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Questions tagged [paradoxes]

Paradoxes are arguments which contradict logic or common sense, often by using false and implicit premises.

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Modified two envelope paradox

This problem is a variation on a two envelope paradox. This time Alice and Bob play the game. Envelopes X and Y, when opened contain money. One envelope has n dollars and the other has 2*n dollars. ...
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1answer
47 views

Limit of bijections not a bijection?

Assume we are working in ZFC. Let $A$ and $B$ be sets of finite ordinals. Assume $f$ defines a bijection between $A$ and $B$. Let $X = A \cap B$ be the intersection of $A$ and $B$. If $X$ is finite ...
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Can an error be found in this proof of Gödel's incompleteness theorem?

Can you find a division by zero error in the following short proof of Gödel's incompleteness theorem? First a little background. $\text{G($a$)}$ returns the Gödel number of the formula $a$. As in ...
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2answers
60 views

Unexpected Asymmetry Between f(sin(θ)) and f(cos(θ - π/2))

I expected to always find cos and sin functions to be identical to each other with the only exception being that their phases will differ by π/2 So what I am trying to say here is that we should ...
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3answers
98 views

In ZFC, are all proper classes paradoxical?

The set of all sets that do not contain themselves, the set of all ordinal numbers, and the set of all sets represent proper classes that would clearly be paradoxical if they were admitted as 'sets' ...
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58 views

Why does Turing-computing (being an inconsistent formalism) has undecidable problems? [closed]

I'd like to apply Church-Turing thesis to Kleene-Rosser paradox: Since untyped lambda-calculus is an inconsistent formalism AND Turing machines are equal in decisive power to lambda-calculus SO We ...
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0answers
74 views

Paradox,shortest proof

I have read somewhere that the shortest proof of a certain formula in the language of natural numbers contains some kind of paradox. I cannot remember what this paradox was nor where I've read it. It ...
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1answer
56 views

Why is the Hilbert-Bernays paradox paradoxical?

The Hilbert-Bernays Paradox is produced by defining h as '(the referent of h) + 1'. Why is this a paradox? It seems strange to believe that we could define h in terms of itself. I suspect I'm missing ...
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4answers
680 views

Adding infinite and finite numbers: why doesn't 0=1?

Okay, so, $$\infty + 1 = \infty$$ subtract infinity from both sides. $$1=0$$ At first I thought, duh, $\infty \neq \infty+1$, but, now, I'm just more confused because my brother rephrased it in ...
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0answers
48 views

Independent events in the context of Simpson's Paradox

I came across this problem in the book "Introduction to Probability" by Dr. Joseph K. Blitzstein and Dr. Jessica Hwang. This problem deals with the concept of independent events in the context of the ...
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1answer
68 views

Wittgenstein's response to Russell's paradox

Can someone explain Wittgenstein's response to Russell's paradox in the Tractatus? Is it possible to cast the response as a mathematical proof? All explanations I have found so far mix logical and ...
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1answer
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Are there still any paradoxes in modern math? [closed]

if I google for paradoxes in math, all I find are ancient paradoxes which already have a hack or solution how to merge them out. Now I'm wondering if there are still any paradoxes in modern math, ...
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1answer
115 views

Have Zeno's Paradoxes been really solved? [closed]

I saw the wiki of Zeno's Paradoxes, and it is not clear on whether Zeno's Paradoxes been solved or not. That wiki article is linking to this article: Why mathematical solutions of Zeno's paradoxes ...
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3answers
68 views

Satisfying explanation of Aristotle's Wheel Paradox.

The paradox: We have a circle and there is another circle with smaller radius. They are co-centeric. If circle make full turn without sliding, both smaller and bigger circle make full turn too. If ...
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Prove a set equidecomposable with a paradoxical set is paradoxical

In Stan Wagon's book The Banach-Tarski Paradox, Proposition 3.4 is written as: "Suppose $G$ acts on $X$ and $E$, $E'$ are $G$-decomposable subsets of $X$. If $E$ is $G$-paradoxical, so is $E'$. I have ...
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Question regarding unexpected hanging paradox [duplicate]

The following is the unexpected hanging paradox: A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to ...
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1answer
48 views

