Consistency-Proof for the Generalized Continuum-Hypothesis

Consistency of the Continuum Hypothesis

One of the basic results in set theory is that the cardinality of the power set of the natural numbers is the same as the cardinality of the real numbers, which is strictly greater than the cardinality of the naturals. In fact Cantor proved a more general theorem: for any set X, the cardinality of X is strictly less than the cardinality of the power set of X. Since there is an infinite set, say...

The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.

1. The axiom of choice (i.e., v. Neumann's Ax. III 3*) 2. The generalized Continuum-Hypothesis (i.e., the statement that 2"= ,, + i holds for any ordinal a) 3. The existence of linear non-measurable sets such that both they and their complements are one-to-one projections of two-dimensional complements of analytic sets (and which therefore are B2-sets in Lusin's terminology2) 4. The existence o...

The Generalized Continuum Hypothesis Revisited Sh460

The Continuum Hypothesis

The continuum hypotheses (CH) is one of the most central open problems in set theory, one that is important for both mathematical and philosophical reasons. The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic number...

continuum hypothesis Preliminary report

In [Sh, chapter XI], Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham’s problem on the existence