Singular cardinals hypothesis

The singular cardinal numbers hypothesis, singular cardinals hypothesis abbreviated according to the English name as SCH, is independent of the usual axioms of set theory statement, which therefore can be neither proved nor disproved. She turns up in the context of studies on the continuum hypothesis.

Formulation

The singular cardinal numbers hypothesis is the following statement: Applies to an infinite cardinal number, the inequality, then.

The cofinality of and successor cardinal number is from. By the theorem of Cantor is always. Follows, therefore, that is, the cardinal number satisfying must be nonsingular. Therefore, the above formulation is made in only a statement about singular cardinals, which explains the name - singular cardinals hypothesis. After the set of king is always so that the minimum possible value for is. The above statement means, then, that assumes for cardinal numbers with the lowest possible value.

From GCH follows SCH

As is true and because of the above-mentioned set of king. If the generalized continuum hypothesis GCH (English: Generalized Continuum Hypothesis ) is true, so is always and follows from the above inequality, that is the conclusion in the SCH applies regardless of any conditions. In particular, the singular cardinal numbers hypothesis follows from the generalized continuum hypothesis.

But there are also models with and SCH, ie in which the continuum hypothesis is violated, but still the singular cardinal numbers hypothesis is true.