The continuum hypothesis was under discussion as an "undecidable statement" at the Princeton University Bicentennial Conference on "Problems of Mathematics" in 1946, the first major international gathering of mathematicians after World War II. Subject: Continuum Hypothesis. A few comments: One set-theorist correspondent said that set-theorists themselves are very loathe to talk about "truth" or "falsity" of such claims. states the proof of Cohen’s theorem that the generalized continuum hypothesis cannot be proved in Zermelo-Fraenkel set theory. S has a 1-1 correspondence to the integers. Some analogies are obvious. I was reading a bit about the Riemann Hypothesis to try and figure out, what it's all about and it got me thinking. As long as his arguments can – Conifold Jan 9 '17 at 23:03 Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Can is be proven that there is no proof of a hypothesis? The working mathematician, unless he is studying the foundations of mathema- tics, usually does not find it necessary to make explicit references to axioms of set theory - except perhaps to invoke the Axiom of Choice or the Continuum Hypothesis. A statement in formal logic is called undecidable if there is no proof or disproof of the statement in formal logic. Georg Cantor's conjecture, the Continuum Hypothesis Without equations, this states that for any set of real numbers, S, one of three things happen: S is finite. For example, many set theorists now believe that the continuum hypothesis (which is known to be undecidable in Zermelo-Fraenkel set theory) is actually false. Whether undecidable statements in general rely on some hidden self-reference is not entirely clear but highly unlikely, e.g. A common misconception is that undecidable statements have no truth value, but this statement is not true. The connection is direct, but still it takes a moment's thought to see to which subset the completeness axiom should be applied assuming a counter-example to the Archimedean axiom. the continuum hypothesis is undecidable apparently for other reasons, something like inherent vagueness of the continuum notion, see Feferman. WHITEHEAD'S PROBLEM IS UNDECIDABLE PAUL C. EKLOF 1. For instance, the proof that the axiom of parallels does not follow from the other Euclid axioms did not close geometry, but made the emergence of non-Euclidian geometries possible, … Thus it seems arbitrary to believe that some undecidable statements have truth value, when it is already accepted that others do not. There is nothing in between the integers and reals. So it appears that Gödel thought that one might be in a position to establish that A and V = L are absolutely undecidable. And as a final point, there exist many undecidable statements, like the continuum hypothesis, that have no agreed-on truth value absent of a formal system. I think the term “independent” is more often used than “undecidable” in this context. The continuum hypothesis (under one formulation) is simply the statement that there is no such set of real numbers. Despite his efforts Cantor could not resolve CH. I just came across your posting about CH and found it quite interesting. The Continuum Hypothesis, originally posed by set theorist Georg Cantor in 1878, states that there is no set whose cardinality is between that of the integers and that of the real numbers. It was through his attempt to prove this hypothesis that led Cantor do develop set theory into a sophisticated branch of mathematics. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. 1.1 The Zermelo-Fraenkel Axioms 2. On the Question of Absolute Undecidability ... ither this proposition is absolutely undecidable or Cantor's continuum hypothesis is demonstrable’ but that he has ‘not yet been able to determine which one of these two possibilities is realized’ (p. 185).

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