Encyclopedia - Cantor's Diagonal Argument: Encyclopedia - Cantor's Diagonal Argument
Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is als...   » Read the article

Encyclopedia - Cantor's Diagonal Argument: Encyclopedia Ii - Cantor's Diagonal Argument - Real Numbers
Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume ...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia - Axiomatic Set Theory
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initia...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - The Origins Of Rigorous Set Theory
The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same numbe...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia - Cardinal Number
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal nu...   » Read the article

Encyclopedia - Crank Person: Encyclopedia - Crank Person
"Crank" (or kook, crackpot, or quack) is a pejorative term for a person who writes or speaks in an authoritative fashion about a particul...   » Read the article

Encyclopedia - Cardinality: Encyclopedia - Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality –...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia - Ontological Argument
In theology and the philosophy of religion, an ontological argument for the existence of God is an argument that God's existence can be p...   » Read the article

Encyclopedia - Uncountable Set: Encyclopedia - Uncountable Set
In mathematics, an uncountable or nondenumerable set is a set which is not countable. Here, "countable" means countably infinite or finit...   » Read the article

Encyclopedia - Computable Number: Encyclopedia - Computable Number
In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the...   » Read the article

Encyclopedia - Constructivism Mathematics: Encyclopedia - Constructivism Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove tha...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Objections To Set Theory
Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming ...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Well-foundedness And Hypersets
In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness: a set, x0, is well founded iff it has no...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Set Theory Zfc Foundations For Mathematics
From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Axioms For Set Theory
The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel se...   » Read the article

Encyclopedia - Countable Set: Encyclopedia - Countable Set
In mathematics the term countable is used to describe the size of a set, i.e. the number of elements it contains. The notion of an infini...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia - Continuum Hypothesis
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept o...   » Read the article

Encyclopedia - Axiomatic Set Theory: Encyclopedia Ii - Axiomatic Set Theory - Independence In Zfc
Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by fo...   » Read the article

Encyclopedia - Mathematics: Encyclopedia - Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematic...   » Read the article

Encyclopedia - Crank Person: Encyclopedia Ii - Crank Person - Topics Typically Associated With The Crank Label
Crank person - Physics computer science and mathematics. Claims to have produced solutions to problems which have been proven to be un...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Major Themes In Mathematics
An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possib...   » Read the article

Encyclopedia - Cantor Set: Encyclopedia Ii - Cantor Set - Properties
Cantor set - The Cantor set is uncountable. It can be shown that there are as many points left behind in this process as there were tha...   » Read the article

Encyclopedia - Definable Number: Encyclopedia Ii - Definable Number - General Facts
The definable numbers form a field containing all the familiar real numbers such as 0, 1, π, e, et cetera. In particular, it contains al...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - The Size Of A Set
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if ther...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - History
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matt...   » Read the article

Encyclopedia - Primitive Recursive Function: Encyclopedia Ii - Primitive Recursive Function - Definition
Primitive recursive functions take natural numbers or tuples of natural numbers as arguments and produce a natural number. A function whi...   » Read the article

Encyclopedia - Countable Set: Encyclopedia Ii - Countable Set - Gentle Introduction
The elements of a finite set can be listed, say { a1, a2, ..., an }. However, insofar as a set is a logical description of the properties...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Objections To Hume's Principle
As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein e...   » Read the article

Encyclopedia - Real Number: Encyclopedia Ii - Real Number - Properties
Real number - Completeness. The main reason for introducing the reals is that the reals contain all limits. More technically, the reals...   » Read the article

Encyclopedia - Cardinality Of The Continuum: Encyclopedia Ii - Cardinality Of The Continuum - Properties
Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers...   » Read the article

Encyclopedia - Countable Set: Encyclopedia Ii - Countable Set - Gentle Introduction
The elements of a finite set can be listed, say { a1, a2, ..., an}. However, insofar as a set is a logical description of the properties ...   » Read the article

Encyclopedia - Pseudomathematics: Encyclopedia Ii - Pseudomathematics - Impossible Problems
Examples of impossible problems include the following constructions in Euclidean geometry using only a ruler and compass: Squaring the c...   » Read the article

