (K 3;4 ). Revision is only triggered when the new information is inconsisten ...
... However, the notion of consistency in the object logic may differ from that of classical logic. The AGM revision relies somehow on the notion of (classical) consistency. This can be easily understood by analysing postulate (K 3;4 ). Revision is only triggered when the new information is inconsisten ...
Hilbert`s Program Then and Now - Philsci
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
... the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new found ...
DISCRETE MATHEMATICAL STRUCTURES
... elements is irrelevant, so {a, b} = {b, a}. If the order of the elements is relevant, then we use a different object called ordered pair, represented (a, b). Now (a, b) = (b, a) (unless a = b). In general (a, b) = (a!, b! ) iff a = a! and b = b! . Given two sets A, B, their Cartesian product A × B i ...
... elements is irrelevant, so {a, b} = {b, a}. If the order of the elements is relevant, then we use a different object called ordered pair, represented (a, b). Now (a, b) = (b, a) (unless a = b). In general (a, b) = (a!, b! ) iff a = a! and b = b! . Given two sets A, B, their Cartesian product A × B i ...
Introduction to Mathematical Logic lecture notes
... we will be able to deduce (or “prove”) formulae from other formulae. Indeed, in real-life Mathematics, a proof is merely a sequence of assertions (alas, in an informal natural ...
... we will be able to deduce (or “prove”) formulae from other formulae. Indeed, in real-life Mathematics, a proof is merely a sequence of assertions (alas, in an informal natural ...
INDEX SETS FOR n-DECIDABLE STRUCTURES CATEGORICAL
... This means that for every Π11 set S there is a uniformly computable sequence of structures {Ai }i∈ω such that i ∈ S ⇐⇒ Ai is computably categorical. Marker in [25] defined ∀- and ∃-extensions, A∀ and A∃ , respectively, of an arbitrary structure A. The main property is that the domain and the basic r ...
... This means that for every Π11 set S there is a uniformly computable sequence of structures {Ai }i∈ω such that i ∈ S ⇐⇒ Ai is computably categorical. Marker in [25] defined ∀- and ∃-extensions, A∀ and A∃ , respectively, of an arbitrary structure A. The main property is that the domain and the basic r ...
CS 208: Automata Theory and Logic
... – We say that a language L is decidable if there exists a program PL such that for every member of L program P returns “true”, and for every non-member it returns “false”. Ashutosh Trivedi – 17 of 19 Ashutosh Trivedi ...
... – We say that a language L is decidable if there exists a program PL such that for every member of L program P returns “true”, and for every non-member it returns “false”. Ashutosh Trivedi – 17 of 19 Ashutosh Trivedi ...
Consequence Operators for Defeasible - SeDiCI
... Proposition 5.3 (Monotonicity). The operator Cwar (¡ ) does not satisfy monotonicity. Proof. A counterexample su±ces. Consider the example given in proposition 4.3. In that case, ¡ j»T pU hence p is warranted. However, in ¡ [f[;; fn1g]:qg there is no argument with conclusion p (and consequently p is ...
... Proposition 5.3 (Monotonicity). The operator Cwar (¡ ) does not satisfy monotonicity. Proof. A counterexample su±ces. Consider the example given in proposition 4.3. In that case, ¡ j»T pU hence p is warranted. However, in ¡ [f[;; fn1g]:qg there is no argument with conclusion p (and consequently p is ...
Chapter 6: The Deductive Characterization of Logic
... sequence σ has a first element σ 1, a second element σ 2, etc. If σ is n-long, then σ n is the last element of σ. Also, to say that σ is a sequence of so-and-so’s is to say that each σ i is a so-and-so. But what is an inference rule? What does it to say that something follows by one? We will largely ...
... sequence σ has a first element σ 1, a second element σ 2, etc. If σ is n-long, then σ n is the last element of σ. Also, to say that σ is a sequence of so-and-so’s is to say that each σ i is a so-and-so. But what is an inference rule? What does it to say that something follows by one? We will largely ...
Yablo`s paradox
... Stephen Yablo has given an ingenious liar-style paradox that, he claims, avoids self-reference, even of an indirect kind, one that is, in fact, ‘not in any way circular’ (Yablo 1993, his italics). He infers that such circularity is not necessary for this kind of paradox. Some others have agreed.1 Th ...
... Stephen Yablo has given an ingenious liar-style paradox that, he claims, avoids self-reference, even of an indirect kind, one that is, in fact, ‘not in any way circular’ (Yablo 1993, his italics). He infers that such circularity is not necessary for this kind of paradox. Some others have agreed.1 Th ...
Logic and discrete mathematics (HKGAB4) http://www.ida.liu.se
... 3. Functions. Discrete structures. • Discrete Mathematics: ...
... 3. Functions. Discrete structures. • Discrete Mathematics: ...
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.