What is the continuum hypothesis?

What is the continuum hypothesis?

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert’s 23 problems presented in 1900.

Is the continuum hypothesis independent of ZFC?

The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.

Is the continuum hypothesis independent of Zermelo-Fraenkel set theory?

The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from combined work of Kurt Gödel and Paul Cohen . Gödel showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted (making ZFC).

What is the system of hyperreal numbers?

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form. Such numbers are infinite, and their reciprocals are infinitesimals.

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.

What is continuity according to quodlibet?

The treatment of continuity in the first book of his Quodlibet of 1322–7 rests on the idea that between any two points on a line there is a third—perhaps the first explicit formulation of the property of density —and on the distinction between a continuum “whose parts form a unity” from a contiguum of juxtaposed things.

What is the nature of continuum?

Continuity connotes unity; discreteness, plurality. While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit.

What is the best book on continuum theory?

Foreman, M. and M. Magidor, 1995, “Large cardinals and definable counterexamples to the continuum hypothesis,” Annals of Pure and Applied Logic 76: 47–97. Foreman, M., M. Magidor, and S. Shelah, 1988, “Martin’s Maximum, saturated ideals, and non-regular ultrafilters. Part i,” Annals of Mathematics 127: 1–47. Gödel, K., 1938a.