Questions tagged [paradoxes]

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

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Paradoxical homework… or am I stupid?

Let $z_1,z_2,z_3\in\mathbb{C}$ such that $|z_1|=|z_2|=|z_3|=1$. If $z_1+z_2+z_3\ne0$ and $z_1^2+z_2^2+z_3^2=0$ then prove $|z_1+z_2+z_3|=2$. What I did $z_1^2+z_2^2+z_3^2=0~~|\cdot(z_1+z_2+z_3)$ $\...
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Set Theory Again

My recent question about Russell's paradox has been closed on the grounds that it is the same as another question asked elsewhere. It is not the same question and the answers to the similar question ...
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Russell's Paradox Again [duplicate]

This is NOT a duplicate question and I don't know why it is marked as one. This is a follow-up to a previous question asked here... Untangling Russell's Paradox It seems that if (as per ZFC) we ...
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Is there a “natural”, non-paradoxical way of defining natural numbers?

What are actually natural numbers? I know that there are Peano's axioms, but they are mostly "abstractly" descriptive. He claims that something like N exists with such properties and yes, my ...
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Untangling Russell's Paradox

I'm pondering the logical connection between Russell's paradox and metaphysics, where it appears as a silghtly different problem. I suspect it appears different only because of the way it's usually ...
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113 views

Does permitting comprehension for all (and only) contingent formulas result in paradoxes?

Does permitting comprehension over all well-formed formulas that are neither contradictions nor tautologies result in paradoxes? I have a hunch that a simple extensional set theory with the "...
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Can we insist on having a universal category (with ZFC and a hierarchy of classes)?

I'm curious whether there is any harm on insisting on the existence of a universal category. I'm also curious if we can make one in an extremely naive way by stapling a hierarchy of classes to $\...
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Paradox of $-i$ seemingly equal to $1$ via the Wallis product for $\pi$ and the Euler sine product

Assuming $x$ is a real variable throughout,$$\frac{\sinh(ix)}{i}=\sin(x)$$ $$\frac{\sinh(\pi ix)}{\pi ix} = \frac{\sin(\pi x)}{\pi x}$$ $$\frac{e^{\pi ix}-e^{-\pi ix}}{2\pi ix}=\prod_{n=1}^{\infty}\...
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Could anyone please help me with this quadratic paradox?

Yesterday I was solving a problem relating to algebraic identities and accidentally proved $a^{2}-b^{2}=b^{2}-a^{2}$. Could anyone tell me how is this wrong (it obviously is) algebraically? This is ...
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Explaining Russell's Paradox in simple minimum set theory notation [closed]

I was trying to learn about Russel's Paradox and got confused about the whole thing. Can it be explained in simple layman terms using a bit (minimum) of set theoretical notation?
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Using R, solve the birthday paradox

The probability that two students in a class have the same birthday is at least 75%. What is the minimum size of the class? I tried, ...
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Could someone please explain this statement about the significance of Russell's Paradox?

This is from an article talking about Russell's Paradox and why it was so significant. Could someone please explain what it means? The reason this conclusion was so groundbreaking was because it ...
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Two envelopes paradox [duplicate]

This is a question about the classical two envelopes paradox problem. The solution posted in https://plus.maths.org/content/two-envelopes-problem-resolution seems to be uniform across all the sources ...
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ZF-{regularity, comprehension} with “reflexive” comprehension

ZF without regularity with "reflexive" comprehension. Can we successfully defuse known paradoxes (and produce a consistent theory) by using a comprehension schema that limits comprehension ...
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Did all axiomatic systems face a crisis with the discovery of Russell's paradox?

When Bertrand Russell outlined his paradox to Gottlob Frege just as his Grundgesetze was going to print, it effectively destroyed the consistency of Frege's theory of arithmetic. But was this the ...
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“St. Petersburg paradox”-like paradox which feels even less intuitive

The original St. Petersburg paradox presents a game whose EV is infinite but no reasonable person would pay large amounts of money to play. Here's a new version: Alice and Bob have a weighted coin ...
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Is the liars paradox actually a paradox at all? I don't think it is.

