Search results
Results From The WOW.Com Content Network
The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from combined work of Kurt Gödel and Paul Cohen. Gödel [ 6 ] [ 2 ] showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted (making ZFC).
The following four independence results are also due to Gödel/Cohen.); the generalized continuum hypothesis (GCH); a related independent statement is that if a set x has fewer elements than y, then x also has fewer subsets than y. In particular, this statement fails when the cardinalities of the power sets of x and y coincide;
In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967. [12]
Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as ...
Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
The continuum hypothesis and the generalized continuum hypothesis; The Suslin conjecture; The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo–Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
Social Security serves as a lifeline for tens of millions of seniors. Today, that number is growing. As of December 2024, the Social Security Administration (SSA) reported that about 65.5 million...
It follows from the independence of the continuum hypothesis, proved in 1963 by Paul Cohen, [5] that the answer to Wetzel's problem is independent of ZFC set theory. [1] Erdős' proof is so short and elegant that it is considered to be one of the Proofs from THE BOOK .