Questions tagged [paradoxes]
Paradoxes are arguments which contradict logic or common sense, often by using false and implicit premises.
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Does this english-language sentence translate to the drinker paradox?
There are (one or) two students such that if they pass the exam, then every student passes the exam.
I'm tasked to formalize the above sentence and give either a proof or counterexample. If we take ...
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A paradox-like consequence of Borel-Cantelli's Lemma
Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables, whose range is $\mathbb{N}$ (e.g., Geo(1)). Consider now the random sequence $X_1,X_2,X_3,...$
What is the probability that each $...
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The Paradox of "Knowing Everything" [closed]
Consider the following statement:
I know everything.
Now, assuming that the above statement is true, what is the truth value of the following statement? Is it true or false?
I know what I don't ...
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A Subjective Probability Paradox When Drawing Balls From an Urn
Suppose that you are randomly drawing balls from an urn without replacement. The urn contains an unknown number of white balls and exactly one black ball. Before starting to draw, your subjective ...
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Intuitively, how does one show Russell's paradox in intuionistic logic?
I am reading the comprehension axiom wiki, where I find this interesting point:
Passing from classical logic to intuitionistic logic does not help, as the proof of Russell's paradox is ...
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Can Zeno's Dichotomy Paradox be used in a proof?
I've been watching multiple videos on the subject of Zeno's Dichotomy Paradox. Specifically, I have been looking into the paradox of halving distances. Where you continuously halve the distance from ...
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Looking for an utility function for which St. Petersburgh paradox becomes unbounded
A professor has explained to my class St. Petersburgh paradox and has introduced the concept of utility function.
The professor then asked us to find an utility function with a positive first ...
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If Flatlanders walk around the Mobius string and we put a new one behind them every few steps, at which point they will be "reversing"?
Let's say that we put a Flatlander on the Mobius strip and make them move forward. Every X feet they travel, we put another one X feet behind them so that they can see each other. Then, after both of ...
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Hard to believe St.Petersburg Paradox after doing Calculations [closed]
A person tosses a fair coin until a tail appears for the first time. If the tail appears on the $n$th flip, the person wins $2^n$ dollars. Would you be willing to pay $1$ million for each game if you ...
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Is Gabriel's Horn possible in four dimensions? [duplicate]
Gabriel's horn, a shape with infinite surface area but finite volume, is one of the most counterintuitive objects in mathematics. However, it is pretty intuitive that we can build a Gabriel's horn in ...
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Effects of multiple trials of the 2 envelopes problem
I'd like to do multiple trials of the 2 envelopes problem so I can see the probabilities play out over multiple repetitions. I've never seen a description of how this happens and how the multiple ...
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Expectation paradox: $E(X)$ is larger than every possible value of $X$?
I have a random number generator that generates a random real number between $0$ and $1$ (not including $0$ or $1$).
I generate a first number. Then I generate more numbers until I get a number that ...
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What is the framework in which we can talk about the procedure of Richard's paradox rigorously?
It seems there are two variants of Richard's paradox: one pertaining to natural language and one pertaining to first-order logic. I will focus on the latter.
Now as pointed out in this post, there are ...
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Deterministic Simpson's Paradox Antidote
Suppose I have eight positive numbers $a_i,b_i,c_i,d_i$ for $i=1,2$ satisfying $$\frac{a_i}{b_i}\le \frac{c_i}{d_i}\hspace{1in}(1)$$ I'm looking for additional conditions that will ensure \begin{align}...
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Logically How Can a Set Contain Itself? [duplicate]
I was thinking about the idea of a set containing itself and that's driving me crazy.
An example of such set is the set of all things that are not turtles, since our set is a "set" then it's ...
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Simpson's Paradox: Inequality Equivalence
I have a question regarding a paper dealing with Simpson's paradox.
The article can be viewed here:
Copositive Matrices and Simpson's Paradox
In the article it is hinted that inequality (1) is ...
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Integral of a derivative of a tangent function, outputting a secant instead of tangent
I am a humble engineer in his first year and I've noticed something curious.
