paper by David Pierce
... In mathematics we use repeated activity in several ways: (1) to define sets; (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; i ...
... In mathematics we use repeated activity in several ways: (1) to define sets; (2) to prove that all elements of those sets have certain properties; (3) to define functions on those sets. These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; i ...
Constructive Mathematics in Theory and Programming Practice
... rather than ontological, one. From now on, when we speak of 'normal mathematical objects', we have in mind the kind of things that are handled by either Heyting arithmetic— the Peano axioms plus intuitionistic logic—or, at a higher level, a formal system such as intuitionistic set theory (IZF), Myhi ...
... rather than ontological, one. From now on, when we speak of 'normal mathematical objects', we have in mind the kind of things that are handled by either Heyting arithmetic— the Peano axioms plus intuitionistic logic—or, at a higher level, a formal system such as intuitionistic set theory (IZF), Myhi ...
From proof theory to theories theory
... In order to design a complete proof search method for this theory, that fails in finite time when searching for a proof of ⊢ ⊥, we may attempt to prove a cut elimination theorem, specialized for this theory, that implies its consistency, in the same way as the cut elimination theorem for arithmetic ...
... In order to design a complete proof search method for this theory, that fails in finite time when searching for a proof of ⊢ ⊥, we may attempt to prove a cut elimination theorem, specialized for this theory, that implies its consistency, in the same way as the cut elimination theorem for arithmetic ...
deductive system
... Some Major Formulations of Deductive Systems in Logic Let us fix a language L (of well-formed formulas). There are four main formulations of deductive systems: • Hilbert system, or axiom system: in this formulation, axioms are the main ingredient, and there are only one or two rules of inference (m ...
... Some Major Formulations of Deductive Systems in Logic Let us fix a language L (of well-formed formulas). There are four main formulations of deductive systems: • Hilbert system, or axiom system: in this formulation, axioms are the main ingredient, and there are only one or two rules of inference (m ...
Chapter 1 Number Sets and Properties
... • A set is a collection of numbers or objects. - If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers. • An element is a member of a set. - 1, 2, 3, 4 and 5 are all elements of A. - means ‘is an element of’ hence 4 A. - means ‘is not an element of’ hence 7 A. - means ‘the emp ...
... • A set is a collection of numbers or objects. - If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers. • An element is a member of a set. - 1, 2, 3, 4 and 5 are all elements of A. - means ‘is an element of’ hence 4 A. - means ‘is not an element of’ hence 7 A. - means ‘the emp ...
Is the Liar Sentence Both True and False? - NYU Philosophy
... lead us to assert both ¬A and ¬¬A. But to assert both a sentence (¬A) and its negation is, in at least one sense of the phrase, to assert a contradiction. Accepting contradictions would be intolerable if contradictions implied everything: we would be logically committed to every imaginable absurdity ...
... lead us to assert both ¬A and ¬¬A. But to assert both a sentence (¬A) and its negation is, in at least one sense of the phrase, to assert a contradiction. Accepting contradictions would be intolerable if contradictions implied everything: we would be logically committed to every imaginable absurdity ...
Cantor - Muskingum University
... Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved ...
... Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved ...
1. Sets, relations and functions. 1.1. Set theory. We assume the
... 1. Sets, relations and functions. 1.1. Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics today. Nonetheless, we shall now give a careful treatment of set theory if only to to allow the reader to become conversant with our notation. Our treatment wil ...
... 1. Sets, relations and functions. 1.1. Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics today. Nonetheless, we shall now give a careful treatment of set theory if only to to allow the reader to become conversant with our notation. Our treatment wil ...
1 The Inclusion-Exclusion Principle
... While finding members of a collection of sets union is a useful context in which to use inclusionexclusion, it’s more commonly used for finding sets of numbers which lack several properties. That Page 2 of 7 ...
... While finding members of a collection of sets union is a useful context in which to use inclusionexclusion, it’s more commonly used for finding sets of numbers which lack several properties. That Page 2 of 7 ...
M131-Tutorial_3-Integers-Division
... smallest positive integer that is a multiple both of a and of b. • Find lcm(6, 10)? Take the maximum power of all factors ...
... smallest positive integer that is a multiple both of a and of b. • Find lcm(6, 10)? Take the maximum power of all factors ...
FOR HIGHER-ORDER RELEVANT LOGIC
... It is time to move up; at the higher-order level, the classical admissibility of Gentzen’s cut-rule is the basic conjecture of Takeuti, whose verification in [4] and [5] is severely non-constructive. A relevant counterpart would be a proof of γ for a suitable higher-order logic. Such logics are wort ...
... It is time to move up; at the higher-order level, the classical admissibility of Gentzen’s cut-rule is the basic conjecture of Takeuti, whose verification in [4] and [5] is severely non-constructive. A relevant counterpart would be a proof of γ for a suitable higher-order logic. Such logics are wort ...
THE LANGUAGE OF SETS AND SET NOTATION Mathematics is
... You are a member of the set of students at your College. The readers of this handout form a subset of the entire set of students or the universal set. The set of those students who do not read this handout form the complement of the set who do. If each person reading this has his or her own copy, th ...
... You are a member of the set of students at your College. The readers of this handout form a subset of the entire set of students or the universal set. The set of those students who do not read this handout form the complement of the set who do. If each person reading this has his or her own copy, th ...
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.