1. A

    Question concerning covering the rationals with open intervals (paradox?)

    The following seems paradoxical. Consider the following countable collection of open intervals: I1 is ( 0, 1/2 ), I2 is ( 1/2, 3/4 ), I3 is ( 3/4, 7/8 ), etc. The sum of the lengths of the intervals is 1. Now cover each rational number with one of the intervals. Either 1) two successive...
  2. R

    The negation of this sentence is false

    I can't tell if the logic problem "The negation of this sentence is false" is a statement or not. If the statement is initially true, then the negation would be false. If the statement is initially false, then the negation would be true. So I thought it wouldn't be a statement because...
  3. S

    Doubt on the Axioma of Separation (Russel's Paradox)

    How can this evade Russel's Paradox? If p is a property, for any X, there exists a set Y = {x E X;P(x)} If P(x) = x ~E x (a set x cannot contain itself) ^ Y = x Then Y E X ^ Y ~E Y A paradox: Y E X and Y ~E Y cannot coexist
  4. R

    Paradox. Extracting probability density function from mean.

    Is it possible to obtain probability density function of continuous distribution from it's mean? 1. No. Probability density function contains more information about distribution than it's mean. Particularly one can obtain same mean for distributions with different probability density...
  5. H

    Cantors "Paradox" of the greatest cardinal number

    This addresses the “proof” of Cantor’s paradox in Suppes (Axiomatic Set Theory), pg 5. Let S1 be the set of all sets EXCEPT S1 with cardinality n. Let S2 be the set of all subsets of S1 EXCEPT S2 with cardinality p. Then every member of S1 is a member of S2 and every member of S2 is a member...
  6. H

    Epimenides “Paradox”: I am a liar

    Is “I am a liar” a paradox? If “I am a liar is true”, then “I am a liar is false.” If “I am a liar is false” then “I am a liar is true.” So of course it is not a paradox. It is simply a variable X which cannot be given a True or False value (they give contradictions). For example “Air Force1...
  7. E

    Banach Tarski Paradox in the Plane

    Hello everyone, Not quite sure that this is the right place for this question, but I couldn't find a topology or functional analysis section anywhere so here goes.. Basically, I have been given an assignment to prove that there is no Banach-Tarski paradox in R^2. I have decided to take the...
  8. H

    Set Membership, Equality, and Russells Paradox

    MEMBERSHIP ϵ is not reflexive, symmetric or transitive. xϵx is not a membership statement, it is an identity: x=x. xϵy →yϵ’x Axiom, 1ϵ{1,2} →{1,2}ϵ’1 (‘ means not) xϵy & yϵz →’xϵz Axiom, 1ϵ{1} & {1}ϵ{{1},2} but 1ϵ'{{1},2} IDENTITY: x=x EQUALITY: x=y, “x” and “y” represent the same...
  9. H

    Russels Paradox doesn't Exist

    Russells Paradox doesn't Exist Russell’s Paradox: (xϵx) iff (xϵ’x) If ϵ is different than =, xϵx is not true for any x and xϵ’x is true for all x. Therefore: 1) The left side of Russell’s Paradox does not exist, and so neither does the paradox. 2) There is no such thing as xϵx. EDIT...
  10. H

    Set Foundations and Russel's Paradox

    =’ not equal to, ϵ’ not a “member” of 1) x=y: undefined except x=x (axiom), x=’x false (logic). 2) xϵy: Principal Primitive Undefined concept* 3) x and y undefined. Theorem: xϵx iff x=x. Proof: x and x are identical so there can be no other relation between them. Theorem: xϵ’x iff x=’x...
  11. H

    Set of All Sets (Russel's) Paradox

    Set of all sets (Russel’s) paradox is not a paradox because there is no such thing as the set of all sets. Proof: Let A, B be sets. Then C={A,B,C} is not a set because it is undefined (circular definition).
  12. P

    Generalised Birthday Paradox - bins problem

    We are putting balls into bins by the Uniform Distribution. Let X be a Random Variable and means moment of collision when two balls fall into the same bin. So minimum X is 2 (2 balls in the same bin) and maximum X is n+1. What is probability P(2 \le X \le n+1) and Expected Value for k bins...
  13. H

    Archimedes Paradox

    For any number r, n exists st n > r. Let r be any member of the set of all integers. Then there is an integer greater than the set of all integers.
  14. A

    Olbers' paradox: why is the night sky dark?

    I believe to have found solution for Olbers' paradox. Treatment originally used to discard inverse square law as the solution was not set up correctly, and if we include sensor surface area in the treatment and model light as photons the result describes what we actually see. I would like to...
  15. S

    Binomial error paradox

    In a binomial distribution if the chance of something happening is p and q=(1-p) and the variance for the event occurring is pq. The variance for the event not occurring is also qp. Since the error within a given confidence interval is related to the variance then this error is the same for both...
  16. M

    Skolem's Paradox (a little philosophy)

    Skolem "showed" that the notion of countability is relative. In set theory, if there is a set consisting of a bijection between a set A and the set of naturals N in the domain some model M, then A is countable in M. Now, suppose that we removed the bijections between A and N from M and...
  17. M

    Help with Skolem's Paradox and Model Theory

    Yo… I'm studying Skolem's Paradox. Cantor's Theorem: for any (arbitrary) set S, no mapping surjects S onto its power set P(S). Consequently, there exist uncountable sets. Skolem's Theorem: if some collection of first-order sentences has any infinite model, then it has a countable model. The...
  18. D

    Variance paradox

    Hello guys, how are you? Say we have to find Var(((a-1)/a) x) which equals with ((a-1)^2/a^2) Var (x) But if we try Var(((a-1)/a) x)= Var((1-(1/a))x) =Var(x)+(1/(a^2))Var (x) = (1+1/(a^2)) Var (x) is a different result from ((a-1)^2/a^2) Var (x). With an arithmetic example we see...
  19. C

    A paradox

    I'm sure there is a name for this paradox, but I haven't studied set theory enough to know what it's called. Can someone please help resolve this? Let N = the set of the natural numbers, and let D = the set of natural odd numbers. N and D have equal cardinalities, and yet N is unequal to...
  20. L

    Russell's Paradox

    Question: Can there exist a book that refers to all those books and only those books that do not refer to themselves? I have reviewed the example in the book, and according to Russell's Paradox, I think the answer is neither yes nor no. If the book refers to itself, then it does not exist...