I'm from Belgium so English is not my home language (Dutch is) and as I went to a technical school, I haven't received much ...
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How to avoid "impossible" linears with trig integrals
Let's say I want to integrate $\int\sec^3xdx$. Due to the way this expression is set up, you must use integration by parts, and not u-sub, etc.
Applying integration by parts, I get $\sec{x}\tan{x}-\...
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Banach-Tarski-like Paradox
This is Problem 9 of Chapter 1 of Stein and Shakarchi's Functional Analysis.
As a consequence of the previous problem one can show that it is not possible to extend Lebesgue measure on $\mathbb{R}^d$ ...
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Is the liar's paradox the principle of explosion in disguise?
Sorry for a high-school-level question, but something is bugging me about the liar's paradox.
The liar's paradox can be formally written as an axiomatic system with an axiom $P \equiv \lnot P$. Due to ...
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The James-Stein estimator seems absurd to me
I do not understand how the James-Stein estimator can perform better (from the point of view of expected Euclidean distance) than the maximum likelihood estimator. I am willing to accept it because I ...
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Modelling subtraction to allow $\infty - \infty$ [duplicate]
A problem (at least, I think it is) by saying that $\infty - \infty = 0$ is the fact that there are numerous examples of (un)countably infinite sets that can be subtracted from each other, and ...
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Using paradox to construct tautology for $p\land \lnot p$
Can a paradox be used to construct a tautology?
$p\land \lnot p$ is logically a contradiction.
Suppose $p$ is the sentence:
"This sentence is a lie"
which we know is paradoxical
is used in ...
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Russell's paradox and the barber paradox example
Russell's paradox deals with the set of all sets that do not contain themselves as a member of that set. It ask whether that set of all sets is also the member of the set or not.
The barber paradox ...
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Algebraic contradiction involving infinitesimals
Let $\Delta$ be infinitesimal, $\hat{e}=(1+\Delta)^{\frac{1}{\Delta}}$, and $log_{\hat{e}}(a)=\hat{ln}(a)$.
$$a^{\Delta}=({\hat{e}^{\Delta}})^{\hat{ln}(a)}=(1+\Delta)^{\hat{ln}(a)}$$
Consider $\frac{a^...
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Russell's paradox explanation
The Russell's paradox deals with the set of all sets that do not contain themselves.
So I want a example of a set that do not contain themselves.
I got a examples of set of turtles.It will contain ...
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Start with $1,000. Suppose you can flip a coin any number of times. Heads you quadruple your money. Tails you lose it all. How many times do you flip? [closed]
Start with $1,000. Suppose you can flip a coin any number of times. Heads you quadruple your money. Tails you lose it all. How many times do you flip?
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Hilberts hotel; mapping one type of infinity to another type
I just finished watching this video and I feel like I'm missing something very obvious. I've read quite a number of the questions here but haven't been able to figure out what.
I assume Hilberts hotel ...
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Define conditional probability on an event given conditional probability on a $\sigma$-algebra?
Let $(\Omega, \Sigma, P)$ be a probability space and $A, B \in \Sigma$ events. If $\Pr (B) = 0$ then there is no coherent definition for $\Pr (A | B)$. As Kolmogorov states, “the concept of a ...
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What 'type' of reflection principle is used in ill-founed set theories, and how does it work?
Recently I have been studying the reflection principle and non well founded set theories. As of lately I have wondered what is there relation? and how does a reflection principle fit in a non-well ...
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How can Gabriel's Horn have a finite volume?
I’m a highschool student who just finished Calc AB and I’m fascinated by the concept of Gabriel’s Horn but I’m confused by the claim that its volume is finite.
Correct me if I’m wrong but isn’t it ...
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I have some questions about the Ross-Littlewood Paradox
TLDR at the end.
Hi, I recently saw this comment given by "completely-ineffable" on the r/badmathematics subreddit. And I just wanted to make sure if I understand it correctly and wanted to ...
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how does Bertrand's paradox challenge the classical definition of probability?
On page 9 of Papoulis's book[Probability, Random Variables, and Stochastic Processes], the classical definition of probability is as follows:
The probability of an event equals the ratio of its ...
