continuum

  1. A

    Solution to the Continuum Hypothesis

    I believe that I have solved the Continuum Hypothesis: https://en.wikipedia.org/wiki/Continuum_hypothesis Assume CH is true. Then $|\mathcal{P}(\mathbb{N})| = |\omega_1|$ and we have a way to prove that my $T$ sequence here cannot contain all of the elements of $\omega_1$ using the...
  2. Z

    Decimals and the Continuum

    To define the real numbers in [0,1) list the natural numbers, put a period after them, and read them in reverse. Call them "decimals" 1. -> .1 2. -> .2 . 10. -> .01 . The decimals can be associated uniquely with points on the line (the continuum) as follows: Given a line segment, label its...
  3. L

    Continuum vs. singularity

    Are there nontrivial properties which a singular number within a continuum has that a singular number outside a continuum does not?
  4. A

    Continuum Hypothesis

    1) Let $q_1, q_2, q_3,$ … be an enumeration of $A = \mathbb{Q} \cap \mathbb{R}(0, 1)$. 2) For any $r \in \mathbb{R}(0, 1)$, let $B_r = \mathbb{Q} \cap \mathbb{R}(-r, 1-r)$. 3) For each $r \in \mathbb{R}(0,1)$, there exists one and only one $k_r \in A$ such that $f:A \rightarrow B_r$ is a...
  5. Z

    Natural Numbers and the Continuum

    Definition Reverse binary/decimal representation of the natural numbers: a0x2^0 + a1x2^1 + a2x2^2 + a3x2^3 + ....., an=0,1 a0x10^0 + a1x10^1 + a2x10^2 + a3x10^3 + ....., an=0,1,.....,9 ALL n. 01523 <-> 32510 (reversed <-> standard) From the definition it follows there is a 1:1...
  6. W

    Question about continuum hypothesis

    Would showing that every element in a set with cardinality 2^aleph-0 can be paired to one real number, and the reverse, be a proof of the continuum hypothesis?
  7. P

    Continuous set and continuum hypothesis

    This article explains why the cardinality of a set must be either Aleph0 or |ℝ|. 1. Rational numbers are discrete 2. Real numbers are continuous 3. Collectively exhaustive and mutually exclusive events 4. Continuum hypothesis 5. Cardinality of discontinuous subsets of real numbers...
  8. B

    Represent continuum betw summary and detail of timed neural

    How to represent the continuum between summary and detail of timed neural There are 6 numbers you need to name a place, time, and how much each of those varies: x y z time space standard deviation time standard deviation These can be represented with hardware bit...
  9. B

    Disproof of Continuum Hypothesis - cardinality is continuous

    http://en.wikipedia.org/wiki/Continuum_hypothesis Found it... an effect of the falseness of the Continuum Hypothesis (which falsely claims there is no set whose size is between the number of integers and number of reals). This core Java function on cardinality shifting (log base 1+x) explains...
  10. E

    Questions on infinities order and Continuum hypothesis.

    Having read a book on the work of Cantor on infinities, I have some questions left unanswered. 1) As Cantor proved, to progress from one infinity to another you take the power set of an infinity A, which is 2^A. 2 is the smallest number for which that works. So 1.9^A = A, and not another...
  11. H

    A word in favor of Continuum Hypothesis

    Hi everyone, you may dream solving an important theorem, indeed I do, but it is a far-off dream, so you may think about one of them a bit while you are "washing dishs". Obviously, I was not able to solve the Continuum Hypothesis, but I am in favor of it and I will expose my reason. I don't know...