Questions tagged [paradoxes]

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

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The Tennessee Waltz paradox [duplicate]

I love to dance, and one of my favorite dances is the Waltz, and a beautiful waltz to dance to is “The Tennessee Waltz” which was a monster hit for Patti Page in 1950. An unusual feature of this song ...
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Banach-Tarski paradox in a countable set?

Let $M$ be a countable set. I can take a finite set $F$ of n-ary operations and construct a minimal set $M'$ for which: $M \subset M'$ For each $f \in F$ and $m_1,m_2…m_n \in M$ where $n$ is the ...
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3answers
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Why does the comparison test fail here?

Someone recently stumped me here. Consider the series $\sum_3^{\infty} \dfrac{1}{x^{1.0001}}$ and the series $\sum_3^{\infty} \dfrac{1}{x\ln{x}}$. The first series in convergent, and the second ...
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1answer
47 views

Contraposition fallacy in raven paradox?

I do not understand how the contraposition "all non-black objects are not ravens" is logically identical to "all ravens are black." I see that IF all ravens are black, THEN all non-black objects are ...
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1answer
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Prove the sequence $\{ 2^n \}$ is unbounded using Russell's paradox

I am really stuck on how to make use of Russel's Paradox. This is where I started. Proof by contradiction. Suppose $\{ 2^n \}$ converges then it is bounded. Let $M$ be an element of the reals and a ...
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Attempting to prove $\int_0^{\infty} \frac{1}{x} dx$ converges

I'm sure countless people has tried to prove this but this is how I came across this, although I do remember seeing this in the past. Background https://www.youtube.com/watch?v=vQ0himyDR2E I was ...
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1answer
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Brainteaser question/solution which reasonment is correct and why?

I am struggling to find some explanation to this: here is my problem: "A cube of ice melts without changing shape at uniform rate 4cm$^3$/min. Find the rate of change of the surface area of the cube ...
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3answers
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The paradox of random occurrence the family problem

we have 100 families: 10 families have no children, 40 families have 1 child for each one, 30 families have 2 children for each one, 10 families have 3 children for each one and 10 families have 4 ...
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1answer
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Mathematical proof of Olber's paradox

Is there a mathematical proof regarding Olber's Paradox? For me this feels a false paradox as I think that it IS possible to have an infinite (in space and in time) Universe with an infinite number ...
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1answer
138 views

Minimal example of Simpson's paradox

Let's say that a finite probability space $(\Omega,\mathscr P(\Omega),P)$ has Simpson's property if you can find events $A,B,C\in\mathscr P(\Omega)$ such that $P(C) \in (0,1)$. $A$ and $B$ are ...
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Are there paradoxical/ counter-intuitive laws in predicate logic? ( beyond the Drinker Paradox)

Preliminary remarks. (1) The term "paradoxical" is not used in a negative sense here. What is " para-doxical" is literally what disagrees with the general and uninformed " opinion": it could be argued ...
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The proof of Banach-Tarski paradox on Wikipedia

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#Step_3 In the part of the proof that deals with the fixed points on $S^2$, the page says "Let $\lambda$ be some line through the origin ...
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1answer
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Self-referential multiple-choice question [duplicate]

I came across this question, which I believe cannot be satisfactorily answered, but I'm not completely sure. Can you tell me if it has a answer? ...
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1answer
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Brandenburger-Keisler paradox

The Brandenburger-Keisler paradox runs as follows. 1) Suppose that A believes that B assumes that A believes that B's assumption is wrong 2) Ask whether A believes that B's assumption is wrong 3.1) ...
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1answer
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Question about the Axiom of Specification and Russell's paradox

I have been reading Halmos' book, Naive Set Theory and while reading the part about Russell's paradox I had the following question: Halmos shows that nothing can contain everything, as he puts it, but ...
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1answer
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Russell's “propositional” paradox

In the Stanford Encyclopedia's page on Russell's Paradox, we get the following anecdote about an additional, lesser-known paradox from Russell: ...in Appendix B Russell also presents another ...
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What are the details of this step in a proof of the Banach-Tarski paradox?

In this exposition of the Banach-Tarski paradox by Terry Tao, Corollary 1.4 says, There exists a partition $S^2 = \Gamma_1 \uplus \dots \uplus \Gamma_8$ and rotation matrices $R_1, \dots, R_8 \in ...
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Complex Number Logarithm Paradox [duplicate]

I think I may have stumbled upon a paradox today: $$\ln(1) = 0$$ $$\ln(1) = \ln(-1^{2}) = 2\ln(-1)$$ $$\ln(-1) = \ln(e^{\pi i}) = \pi i$$ $$2\pi i = 0 \rightarrow i = 0$$ I can't seem to find the ...
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1answer
236 views

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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2answers
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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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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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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
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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
40 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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1answer
46 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
261 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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1answer
76 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
140 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
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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
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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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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
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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
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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
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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
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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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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
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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
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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
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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
131 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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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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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
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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
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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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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
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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
104 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
418 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 ...