1. Z

    Cantor's Diagonal Argument in a Nutshell

    Cantor's Diagonal Argument in a Nutshell CDA allows every member of a list of infinite binary sequences to be counted except the last one. The last member doesn't exist. The list can be counted.
  2. Z

    Cantor's Diagonal Argument is Wrong

    This is a slight refinement (simplification and condensation) of my previous expositions which evolved in the quoted thread. I put it here because it was being buried, and because it addresses the subject directly.
  3. happy21

    About argument of complex numbers

    Hi, Let, z, z{_{1}}, z{_{2}} be three complex numbers. How to prove the following two properties: (i) arg(z) - arg(-z) = \pm \pi (ii) \mid z{_{1}}+z{_{2 }}\mid=\mid z{_{1}}-z{_{2 }}\mid \Leftrightarrow arg(z{_{1}}) - arg(z{_{2}})=(\pi /2) Thx.
  4. Z

    Cantors Diagonal Argument Automatically Fails

    Cantors Diagonal Argument Aotomatically Fails Even if you "imagine" reaching the end of an infinite binary sequence, it doesn't matter since you always have more sequences than digit places no matter how many digit places there are, and CDA automatically fails. For Ex, two digit places give...
  5. Z

    Cantor's Diagonal Argument and Infinity

    If infinity means endless, you can never get to the end of CDA.
  6. Z

    Cantor's Diagonal Argument

    Suppose I have a set of all infinite sequences of binary digits. Either I know the digits of each sequence or I don't. If I don't, how do I know they are all different? If I do, I can arrange them in numerical (countable) order. The list has a smallest member 0, and I can decide which of any...
  7. A

    Oh Boy, A Crazy Challenge to Cantor's Argument

    Let g(1) = 0 and g(0) = 1. For any infinite binary string $x = x_1x_2x_3\dots$, let $f( \,x) \, = g(x_1)g(x_2)g(x_3)\dots$. If for each element $x$ of a countable set $X$ of infinite binary strings, $f( \,x) \,$ is also in $X$, then let $X$ be considered ‘balanced.’ Let V be a...
  8. SenatorArmstrong

    Modulus and Argument of z

    Hello, I am studying complex numbers. I am a little confused since I see if you have $$z=re^{i\theta} $$ I see you can find $r$ by $$ r = |z| = \sqrt{x^2+y^2}$$ and $$\theta = \arctan(x,y)$$ I have never seen trigonometric functions with more than one variable. Especially in this case where...
  9. Z

    Cantors Diagonal Argument for n digits.

    The number of binary sequences for n digits is always greater than n, for all n. Ex, n=2 10 01 11 00 11=00 is in the list. 00 01 10 11 01=10 is in the list.
  10. Z

    Cantors Diagonal Argument, Logic

    A \rightarrow NotA
  11. M

    Directional derivatives do not guarantee continuity : my argument

    Hi, I've reached the conclusion that just because all of the directional derivatives at a point on a function f (as defined in the image attached) exist, that does not imply that the function is continuous at that point. In turn, that does not guarantee differentiability. Please let me know...
  12. M

    I don't see the significance of Cantor's diagonal argument

    I know it shows that we can exhaust all of the naturals leaving out a real number by implementing a rule. But we can do the same thing with matching the naturals with rationals. For example, we can exhaust all elements of N by just matching each of them to each element in the infinite set of...
  13. B

    composite argument properties

    sin(a+b) = sin(a)cos(b) + cos(a)sin(b) Of course there are more for cos etc. I am wondering if any of you fine gentlemen or ladies have any information on the history of this property: I.E: who discovered it and perhaps a location of their notes? I am also in search of a concise proof...
  14. M

    Reals are uncountable - without Cantor diagonal argument

    Gist of proof: length of unit interval = 1, if the reals are countable, the length = 0, contradiction. Proof: Arrange reals into countable list. Let x be an arbitrarily number > 0. Cover first point on list by interval of length x/2, second point by interval of length x/4, third point by...
  15. V

    Cantors Diagonal Argument is Not Diagonal

    In my view, Zylo's problem is that he thinks of infinite in terms of some sort of limit of finite cases.
  16. Z

    Cantor's "Diagonal" Argument, Last Member By definition, Cantor's s is different from every member of the list, so it can only be at the end of the list. But the list is countably infinite and so has no end. So s doesn't exist and Cantor's "Diagonal" Argument fails.
  17. Z

    Cantor's Diagonal Argument Fails by Inductiou

    The only mathematical tool for dealing with infinity is Peano's axiom on Induction. For all n, there are more binary sequences than there are places. Cantor's diagonal construction is impossible. Example, for 2 places there are 4 sequences: 00 01 10 11
  18. Z

    Cantor's Diagonal Argument and Square Arrays

    In order for Cantor's construction to work, his array of countably infinite binary sequences has to be square. If si and sj are two binary sequences in the array, and each has countably infinite many digits, are they the same length? Ref:
  19. Z

    Cantor's Diagonal Argument and Binary Sequences

    Countably Infinite Reversed Binary Sequence Definitions: Binary Number, Binary Sequence, Reversed Binary Number, Reversed Binary Sequences BN=p_{n}2^{n}+p_{n-1}2^{n-1}+...p_{0}2^{0}\\ BS=p_{n}p_{n-1}....p_{2}p_{1}p_{0}\\ RBN=p_{0}2^{0}+p_{1}2^{1}+p_{2}2^{2}+...p_{n}2^{n}\\...
  20. Z

    Cantor's Diagonal Argument and Infinity

    Cantor's argument fails because there is no natural number greater than every natural number.