Continuum hypothesis - The size of a set

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is ("aleph-null") and the cardinality of the real numbers is , the continuum hypothesis says: ...

Including:

• Continuum hypothesis - The size of a set
• Continuum hypothesis - Impossibility of proof and disproof
• Continuum hypothesis - Arguments pro and con
• Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia - Continuum hypothesis

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 there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}. With infinite sets such as the set of intege ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » 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 there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, appl ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - The size of a set

Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. More recently, some experts (e.g ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - Arguments pro and con

The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. An equivalent condition is that for every ordinal α. Another equ ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - The generalized continuum hypothesis

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 list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris. Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is ind ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - Impossibility of proof and disproof

It is interesting to note that Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. M ...

See also:

Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

Read more here: » Continuum hypothesis: Encyclopedia II - Continuum hypothesis - Arguments pro and con