Note - I am not versed in logic notation or whatever the particular notation is for this kind of thing. Sorry in advance. Edit - This is to address the argument about taking the statement as a whole ...
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A confusion in understanding Lebesgue measure

Any open subset $G$ of the real line can be written as a countable disjoint union of open intervals say $G=\bigcup_{i=1}^{\infty} (a_i,b_i)$ which its closure is $\operatorname{cl}(G)=\bigcup_{i=1}^{\...
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Help me to understand the potato paradox

From wiki - Fred brings home $100$ kg of potatoes, which (being purely mathematical potatoes) consist of $99\%$ water. He then leaves them outside overnight so that they consist of $98\%$ water. What ...
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Monty Hall Problem with unknown probabilities

Does someone know a solution to the following generalization of the Monty Hall Problem: The Problem: Assume you are on Let's Make a Deal and are presented with the regular dilemma of the Monty Hall ...
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How to disprove Russell's paradox in ZF?

How to avoid a Russell-style paradox in ZF? The Zermelo-Fraenkel system, by not using a theory of types, does not prevent us from posing Russell's paradoxical scheme. Only when the argument is ...
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How to solve $\int_0^x f(t) dt = g(f(x)) $?

How to solve an equation of this type : $$\int_0^x f(t) dt = g(f(x)) $$ for a given function $g$. Im not so good with differential equations and integral equations. I know how to solve $h'(x) = j(h(x))...
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Class size paradox vs. length-biased sampling

According to Introduction To Probability by Blitzenstein and Hwang p. 244, one example of length-biased sampling is the following: "For example, asking randomly chosen mothers how many children ...
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Where's the flaw in this construction of a set that's both countable and uncountable?

Say we constructed a set of all finite math expressions, which is countable. Then we take the subset of expressions that evaluate to a single real number. That subset should still be countable. Now we ...
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Vector component paradox

This is taken from the famous 3 polarisation filter experiment on light and got my thinking about coordinate transformations. and vector component consistancy and would really like an answer. Suppose ...
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Interesting/Paradoxical Math Problems for High School Students

I am creating a math course for high school students (not necessarily advanced) about interesting/paradoxical math problems (think Monty Hall and Hilbert's Hotel). What are some other interesting math ...
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Set of Infinite Sets Contradiction?

Consider the set $S$ that contains all sets $X$ such that $|X|\geq |\Bbb N|$. The whole reason ZFC was made was because of Russel’s paradox right? And so we know that we can't have a set that contains ...
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choice set for equivalence classes

I'm trying to understand the proof of the Paradox of Banach-Tarski. I was reading Wagon's work about the paradox and tried to understand the proof of Theorem 1.5, only I haven't warmed up to group ...
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Another take at a Debt 'paradox'

I use quotes around paradox because this is certainly not a mathematical paradox but only used in common usage. The situation goes as follows A tourist $\beta$ visits hotel $\nabla$ in a poverty-...
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41 views

Negative equal positive paradox

I was just bored and started practicing even more the exponentiations, and as I was working, I went ahead and did this: $(-1)^{2} = ((-1)^{\sqrt{2}})^{\sqrt{2}}$ So, I entered in my calculator of what ...
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Creating the Liar's Paradox with one truth and one lie (that aren't meta).

Some friends were playing a game where you say 1 truth and 1 lie about yourself, and the others have to guess which is which. Just for fun, I was wondering if there was a reasonable way to give 2 ...
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61 views

Extended Bertrand's Boxes Problem - 3 boxes (2 coins per box)

Apologies for the long text, but it is to give context to the exercise. I am having some trouble understanding an extended version of Bertrand's paradox. It comes in a few questions I have answered ...
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My paradox in determining the probability of distributing similar balls into different boxes

Inspired by the discussion between @Matthew Pilling and me in this post What is the probability that there more rabbits than chickens in each of these three cages?, I am trying to find and understand ...
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Is it possible that individually, pieces of evidence increase the chance of guilt but together they decrease the chance of guilt?

Let G be the event that a certain individual is guilty of a certain robbery. In gathering evidence, it is learned that an event $E_1$ occurred, and a little later it is also learned that another event ...
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Does ``Converge to'' and ``Strict Equality'' Always Mean the Same Thing? If Not, is This a Paradox?