Encyclopedia - Reductio Ad Absurdum: Encyclopedia Ii - Reductio Ad Absurdum - In Philosophy
The following dialogue is an example of reductio ad absurdum: A — You should respect C's belief, for all beliefs are of equal validity...   » Read the article

Encyclopedia - Constructivism Mathematics: Encyclopedia Ii - Constructivism Mathematics - Constructivist Mathematics
Constructivist mathematics use constructivist logic, which is essentially a removal of the law of the excluded middle from classical logi...   » Read the article

Encyclopedia - Computable Number: Encyclopedia Ii - Computable Number - Computing Digit Strings
Turing's original paper defined computable numbers as follows: A real number is computable if its digit sequence can be produced by some...   » Read the article

Encyclopedia - Cardinality: Encyclopedia Ii - Cardinality - Comparing Sets
We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia Ii - Cardinal Number - Motivation
In informal use, a cardinal number is what is normally referred to as a counting number. They may be identified with the natural numbers ...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - The Size Of A Set
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if ther...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Criticisms And Objections
Ontological argument - Gaunilo's island. One of the earliest recorded objections to Anselm's argument was raised by one of Anselm's con...   » Read the article

Encyclopedia - Crank Person: Encyclopedia Ii - Crank Person - Related Terminology
"Kook" is a somewhat similar pejorative term that is usually used to describe a person whose areas of interest are perceived to be eccent...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration, Pure And Applied Mathematics, And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation, Language, And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article

Encyclopedia - Countable Set: Encyclopedia Ii - Countable Set - Definition
A set S is called countable if there exists an injective function If f is also bijective then S is called countably infinite or denume...   » Read the article

Encyclopedia - Countable Set: Encyclopedia Ii - Countable Set - Definition
A set S is called countable if there exists an injective function If f is also bijective then S is called countably infinite or denume...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Anselm's Argument
The ontological argument was first proposed by Anselm in Chapter 2 of the Proslogion. While Anselm did not propose an ontological system,...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Footnote
The quote "Later generations will regard set theory as a disease from which one has recovered" is from Kline[1982], and is apparently his...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Philosophical Assumptions Underlying The Argument
In order to understand the place this argument has in the history of philosophy, it is important to understand the essence of the argumen...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Reception Of The Argument
From the start, Cantor's Theory was controversial among mathematicians and (later) philosophers. I don't know what predominates in Cantor...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Cantor's Argument
Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numbers), which has a larger number...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Naïve Objections
Objections to Cantor's proof (together with objections to Gödel's theorem) are a standard feature of mathematical Usenet discussions. Th...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Objections To Cantor's Theorem
As shown above, most objections to Cantor's theorem (i.e. the theorem that no set can be correlated one-one with the set of all of its su...   » Read the article

Encyclopedia - Definable Number: Encyclopedia Ii - Definable Number - Other Notions Of Definability
The notion of definability treated in this article has been chosen primarily for definiteness, not on the grounds that it's more useful o...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Objection To The Axiom Of Infinity
One of the most common (and also the most respectable) objections to Cantor's theory of infinite number involves the axiom of infinity. I...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation Language And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Mathematical Tools
Old: Abacus Napier's bones, slide rule Ruler and compass Mental calculation New: Calculators and computers Programming languages Compu...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Common Misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. ...   » Read the article

Encyclopedia - Primitive Recursive Function: Encyclopedia Ii - Primitive Recursive Function - Examples
Primitive recursive function - Addition. Intuitively we would like to define addition recursively as: add(0,x)=x add(n+1,x)=add(n,x)+1...   » Read the article

Encyclopedia - Primitive Recursive Function: Encyclopedia Ii - Primitive Recursive Function - Limitations
Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be. Certainly the ini...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Major Themes In Mathematics
An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possib...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Overview Of Fields Of Mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the re...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration Pure And Applied Mathematics And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Introduction
Georg Cantor's argument that there are sets that have a cardinality (or "power" or "number") that is greater than the (already infinite) ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Is Mathematics A Science?
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". If one considers science to be strictly about the physical w...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - Arguments Pro And Con
Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gö...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - A Modern Description Of The Argument
Here's a short, and very general description of the ontological argument: 1) God is the greatest possible being and thus possesses all p...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation Language And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article

