List of statements independent of ZFC information

mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel...

the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC. Statements in this class include the statement that P = NP...

group is free. List of statements independent of ZFC Statements true in L Akemann, Charles; Weaver, Nik (2004). "Consistency of a counterexample to Naimark's...

truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum...

non-measurable set of real numbers, all of which are independent of ZFC. The axiom of constructibility implies the non-existence of those large cardinals with consistency...

group? Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory. Assume that A is an abelian group such that...

axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means...

independent from ZFC. Starting in 1935, the Bourbaki group of French mathematicians started publishing a series of books to formalize many areas of mathematics...

existence of an ℵ2-Suslin tree, is a longstanding open problem. Glossary of set theory Kurepa tree List of statements independent of ZFC List of unsolved...

the continuum is independent from the axioms of ZFC). This existence of proofs of relative consistency implies that the consistency of modern mathematics...

axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable...

advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem...

axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC. Gödel's...

with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Mathematical...

this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic...

axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von...

problems List of lemmas List of theorems List of statements undecidable in ZFC Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press...

language of ZFC set theory by axioms, analogous to Peano's. See construction of the natural numbers using the axiom of infinity and axiom schema of specification...

scientists to suggest the P versus NP problem may be independent of standard axiom systems like ZFC (cannot be proved or disproved within them). An independence...

between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that...