Consider the geometric series with a = 1 and r = 1/10. Then, we have $$ \sum_{n=1}^{\infty}\left(\frac{1}{10}\right)^{n} = \sum_{n=0}^{\infty}9\left(\frac{1}{10}\right)^{n} - 9 = 9\left[ \lim_{k\to\...
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Understanding how different groupings of terms in an Infinite series can lead to different answers.

One of the most, conceptually speaking, for me to understand is the topic of infinite series. I have always had a hard time proving that an infinite series diverges or even finding a solution for the ...
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Formally writing the Bertrand's paradox in logic?

I was trying to formalise the different methods of calculating probability in bertrand's paradox in logic, this is my attempt: \begin{align} \Phi \equiv x^2 + y^2 = 1\\ \Psi(m,c) \equiv y = mx + c\\ \...
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Solving $\int_0^{\pi/2}x\cot x\,\mathrm{d}x$ while running into a $0\times\infty=0$ problem

While i have been trying to solve the integral $ \int_0^{\pi/2} x\cot x \, \mathrm{d}x $ i have noticed that by trying integrating by parts using $u = x$ and $\mathrm{d}v = \cot x \, \mathrm{d}x$, i ...
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Notation used in Russell's paradox

I am slightly confused by the notation used in Russell's paradox. I am following this text. I understand that $\phi (x)$ is this boolean function, which outputs either ...
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Couples are equally likely to have 1 or 2 children. How likely does a randomly chosen person have a sibling?

Assume that every couple can only have exactly 1 child or two children, with those outcomes being equally likely. Ignore any silly extra factors (e.g. 1 child dying, another being alive). If I choose ...
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Is $a$ bigger than $0$ or not?

Consider Goldbach's original conjecture (no need worry, because we don't talk about "Goldbach" itself): every integer $n> 2$ could be detached as a sum of three primes. (In Goldbach's ...
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I get $ \int fg' \, dx = 2fg + \int fg'\,dx + C $ while performing integration by parts. What did I do wrong?

I am reaching a paradox by using integration by parts. I must be going wrong but unable to figure where I went wrong. Can you help me understand what I did wrong here? We have two differentiable ...
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Why wasn't the Russell paradox solved by creating a third kind of pertinence, say $\rlap {~\small+} \in$, beyond $\in$ and $\notin$?

[I am trying to understand the historical process of math, and also the principles in a larger scope than just learning the axioms. So I think this question is a good exercise in that direction.] I ...
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Why do we assume the set of all sets that do not contain themselves exists?

It seems to me that the set of all sets that do not contain themselves is very similar to the set \begin{equation} X = \{x | \ x \not\in X\} \end{equation} which is again very similar to the number $x$...
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One-sample T-test p-value paradox?

Let $X$ be a random variable. According to CLT, the distribution of means of samples of $X$ converges to normal as the sample size grows. Suppose we sample $X$ $n$ times and observe data $[x, x, x, ......
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Paradox in Theory of probability of transition to polar coordinates

Let there are two independent random variables $X$,$Y$ with normal distribution. Vector $(X, Y)$ can be considered as a random point on the plane. Let $R$ and $\phi$ polar coordinates of this point. ...
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Probability of picking a real number randomly

If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But ...
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Can someone explain the relation between “Achilles chasing turtle” paradox and monotonic, bounded sequence to me?

Recently I have read a book called "Caculus - Basic Concepts for High School" The possibility of infinite but bounded sets was not known, for example, to ancient Greeks. Suffice it to ...
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60 views

Paradox in definite integrals

I don't know whether this question is suitable or not for this forum because it is not exactly rigorous mathematics. We generally descibe geometrically absolute value of a definite integral as area ...
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Hosting guests coming in infinite lines, each line infinite in length and compossed of infinitely large buses, each bus with infinite guests?

I'm refering to Hilbert's hotel problem. I know it's possible to host an infinite line of infinitely large buses where each one brings infinite passengers this way. As primer numbers are infinite we ...

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