The term weak continuum hypothesis can be used to refer to the hypothesis that , which is the negation of the second continuum hypothesis.[1]: 80 [2]: Lecture 7 [3]: 3616 It is equivalent to a weak form of ◊ on .[4]: 2 [5] F. Burton Jones proved that if it is true, then every separable normal Moore space is metrizable.[6]: Theorem 5
Weak continuum hypothesis may also refer to the assertion that every uncountable set of real numbers can be placed in bijective correspondence with the set of all reals. This second assertion was Cantor's original form of the Continuum Hypothesis (CH). Given the Axiom of Choice, it is equivalent to the usual form of CH, that .[7]: 155 [8]: 289
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and 25 Related for: Weak continuum hypothesis information
The term weakcontinuumhypothesis can be used to refer to the hypothesis that 2 ℵ 0 < 2 ℵ 1 {\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}} , which is...
in the aleph number hierarchy, but it follows from ZFC that the continuumhypothesis (CH) is equivalent to the identity 2ℵ0 = ℵ1. The CH states that there...
generalized continuumhypothesis states that κ + = 2 κ {\displaystyle \kappa ^{+}=2^{\kappa }\,} for every infinite cardinal κ. Under this hypothesis, the notions...
assume the generalized continuumhypothesis. If the continuumhypothesis holds, all real closed fields with cardinality of the continuum and having the η1...
of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic...
cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuumhypothesis holds, then a...
in applications to algebra include the finding that under the weakcontinuumhypothesis there is no universal object in the class of uncountable locally...
paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an...
describes how he considers the original Multiregional hypothesis to have been modified over time into a weaker variant that now allows a much greater role for...
of the form ωα GCH Generalized continuumhypothesis generalized continuumhypothesis The generalized continuumhypothesis states that 2אα = אα+1 generic...
invalidate it. The continuumhypothesis and the axiom of choice, are examples of possible transcendental decision points. Solipsism in its weak form is characterized...
choice, which is weaker than AC but sufficient to develop most of real analysis. In all models of ZF¬C, the generalized continuumhypothesis does not hold...
induction is sometimes known as weak induction), makes the induction step easier to prove by using a stronger hypothesis: one proves the statement P ( m...
change on the "elaboration continuum" ranging from low to high. When the operation processes at the low end of the continuum determine attitudes, persuasion...
weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is...
with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful...
uniformization Axiom of real determinacy Von Neumann–Bernays–Gödel axioms Continuumhypothesis and its generalization Freiling's axiom of symmetry Axiom of determinacy...
that holds in the constructible universe (L) and that implies the continuumhypothesis. Jensen extracted the diamond principle from his proof that the axiom...
can well order the continuum, and we can do so in such a way that any proper initial portion has lower cardinality than the continuum. We use the obtained...
The nebular hypothesis is the most widely accepted model in the field of cosmogony to explain the formation and evolution of the Solar System (as well...
universe of set theory in which the continuumhypothesis must hold. In 1963, Paul Cohen showed that the continuumhypothesis cannot be proven from the axioms...
_{2}} -Aronszajn trees is undecidable in ZFC: more precisely, the continuumhypothesis implies the existence of an ℵ 2 {\displaystyle \aleph _{2}} -Aronszajn...
ZFC+CH and ZFC+¬CH are all equiconsistent (where CH denotes the continuumhypothesis). When discussing fragments of ZFC or their extensions (for example...
of undecidable statements (in the first sense of the term): The continuumhypothesis can neither be proved nor refuted in ZFC (the standard axiomatization...
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