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Can braess routes change dynamically with respect to changing demand in a network
Braess Paradox is a counter intuitive phenomenon where removing a link from a network increases the network efficiency. Usually these links are detected over long period of time in an average ...
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A cone has a height of $10\;\mathrm{ cm}$ and base with radius of $4\;\mathrm{ cm}$. Find the volume of the cone.
A cone has a height of $10\;\mathrm{ cm}$ and base with radius of $4\;\mathrm{ cm}$
(a)Find the volume of the cone.
(b) A cone frustum is formed when you eliminate the superior part of the cone with ...
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Can Hilbert's Hotel be explained by a difference between ordinal numbers and cardinal numbers
In taking a philosophy of maths course I have been very curious about the notion of infinity, and whether or not it is paradoxical. One thing I have frequently thought is that "infinity" as ...
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Perimeter of Inscribed Square - Paradox?
Imagine a simple X / Y coordinate graph. A circle surrounds the point of origin. Let's say the radius = 3.
We want to know how many points exist on the circumference of the circle through which a ...
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Is there an equivalent to Godel's theorem that looks like "This statement is provable."? [duplicate]
I've been thinking about Godel's thoerem and the liar's paradox. The liar's paradox, when flipped around, stops being a paradox and becomes valid logically whether the statement is true or not. "...
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Paradox: Derivative w.r.t. basis element
Let $z_1\in \mathbb{R}\setminus\{0\}$ and $z_2\in i\mathbb{R}\setminus\{0\}$. Then $\{z_1,z_2\}$ form a basis for $\mathbb{C}$. This means that any $z\in\mathbb{C}$ can be written as a linear ...
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Does Bertrand's Paradox depend on the Axiom of Choice?
This is the set up to Bertrand's Paradox:
Randomly choose two points on a circle. Construct a line segment (circle chord) between them. Construct an inscribed equilateral triangle within the circle. ...
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Paradox of proof by induction
I'm having trouble understanding the following situation:
Given an identity P(n) that is wrong (but I don't know whether it's right or wrong), I am trying to check whether this identity can be proven ...
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Axiom of replacement vs. Axiom of separation and Galileo paradox
Galileo's paradox says that on one side there are fewer square numbers (second powers) among natural numbers than all numbers because only some numbers are squares. On the other side, there are as ...
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Two people Monty Hall paradox
We're on a game show, and we have to select between three doors, one of which has a Lamborghini behind it while the others have goats. After we've decided, the host opens one of the other two doors, ...
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A variation on the three prisoners problem
Three prisoners hear that one of them will be executed (the exact person who will be executed is determined upfront, and cannot be changed), while the other two will be released. Prisoner A asks the ...
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What is the real potential energy of an alternating q and -q infinity system?
We can create a model of an infinite one-dimensional ionic crystal. Considering a system of $N\gg1$ alternating point charges $Q$ and $-Q$, that are distributed as the distance between two neighboring ...
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The Engagement paradox
Firstly, I should say that I came up with this paradox after reading of the Grimm Reapers paradox, but I’m not quite sure how this should be resolved. Nevertheless here is the problem:
Suppose a lady ...
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Dirac delta distribution in $1D$
We know that $$\frac{d^2}{dx^2}\left(\frac{1}{|x|}\right) =-4\pi\delta(x)$$
where $\delta(x)$ is Dirac delta distribution.
$$\Rightarrow \lim_{x\to 0}\frac{d^2}{dx^2}\left(\frac{1}{|x|}\right) =-\...
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Question regarding Bertrand's paradox
The classic example of Bertrand paradox deals with the case where we count the uncountable set of chords in a circle in different ways and ends up getting different probability each time. The ...
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In the Staircase paradox, where does the limit function differ from the hypothenuse?
I am aware of the answers to the Staircase paradox here
Now for the example of the unit square and the approximation of the hypothenuse by the staircase function, surely the limit function differs ...
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would a set of all countable sets have any paradoxical properties?
I recently talked with a friend about set theory and he mentioned "set of all countable sets". I think that such set does not exist (just like "set of all sets" does not exist) and ...