Encyclopedia - Reductio Ad Absurdum: Encyclopedia Ii - Reductio Ad Absurdum - As A Figure Of Speech
Among some people, there is a misconception that reductio ad absurdum just means "a silly argument". In general practice, a reductio ad a...   » Read the article

Encyclopedia - Computable Number: Encyclopedia Ii - Computable Number - Properties
The computable complex numbers form an algebraically closed field, and for many purposes is large enough already without requiring the no...   » Read the article

Encyclopedia - Computable Number: Encyclopedia Ii - Computable Number - Formal Definition
A real number a is said to be computable if it can be approximated by some algorithm (or Turing machine), in the following sense: given a...   » Read the article

Encyclopedia - Cardinality: Encyclopedia Ii - Cardinality - Cardinal Numbers
Note that, up until this point, we have only defined the term "cardinality" in a strictly functional role: we have not actually defined t...   » Read the article

Encyclopedia - Cardinality: Encyclopedia Ii - Cardinality - Examples And Other Properties
Such a property allows for the comparison of how many elements are contained in two or more sets without resorting to an intermediate set...   » Read the article

Encyclopedia - Constructivism Mathematics: Encyclopedia Ii - Constructivism Mathematics - Attitude Of Mathematicians
Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because o...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Common Misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration Pure And Applied Mathematics And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - History
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matt...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Is Mathematics A Science?
Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. If one considers science to be strictly about the physical wor...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Overview Of Fields Of Mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the re...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - Arguments Pro And Con
It is interesting to note that Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent s...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia Ii - Cardinal Number - Cardinal Arithmetic
We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. If X and Y are disjo...   » Read the article

Encyclopedia - Cardinality Of The Continuum: Encyclopedia Ii - Cardinality Of The Continuum - The Continuum Hypothesis
The famous continuum hypothesis asserts that c is also the first aleph number ℵ1. In other words, the continuum hypothesis states that ...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - The Generalized Continuum Hypothesis
The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that o...   » Read the article

Encyclopedia - Cantor Set: Encyclopedia Ii - Cantor Set - What's In The Cantor Set?
Since the Cantor set is defined as the set of points not excluded, the proportion of the unit interval remaining can be found by total le...   » Read the article

Encyclopedia - Continuum Hypothesis: Encyclopedia Ii - Continuum Hypothesis - Impossibility Of Proof And Disproof
Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first on David Hilbert's...   » Read the article

Encyclopedia - Cantor Set: Encyclopedia Ii - Cantor Set - Variants Of The Cantor Set
Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percenta...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Revisionists
Obviously Anselm thought this argument was valid and persuasive, and it still has occasional defenders, but many, perhaps most, contempor...   » Read the article

Encyclopedia - Real Number: Encyclopedia Ii - Real Number - Definition
Real number - Construction from the rational numbers. The real numbers can be constructed as a completion of the rational numbers. For ...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia Ii - Cardinal Number - History
The cardinal numbers were invented by Georg Cantor, when he was developing the set theory now called naive set theory in 1874–1884. He ...   » Read the article

Encyclopedia - Cardinal Number: Encyclopedia Ii - Cardinal Number - Formal Definition
Formally, the order among cardinal numbers is defined as follows: | X | ≤ | Y | means that there exists an injectiv...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Plantinga's Modal Form And Contemporary Discussion
Alvin Plantinga has given us a another version of the argument, one where the conclusion follows from the premises, assuming axiom S5 of ...   » Read the article

Encyclopedia - Ontological Argument: Encyclopedia Ii - Ontological Argument - Descartes' Ontological Arguments
Descartes composed a number of ontological arguments which differed from Anselm's formulation in important ways. Generally speaking, it i...   » Read the article

Encyclopedia - Pseudomathematics: Encyclopedia Ii - Pseudomathematics - Current Trends In Pseudomathematics
In recent years, pseudomathematicians have devoted their energies to disproving Gödel's second incompleteness theorem (efforts that fall...   » Read the article

Encyclopedia - Controversy Over Cantor's Theory: Encyclopedia Ii - Controversy Over Cantor's Theory - Preface
The pure mathematicians and applied mathematicians who object to Cantor's theory of sets claim that Cantor introduced into mathematics an...   » Read the article