All Topics  
Well-ordering theorem

 

 
   Email Print
   Bookmark   Link
 

 

Well-ordering theorem
 
 
  Topics Well-ordering theorem Topic Home

The well-ordering theorem (not to be confused with the well-ordering principle
Well-ordering principle

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element.The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem"....
) states that every set can be well-order
Well-order

In mathematics, a well-order relation on a Set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering....
ed.

This is important because it makes every set susceptible to the powerful technique of transfinite induction
Transfinite induction

Transfinite induction is an extension of mathematical induction to well-order, for instance to sets of Ordinal number or cardinal number....
.

Georg Cantor
Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....
 considered the well-ordering theorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set R of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s; in 1904, Gyula Konig claimed to have proven that such a well-ordering cannot exist.






Discussion
Ask a question about 'Well-ordering theorem'
Start a new discussion about 'Well-ordering theorem'
Answer questions from other users
Full Discussion Forum



Encyclopedia


The well-ordering theorem (not to be confused with the well-ordering principle
Well-ordering principle

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element.The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem"....
) states that every set can be well-order
Well-order

In mathematics, a well-order relation on a Set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering....
ed.

This is important because it makes every set susceptible to the powerful technique of transfinite induction
Transfinite induction

Transfinite induction is an extension of mathematical induction to well-order, for instance to sets of Ordinal number or cardinal number....
.

Georg Cantor
Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a Germany mathematician, born in Russia. He is best known as the creator of set theory, which has become a foundations of mathematics in mathematics....
 considered the well-ordering theorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set R of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s; in 1904, Gyula Konig claimed to have proven that such a well-ordering cannot exist. A few weeks later, though, Felix Hausdorff
Felix Hausdorff

Felix Hausdorff was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis....
 found a mistake in the proof. Ernst Zermelo
Ernst Zermelo

File:Ernst Zermelo.jpegErnst Friedrich Ferdinand Zermelo was a Germany mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy....
 then introduced the axiom of choice
Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinite set many bins and there is no "rule" for which object t...
 as an "unobjectionable logical principle" to prove the well-ordering theorem. It turned out though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms is sufficient to prove the other, in first order logic. (Incidentally, the same applies to Zorn's Lemma
Zorn's lemma

Zorn's lemma, also known as the Kuratowski?Zorn lemma, is a proposition of set theory that states:Every partially ordered set in which every Total order has an upper bound contains at least one maximal element....
.) In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.

The well-ordering theorem has consequences that may seem paradoxical, such as the Banach–Tarski paradox
Banach–Tarski paradox

The Banach?Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball....
.

See also

  • Well-ordering principle
    Well-ordering principle

    In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element.The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem"....