Cantor's theorem

In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Cantor's theorem - The proof. Including:

• Cantor's theorem - The proof
• Cantor's theorem - A detailed explanation of the proof when X is countably infinite
• Cantor's theorem - History

Read more here: » Cantor's theorem: Encyclopedia - Cantor's theorem

To get a handle on the proof, let's examine it for the specific case when X is countably infinite. Without loss of generality, we may take X = N = {1, 2, 3,...}, the set of natural numbers. Suppose that N is bijective with its power set P(N). Let us see a sample of what P(N) looks like: P(N) contains in ...

See also:

Cantor's theorem, Cantor's theorem - The proof, Cantor's theorem - A detailed explanation of the proof when X is countably infinite, Cantor's theorem - History

Read more here: » Cantor's theorem: Encyclopedia II - Cantor's theorem - A detailed explanation of the proof when X is countably infinite

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia - Axiomatic set theory

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a musician of German descent, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and ...

See also:

Georg Cantor, Georg Cantor - Biography

Read more here: » Georg Cantor: Encyclopedia II - Georg Cantor - Biography

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 number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory

In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one can speak about one being greater or less than another. Then Cantor's paradox is: Theorem: There is no greatest cardinal number. This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set. Proof: Assume the contrary, and let C be the largest cardinal number. Then (in the von Neumann formulation of cardinality) C is a set ...

See also:

Cantor's paradox, Cantor's paradox - Statement and Proof, Cantor's paradox - Discussion and Consequences, Cantor's paradox - Historical Note, Cantor's paradox - Sources

Read more here: » Cantor's paradox: Encyclopedia II - Cantor's paradox - Statement and Proof

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and became the first president of the society. Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountab ...

See also:

Georg Cantor, Georg Cantor - Biography

Read more here: » Georg Cantor: Encyclopedia II - Georg Cantor - Biography

Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow ...

Including:

• Logic - Nature of logic
• Logic - Informal formal and symbolic logic
• Logic - Rival conceptions of logic
• Logic - History of logic
• Logic - Relation to other sciences
• Logic - Deductive and inductive reasoning
• Logic - Topics in logic
• Logic - Syllogistic logic
• Logic - Predicate logic
• Logic - Modal logic
• Logic - Deduction and reasoning
• Logic - Mathematical logic
• Logic - Philosophical logic
• Logic - Logic and computation
• Logic - Controversies in logic
• Logic - Bivalence and the law of the excluded middle
• Logic - Implication: strict or material?
• Logic - Tolerating the impossible
• Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia - Logic

In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter (beth) is also used. To define the beth numbers, start by letting be the cardinality of countably infinite sets; for concreteness, take the set of natural numbers to be the typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define = the cardinality of the power set of A if is the cardina ...

Read more here: » Beth number: Encyclopedia - Beth number

Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

Including:

• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

Read more here: » Cantor's diagonal argument: Encyclopedia - Cantor's diagonal argument

Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC

The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC. An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory

In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness: a set, x0, is well founded iff it has no infinite descending membership sequence: · · · In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets

From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc. For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics

Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...

See also:

Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory

Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory

Because of its fundamental role in philosophy, the nature of logic has been the object of intense disputation; and it is not possible to give a clear delineation of the bounds of logic in terms acceptable to all rival viewpoints. Nonetheless, the study of logic has, despite this controversy, been very coherent and technically grounded. Here we characterise logic, first by introducing the fundamental ideas about form and then by outlining some of the different schools of thought as well as giving a brief overview of its history, an account of its relationship to other sciences, and--finally--an expositi ...

See also:

Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia II - Logic - Nature of logic

Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. c is strictly greater than the cardinality of the natural numbers, ℵ0 (aleph-null) In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in a couple of different ways. See Cantor's ...

See also:

Cardinality of the continuum, Cardinality of the continuum - Properties, Cardinality of the continuum - The continuum hypothesis, Cardinality of the continuum - Sets with cardinality c

Read more here: » Cardinality of the continuum: Encyclopedia II - Cardinality of the continuum - Properties

Because of its fundamental role in philosophy, the nature of logic has been the object of intense dispute: it is not possible clearly to delineate the bounds of logic in terms acceptable to all rival viewpoints. Despite that controversy, the study of logic has been very coherent and technically grounded. In this article, we first characterise logic by introducing fundamental ideas about form, then by outlining some schools of thought, as well as by giving a brief overview of logic's history, an account of its relationship to other sciences, and finally, an exposition of some of logic's essential concepts. Logic - I ...

See also:

Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia II - Logic - Nature of logic

An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p ⊥ q), meaning there is no r in P such that r ≤ p and r ≤ q. In the Borel sets example, incompatibility means p∩q has measure zero. In the finite partial functions example, incompatibility means that p∪q is not a function, in other words ...

See also:

Forcing mathematics, Forcing mathematics - Forcing posets, Forcing mathematics - Countable transitive models and generic filters, Forcing mathematics - Forcing, Forcing mathematics - Consistency, Forcing mathematics - Cohen forcing, Forcing mathematics - The countable chain condition, Forcing mathematics - Easton forcing, Forcing mathematics - Random reals, Forcing mathematics - Boolean-valued models, Forcing mathematics - Meta-mathematical explanation

Read more here: » Forcing mathematics: Encyclopedia II - Forcing mathematics - The countable chain condition

The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick. Russell's paradox - Russell-like paradoxes. As illustrated above for Barbers and Lists, the Russell paradox is not hard to extend. Needed is A transitive verb < ...

See also:

Russell's paradox, Russell's paradox - History, Russell's paradox - Applied versions, Russell's paradox - Set-theoretic responses, Russell's paradox - Responses illustrated, Russell's paradox - Applications and related topics, Russell's paradox - Russell-like paradoxes, Russell's paradox - Reciprocation, Russell's paradox - Independence from excluded middle, Russell's paradox - Other related paradoxes

Read more here: » Russell's paradox: Encyclopedia II - Russell's paradox - Applications and related topics