Paradoxes are arguments, which are contradicting with logic or common sense, mostly using a false and implicit premises.

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On Proving that the first euclidean axiom is wrong [on hold]

Well, The first axiom in the euclidean geometry is "A straight line segment can be drawn joining any two points". But I think that there are points that can't be joined: In the image below, We have ...
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Fallacy - where is the mistake?

Could anyone help me to find the mistake in this fallacy? Because the actual result for $I$ is $\pi/2$ \begin{equation} I = \int_{0}^{\pi} \cos^{2} x \; \textrm{d}x \end{equation} \begin{equation} I ...
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How would you explain this graph illustration of Simpson's paradox?

I need your help for understanding WHAT in the graph you find in the following link proves Simpson's Paradox. For those who don't know about Simpson's paradox, it is the inversion of the inequalities ...
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Points and real intervals [closed]

The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, ...
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If you have two envelopes, and …

Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch ...
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Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
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Why is Banach–Tarski's paradox so interesting?

Here is how I understand the Banach–Tarski paradox, based on the Wikipedia article : with a clever partitioning, one can decompose a solid ball into two solid balls, each identical to the first one. ...
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Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
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Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. ...
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If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

The title pretty much says it all: If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true? Edit: Let me attempt to be a little more precise: ...
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Dichotomy Paradox for the Running Man.

This question is inspired by the Dichotomy Paradox but with a twist: Let's say that Telemachus is running between two light posts, distance L length units apart. He starts at the first light post with ...
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Fixed point combinator and functions with no fixed point

In lambda calculus the fixed point combinator is defined as: It is very easy to see how $Yg =g(Yg)$ for any $g$ by using $\beta$-reduction. At the same time I wonder what is the meaning of ...
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Constructing a circle from a square [duplicate]

I have seen a [picture like this] several times: featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and ...
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Demonstration that 0 = 1 [duplicate]

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e ...
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Explain probability paradox

I was planning my cycling schedule when I thought of this question... Can anyone explain why this is not true??? Suppose there is a 1% chance of a person getting knocked down by a vehicle each ...
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“Waiter's paradox” - what's wrong with this reasoning? [duplicate]

Here's a puzzle I just heard and while I know that this reasoning is fundamentally wrong, I can't explain why: Three people bought a dish for, say, 25\$ and paid 30\$ The waiter didn't want to ...
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Problems with nesting proof predicates in first order logic.

Whenever I start nesting proof predicates, I always seems to run into these bizarre situations. I was wondering if anyone knows about this and could shed some light on it or provide me with some ...
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Is there such a classification as Co-Paradox?

So, my line of thinking is that a set that contains all sets that do not contain themselves is a paradox. And the opposite of that is a set that contains all sets that contain themselves, and, while ...
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Understanding the Banach-Tarski Paradox

How is it possible to prove a paradox? Also, can someone explain the Banach-Tarski paradox in layman's terms (for someone up to calc 3 and ODEs knowledge)?
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How to make a good definiton

The reason why I come up this idea may due to Banach–Tarski paradox. The process we make a definition may consist of several steps. First step is that we observe a phenomenon. Second is to make a ...
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Formal approach to (countable) prisoners and hats problem.

I've found this nice puzzle about AC (I'm referring to the countable infinite case, with two colors). The puzzle has been discussed before on math.SE, but I can't find any description of what is ...
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Why is time important in the Ross-Littlewood paradox?

I have read many defferent versions of the Ross-Littlewood Paradox. This post: Fun quiz: where did the infinitely many candies come from? This post: Paradox: increasing sequence that goes to $0$? ...
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Fun quiz: where did the infinitely many candies come from?

Story 1: Let there be a bowl $A$ with countably infinite many of candies indexed by $\mathbb{N}$. Let bowl $B$ be empty. After $1/2$ unit of time, we take candy number 1 and 2 from $A$ and put ...
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St. Petersburg and the law of large numbers

Recently I learned about the discussion around the St. Petersburg paradox and how people try to explain why the calculated expected value differs so much from most people's intuition. My question: ...
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Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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Is the pseudomenon a statement?

I'm asking this because I'm teaching a class on paradoxes for kids, and I realized I have no idea what the answer to this question is. It is a research question in the pedagogical sense, I suppose. ...
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Does this paradox have a name?

