Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... This document comprises a set of lecture notes that I prepared in the fall of 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that d ...
... This document comprises a set of lecture notes that I prepared in the fall of 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that d ...
Gödel`s Incompleteness Theorems
... he visited Alonzo Church (1903–1995) in Princeton. In 1934 he gave a series of lectures at Princeton. Church’s two brilliant students Stephen Kleene (1909–1994) and Barkley Rosser (1907–1989) took notes, which were subsequently published in [17]. At that time Church was contemplating how to formaliz ...
... he visited Alonzo Church (1903–1995) in Princeton. In 1934 he gave a series of lectures at Princeton. Church’s two brilliant students Stephen Kleene (1909–1994) and Barkley Rosser (1907–1989) took notes, which were subsequently published in [17]. At that time Church was contemplating how to formaliz ...
Acta Mathematica et Informatica Universitatis Ostraviensis - DML-CZ
... Diophantus studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: -—, ff, T ' W ( s e e -7-)* T h e ^ r s t s e t ° ^ o u r positive integers with the above property was fou ...
... Diophantus studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: -—, ff, T ' W ( s e e -7-)* T h e ^ r s t s e t ° ^ o u r positive integers with the above property was fou ...
Solution for Fermat`s Last Theorem
... The demonstration of the Taniyama-Shimura conjecture was already on a challenge of the utmost importance, because that was one of the points of the so-called Langlands program, whose goal is to unify areas of mathematics which apparently have no unrelated. Wiles spent 8 years following the demonstra ...
... The demonstration of the Taniyama-Shimura conjecture was already on a challenge of the utmost importance, because that was one of the points of the so-called Langlands program, whose goal is to unify areas of mathematics which apparently have no unrelated. Wiles spent 8 years following the demonstra ...
Topological conjugacy and symbolic dynamics
... One way to describe orbits of the tent map (and hence also orbits of the logistic map f4 (x) = 4x(1 − x)) is by their ‘left-right’ itineraries: to each x0 ∈ [0, 1] we assign a sequence of L’s and R’s by writing ‘L’ if the nth point of the orbit of x0 is in [0, 1/2] and ‘R’ if it is in [1/2, 1]. This ...
... One way to describe orbits of the tent map (and hence also orbits of the logistic map f4 (x) = 4x(1 − x)) is by their ‘left-right’ itineraries: to each x0 ∈ [0, 1] we assign a sequence of L’s and R’s by writing ‘L’ if the nth point of the orbit of x0 is in [0, 1/2] and ‘R’ if it is in [1/2, 1]. This ...
Lecture Notes - Alistair Savage
... mathematics courses. We will discuss the axiomatic method and various methods of proof (proof by contradiction, mathematical induction, etc.). We do this in the context of some fundamental notions in mathematics that are crucial for upper-level mathematics courses. These include the basics of naive ...
... mathematics courses. We will discuss the axiomatic method and various methods of proof (proof by contradiction, mathematical induction, etc.). We do this in the context of some fundamental notions in mathematics that are crucial for upper-level mathematics courses. These include the basics of naive ...
1. Sets, relations and functions. 1.1. Set theory. We assume the
... 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 will be naive and not axiomatic. For ...
... 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 will be naive and not axiomatic. For ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.