Russell's paradox in ZF theory : Enderton's Elements of set theory : Ch.2

I am reading chapter 2 of Elements of set theory by Herbert Enderton and I have a confusion. Can we contruct a set from subset axiom of ZF set theory, such that the set of all sets which does not ...
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1answer
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Two-child probability paradox, a nuanced explanation

You have a co-worker named Jill. You know Jill has two children, but know nothing more about them. Jill invites you and your family to a holiday party at her house. When you arrive, you knock on the ...
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1answer
52 views

Weird points about validity of argument

I'm just wondering property about validity of argument so I made this weird argument. But it seems that I still do not understand these things... Suppose that An argument must be either valid or ...
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2answers
85 views

Paradox involving Euler's identity [duplicate]

There is something I fail to understand involving Leonhard Euler's identity: It is well known that $(e^{2π})^i = 1$. That means $\sqrt[i]{1} = e^{2π} ≈ 535.49 $. But there's a rule that states $ \...
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1answer
116 views

Paradoxical result from the chain rule

I noticed a very simple problem, yet paradoxical when I was solving a different problem. It would be great if you help me understand which of the following lines lead to the paradoxical result and why ...
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3answers
318 views

Skolem's paradox showing us that we might be trapped in our view of the world

According to Skolem's Paradox, ZFC as a first order axiomatization of set theory has a countable model, but allows a proof that uncountable sets exist in every model of ZFC. It becomes counter-...
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238 views

Unexpected hanging of Mr. Fitch

Judge's statement S: The prisoner will be hanged next week and its date will not be deducible the night before using this statement as an axiom. Using an equivalent form of the paradox which reduces ...
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2answers
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Birthday Paradox Application

I learned about the birthday paradox or birthday problem in school, and it was pretty intriguing. I finished all my homework for said class but I am stuck one specific question, which is supposed to ...
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3answers
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Friday analysis of the unexpected hanging paradox [closed]

The judge told me: A1. You will be hanged on day X. (X is some day from Monday to Friday) B1. You can't deduce what X is. It's Friday morning and I'm still alive. My first deduction is (please tell ...
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Interpolation error and it's weird behaviour

I am trying to find the value of $f(x)=\ln(1-3x)$ at $x=-1.5$ using Newton's interpolation method. I am given the points $-3$, $-2.4$, $-2.2$, $1.8$, $-0.5$, $0$, and using Newton's method gives $f(x)\...
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1answer
40 views

Why do we assume there are ordinary and extraordinary sets?

The Wikipedia page on Russell's paradox states if $R$ were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if $R$ were abnormal, it would not ...
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Koch snowflake paradox: finite area, but infinite perimeter

The Koch snowflake has finite area, but infinite perimeter, right? So if we make this snowflake have some thickness (like a cake or something), then it appears that you can fill it with paint like ...
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1answer
45 views

Explanation of Skolem's Paradox in Enderton's book

In the book 'A Mathematical Introduction to Logic' by Enderton, he stated the Skolem's paradox at page $152:$ Let $A_{ST}$ be your favourite set of axioms for set theory. We certainly hope these ...
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2answers
44 views

Calculating the expected gain in St. Petersburg Paradox

I'd like to understand why can't we calculate the expected gain in St. Petersburg paradox as follows. Let $G$ denote our gain and let $R$ denote the number of rounds that game proceeds. Then we have, ...
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4answers
75 views

Infinite series with finite sum [duplicate]

Assume that I have 1 unit of something and then I add ½ unit resulting in a total of 1.5 units. Then I add half of the half (0.25 units) for a total of 1.75 units. Then I add the half of the half of ...
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4answers
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Why is $((-8)^2)^{1/6} > 0 \text{ and } -2 = (-8)^{1/3}$? [closed]

Why is $((-8)^2)^{1/6} > 0 \text{ and } -2 = (-8)^{1/3}$? Doesn't this contradict the exponentiation rule (power of power)?
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1answer
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When does the Bertrand paradox apply?

Link to Wikipedia article on the Bertrand paradox There's another question asked recently that superficially looks like Bertrand's paradox. Both involve picking random points/chords and then ...
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1answer
78 views

Find a perfect strategy algorithm for finding another person in a shop

'There is a row of 9 consecutive shops, John will visit a shop for 14 consecutive days. John moves venues daily to a shop directly left or directly right (end of row means forced move). John moves ...
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1answer
72 views

If and only if condition for Simpson's paradox

Suppose that female and male students apply to schools A and B. Given that $p>q$ and $r>s$ where $p$ is the ratio of female students accepted to A, $q$ is the ratio of males accepted to A, $r$ ...
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1answer
59 views

Is $\approx$ actually an entourage?