As a student many years ago I learned of a paradox of something that is almost a certainty, while at the same time being highly improbable. For example, if you flip 10 coins the chances of all being ...
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Could you explain Perron's paradox to me, please?

This is Perron's paradox: Let $N$ be the largest integer. If $N > 1$, then $N^2 > N$, contradicting the definition of $N$. Hence $N = 1$. What does it mean? I get from it that a very large ...
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Russell's paradox with bounded comprehension

Consider the set $S = \{A, \varnothing\}$ and define $A = \{x \in S|x \not\in x\}$; this is the same as Russell's paradox except with bounded comprehension, ie $A\in A\iff A\not\in A$. I think the ...
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Could the birthday paradox be interpreted also about deaths?

Is the probability from the birthday paradox also true about deaths? If so, why? Or why not? I would think that it is also true about deaths, but it doesn't say so.
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Explain the Birthday Paradox

this is my first question at Maths Stackexchange, so, first of all, hello world! My question is, I recently read about the Birthday Paradox which states that in a group of 23 people, there's a ...
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World cup birthday paradox for two pairs

The BBC report on the world cup squads The birthday paradox at the World Cup shows the 50:50 prediction for there being 16 teams out of the 32 that meet the pair birthday criteria. However my ...
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An exercise about Borel paradox

If $X$ and $Y$ are independent standard normals, what is the conditional distribution of $Y$ given that $Z=1$, where $Z=I(X=Y)$?
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Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function y=1/x, x>0 (say x and y both measure meters) about the x axes by 2pi you get a shape which has infinite surface area and finite ...
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Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
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Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
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Buffon needle in higher dimensions

Imagine a stick of length 1, and also a floor with evenly spaced lines, such that the distance between neighboring lines is also 1. If one throws the stick on the floor, there will be certain ...
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St. Petersburg Paradox. Expected value seems wrong.

Related: St. Petersburg Paradox I was reading today the Wikipedia page on the St. Petersburg Paradox. The posted expected value is: $ 1/2 * 1 + 1/4*2 + 1/8*4 ... $ This seems very wrong to me. ...
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Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
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Bus arrival poisson paradox

I have a question about the waiting time paradox for poisson processes(in this case in terms of bus arrivals). Suppose I know that buses arrive with poisson distribution(lambda). I arrive at fixed ...
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What exactly is the paradox in Zeno's paradox?

I have known about Zeno's paradox for some time now, but I have never really understood what exactly the paradox is. People always seem to have different explainations. From wikipedia: In the ...
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Very strange “fact” regarding movement

Perplexing (for me at least) statement from the site: http://www.quora.com/Mathematics/What-are-some-of-the-most-counterintuitive-mathematical-results "Fact: You can have a car stand still for ...
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Solving a version of the liar paradox

Given two people $Alice ,Bob$ are either lying or telling the truth Now suppose $Alice$ says "At least one of us is lying." We have two cases: $Alice$ is telling the truth $\implies$ $Bob$ is ...
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Elementary proof that there is no paradoxical decomposition using triangular pieces

I am teaching a geometry course and I am trying to understand two definitions in the textbook ("Geometry with Geometry Explorer" by Michael Hvidsten.) Definition: The area of a rectangle is its base ...
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Infinity and Hilbert's hotel paradox

I did some infinite series calculations while studying Fourier analysis and the concept of infinity really bugs me. I haven't read or heard not one sensible explanation yet (for me), what infinity ...
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Paradox with function representation

Let assume the function $\eta(E)$ has the following representation: $$\eta(E) = \sqrt{\frac{a}{E}}$$ where $a$ is the known positive constant, and $E \in [-\infty, +\infty]$. I know that $\sqrt{a} = ...
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Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
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Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
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Why $\bigcap \emptyset $ isn't defined? [duplicate]

Let us define: $\bigcap \emptyset = \{ x|\forall A(A \in \emptyset \Rightarrow x \in A)\} $ I understood that this cannot be defined. Somehow it enables Russell's paradox to exist. Why is that?
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Why is this inclusion of dual of Banach spaces wrong?

Ive been struggling the last days on this paradox, please I need help! Let $$E\subset F$$ be two Banach spaces equipped with the same norm. Some people told me that $$F^* \subset E^*$$ with $E^*$ ...