What is Continuum hypothesis: Definition and 15 Discussions
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers:
2
ℵ
0
=
ℵ
1
{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
.
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. 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.The name of the hypothesis comes from the term the continuum for the real numbers.
Hi,
Just to test my understanding. Is it the case that the Continuum Hypothesis( CH) is not considered true within ZFC because there are both models/interpretations of ZFC where CH holds , as well as models/interpretations of ZFC where it doesn't, whereas truth of a ( statement?)in FOL...
Let ##\omega_1## be the first uncountable ordinal such that ##x## is an element of ##\omega_1## if and only if it is either a finite ordinal or there exists a bijection from ##x## onto ##\omega##.
I want to define a matrix such that the matrix contains each element of ##\omega_1## only once.
To...
Introduction: Making a Sequence ##T## based on “The Rule of Three”
The primary means of generating the sequence ##T## is through the use of a function ##f##. In general, function ##f## is going to be a function that takes as input a three-member sequence of ordinal numbers (an ordered triplet)...
Summary: The Continuum Hypothesis and Number e
Now, I must ask a very stupid question:
When taking: $$\lim_{_{n \to \infty} } (1+\frac{1}{n})^n=e\\$$ the ##n## we use take its values from the set: ## \left\{ 1,2,3 ... \right\} ## which has cardinality ## \aleph_0 ##, which is equivalent...
Is it possible to calculate this :
Suppose the iterative root of ##2^x## :
##\phi(\phi(x))=2^x## (I suppose the Kneser calculation should work, it affirms that there is a real analytic solution)
Then how to compute ##\phi(\aleph_0)## ? (We know that ##2^{\aleph_0}=\aleph_1##).
Could this be...
Are undecidable statements, such as the provability of the continuum hypothesis, natural examples of statements that require a multivalent logic in order for them to be adequately described and/or even properly understood? (NB: by properly I am taking this to mean that undecidable matters such...
Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ0 and cardinality of reals = ℵℝ; then there is no ℵ such that ℵ0 < ℵ < ℵℝ . But what about...
Cardinality of the set of binary-expressed real numbers
This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis.
1. Counting...
If we accept that there does indeed exist a set whose cardinality is between \aleph_0 and \aleph_1, what would such a set look like?
I know that in ZM-C we can choose to either add the continuum hypotheses or not, but if we chose to negate it, that means that there definitely is a set greater...
Continuum approximation of fluid mechanics (& relativistic fluids)
I have a few 'foundational' questions on fluid mechanics which I haven't been able to find quick answers to, any help would be appreciated.
At the start of any course on fluids, one is told of the continuum hypothesis...
http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html?full=true
Above reference describes the history of the continuum hypothesis. Among the items of interest is the development of axiom systems in which the continuum hypothesis is true.
Suppose we assume that the Continuum Hypothesis is false. Then there must be a subset of the real numbers that has the cardinality of Aleph 1. What is an example of such a subset?
Did Paul Cohen "settle" the Continuum Hypothesis?
Paul Cohen proved the Continuum Hypothesis is independent of ZFC and concluded that it's truth or falseness is undecidable (1963). Is this still the case today? These links suggest there is still interest in proving or disproving it...
R=2^\alepha 0 vs Continuum hypothesis! A result in "a taste of topology"
A year ago or so I read a proof in A Taste Of Topology, Runde that the cardinality of the continuum equals the cardinality of the powerset of the natural numbers. But a few hours ago I found Hurkyl making that statement...