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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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Does absolute infinity invoke Cantor's Paradox? [closed]

Let $\mathcal{P}(X)$ denote the powerset of $X$, $\mathcal{P}^2(X)=\mathcal{P}(\mathcal{P}(X))$, and $\mathcal{P}^n(X)=\mathcal{P}(\mathcal{P}^{n-1}(X))$; $\mathcal{P}^0(X)=X$. It is trivial to show ...
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The ant and the rubber string.

We have an ant on the tip of a horizantal rubber string of length say $\text{10 cm}$. The ant moves $\text{5 cm}$ each second, and the rubber string is stretched $\text{100 cm}$ each second. Will ...
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Finding the first program/algorithm with more than 1 million symbols in length which produces a new finite sequence within 1 trillion operations

Edit: There has to be an adjustment to my question as the number of permutations far exceeds 1 trillion which would lead to the main program doing way more than 1 trillion operations. Therefore, the ...
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0answers
55 views

Flip $n$ coins on a circle. Assume a coin has been chosen from among those whose neighbors are both heads. What's the probability it is heads?

This is a generalization of the problem below (first appeared here) I am particularly curious to know if there is a closed-form formula to calculate the probability for any $n$ and any probability of ...
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31 views

Forex rate: Expected value paradox

Let us suppose at present 1 dollar = 1 euro After 1 year There is 50% chance that 1 dollar = .80 euro ...[1] And there is 50 % chance that 1 dollar = 1.25 euro ...[2] Therefore expected value ...
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1answer
20 views

Why can('t) I map this closed subset of a product of compact sets to a non-compact set?

Let $S$ be a countable set of real numbers that is bounded but has neither a maximum nor a minimum. Next we create the product topology $[0, 1]^S$ (using the usual topology on $[0, 1]$). This should ...
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1answer
33 views

Gödel diagonalization and formulas not holding for themselves

Is there a formula $\varphi (n)$ in one free variable $n$ in ZFC (PA etc.) such that for every formula $\psi(n)$ in one variable the equivalence $$ \varphi ( \ulcorner\psi\urcorner) \leftrightarrow \...
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40 views

Aristotle's wheel [duplicate]

two wheels of different diameter attached to each other at the center, roll along a straight line. since they have different diameters, the smaller wheel must slip. (see other stackexchange answers) ...
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6answers
252 views

Probability of a single trial within binomial experiment vs. stand-alone bernoulli experiment

When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. ...
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39 views

Tennis Paradox Probability

Under certain circumstances, you have your best chance of winning a tennis tournament if you play most of your games against the best possible opponent. Alice and her two sisters, Betty and Carol, ...
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2answers
103 views

Koch curve from Cantor sets (paradox)

The Koch curve is normally constructed by taking a line segment, replacing the middle third with two copies of itself forming legs in an equilateral triangle, and then repeating this recursively for ...
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1answer
34 views

Contradiction: Portion of our area is greater than our full area

I was reading an answer to a stack exchange post titled Is the electric field of a volume charge distribution well defined? . That answer is shown in the image below: Now I make a comparison ...
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1answer
28 views

Car-garage paradox with just one door

Special relativity implies the possibility of some apparently paradoxical situations, which can ususally be made sense of if one applies the theory rigorously. One of these is the car-garage paradox: ...
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0answers
101 views

Causal Inference A Primer Study Question

I am reading Pearl's Causal Inference book and attempted at solving study question 1.2.4. Here is the entire problem: In an attempt to estimate the effectiveness of a new drug, a randomized ...
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1answer
56 views

When to quit a coin toss doubling game?

The game is as follows: I put in a dollar and if I get heads, I double my money. I can then continue playing and double my $2. Basically, I'm always allowed to continue playing and double the previous ...
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4answers
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Spheres cause contradictions in dimensions $10$ and more?

According to this Numberphile video, if you tightly pack hyper-spheres into a hyper-box and then find the radius of the largest hyper-sphere that could possibly fit in the remaining space, the ...
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2answers
110 views

Why Hilbert changes the property of a set in his Infinite hotel?

I'm not a mathematician, so my question may look a bit lame to most of you. In the Infinite Hotel paradox we are dealing with infinite set of pairs (room/guest). A main property of a pair is 50/50 ...
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2answers
68 views

Backwards Zeno's Paradox [closed]

As I understand it, correct me if I am wrong, walking a finite distance will not take an infinite amount of time, because although you have to travel an infinite number of finite distances, the time ...
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1answer
78 views

How can adding new numbers overflow $\mathbb{N}$ in Cantor's diagonal argument?

I've been thinking and asking around about this for a while. So I think Cantor's diagonal argument basically said that you can find one new number for every attempted bijection from $\mathbb{N}$ to $\...
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0answers
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expected utility, St Petersburg Paradox, effect of initial wealth on limit price

I have a question about the St-Petersburg paradox. In the case of expected utility with log utility function, how can we show analytically that, for $w > 2$, $c$ (limit price) is increasing in $w$ (...
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1answer
40 views

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
125 views

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
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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
114 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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1answer
78 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
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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
72 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
715 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
115 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
88 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
98 views

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
281 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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242 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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32 views

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
72 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
55 views

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
55 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
124 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 $ \...
3
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1answer
124 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
337 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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1answer
244 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
265 views

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
132 views

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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0answers
23 views

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
58 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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7answers
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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
58 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
82 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
82 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 ...