Continuous assumption

concept

Continuum Hypothesis (continuum hypothesis), the mathematical assumptions about the continuum potential. Often referred to as CH. The assumption is that the infinite set, in addition to the base set of integers, real numbers cardinality is minimal.

issues raised

is usually called real numbers i.e. the set of points on the straight line is a continuum, and the cardinality of the continuum (size) referred to as C1.

2000 years, people have always thought that any two infinite sets are the same size. Until 1891, G Cantor proof: any power set of a set (ie the set of all its subsets composed of) potential is greater than the cardinality of the set, people realized the infinite set can also compare the size.

is the smallest set of natural numbers infinite set, the potential of the set of natural numbers denoted aleph zero. Cantor continuum demonstrated the potential of the potential set of natural numbers equal power set. Is there an infinite set, its potential is greater than the potential of the set of natural numbers is smaller than the continuum potential? This problem is called continuum problem.

Cantor guess answer to this question is negative, i.e. the continuum potential is a minimum larger than the potential of the potential set of natural numbers in an infinite potential, referred to as a C1; potential set of natural numbers is designated as C0 . This conjecture is called the continuum hypothesis.

1938 Nian, K. Gödel proved CH ZFC axioms of the system (see axiomatic set theory) are coordinated, in 1963, PJ Cohen proved CH ZFC axiomatic system of independent, it is impossible to determination of true and false. Thus, the system ZFC axiom, CH true and false determination is impossible. This is one of the greatest advances in the 1960s set theory. However, in the 21st century, previous conclusions began to be shaken.

Cantor continuum proof base is a natural number set in the power set base, and it referred to as 2 ^ ℵ0 (wherein ℵ0 pronounced Aleph zero). Cantor endless base also arranged in an ascending order of ℵ0, ℵ1, ... ℵa ...... where a is an arbitrary ordinal numbers, Cantor conjecture, 2 ^ ℵ0 = ℵ1. This is the famous continuum hypothesis (abbreviated CH). Generally, any sequence of numbers a, concluded that 2 ^ ℵa = ℵ (a + 1) established, called generalized continuum hypothesis (abbreviated GCH). In the ZF, CH, and the axiom of choice (abbreviated to AC) are mutually independent, but can be introduced by the GCH AC. ZF can be configured together axiom (abbreviated V = L) can be introduced GCH, CH course be introduced and AC.

Generalized Continuum Hypothesis

Generalized Continuum Hypothesis (Generalized continuum hypothesis, referred GCH) means that: If A is an infinite set of another base and its power set infinite set S < between section> , and the base thereof is necessarily a power set

CH GCH and independent of the ZFC, but proved Sierpiński ZF + GCH can be deduced from the axiom of choice, in other words, ZF + GCH AC is not satisfied, but postulated system does not exist.

in any of an infinite set of A and B, if there is a single exit from A to B, then there is a sub-set of A to B is a subset of a single shot. Therefore, for any finite sequence numbers A and B,

If A and B is a finite set, then we can get stronger inequality:

GCH means that the strict inequality for infinite and finite ordinal ordinals are established.