I was looking at applying the ideas in the paper On Nonstandard Topology to Uniform spaces. Given a uniform space $(X,\Phi)$, we can define the relation $\approx$ on ${}^*X$ as follows $$\approx \, = \...
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1answer
397 views

Misunderstanding Löwenheim–Skolem

The Löwenheim–Skolem theorem shows that we can find a countable elementary submodel of $V$ that satisfies $ZFC$. [assuming, Con$(ZFC$)]. Call this set $U$. Then by the definition of elementary ...
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4answers
44 views

What's the square root of i to the power of 4?

This is not a homework question $\sqrt{i^4} = \sqrt{1} = 1$ $\sqrt{i^4}=i^{\frac{4}{2}}\ =\ i^2=-1$ So what did I do wrong?
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1answer
118 views

Seeming contradiction of the tertium non datur principle through a logic problem

The problem is as follows. There is a group of three people (A,B,C) who are perfect logicians, and A is a thief. We say that a person recognises another one if the former knows whether the latter is ...
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1answer
58 views

What is the set $\{M \subset \mathbb{R}^n| M^{af} \subsetneq M^f\}$?

I am reading an elementary general topology book now. I found this formula in the book: $M^{af} \subset M^f$ for any $M \subset \mathbb{R}^n$ $M^f$ is the set of boundary points of $M$. $M^a$ is ...
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1answer
156 views

Is this a commonly known paradox?

I would like to know if the paradox below is commonly known and has a name. Graham Priest, in his book Logic: A Very Short Introduction, at the end of chapter 12 “Inverse Probability“, asks the ...
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1answer
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What is the maximum number which can be written with fewer less than sixty five symbols?

If you try to write down all numbers with the symbols of any alphabet using at most 65 characters, then among them there must be a maximum. Let's call it $N$. But then the number described by the ...
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1answer
55 views

What does it mean for a set to be SO(3) paradoxical?

I am trying to understand the Hausdorff Paradox for later use in the Banach-Tarski paradox. The Hausdorff Paradox states that: "There is a countable subset: $D$ of $S^2$ such that $S^{2}$ \ $D$ is SO(...
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1answer
66 views

How did I arrive at a logical contradiction?

Assume we've defined a solid cylinder $ x^2+y^2\leq 2y$ It follows that $ x^2+(y-1)^2 \leq 1$. We have a solid cylinder of radius one. Now let $ y-1$ = $ \sin(\theta) $. Would it be a contradiction ...
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Original text by Bertrand Russell describing his Barber Paradox?

I am looking for an authoritative online source that gives the original text by Bertrand Russell describing his Barber Paradox. Quine described it like this: In a certain village there is a man, ...
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What if the frequencies do not define a probability space?

In the frequentist interpretation of probability, we define probability as follows $$P(x) = \lim_{n_t \to \infty}\frac{n_x}{n_t}$$ Where $n_t$ is the number of trials we preform of a certain ...
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1answer
29 views

Does there exist a mathematical expression consisting of all mathematical expressions? [closed]

The set of all mathematical expressions consist of all analytic expressions, closed-form expressions, algebraic expressions, polynomial expressions, and arithmetic expressions. Do we run into ...
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1answer
82 views

You are tossing a coin, and rolling a dice. What is the probability that you get a head in toss or an odd number in die?

Probability for tossing on heads$=0.5$ Probability of rolling on odd number on die (1 or 3 or 5)$= 0.5$ As per addition rule (A union B, A or B) that is $0.5 + 0.5 = 1$ that seems impossible. How ...
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“Real-world” application for the infinite Urn problem?

Relative to this question : https://stats.stackexchange.com/questions/315502/at-each-step-of-a-limiting-infinite-process-put-10-balls-in-an-urn-and-remove-o We consider an infinite process where at ...
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2answers
174 views

Russell's paradox and the barber example

In his reply, Peter Smith provided a clear explanation of the barber's paradox. However, I am having some hard time to understand the link between the established theorem and Russell's paradox. In ...