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

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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 $E^*$ ...
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How to ensure that you haven't run into a paradox proving a theorem e.g. by proof by contradiction?

While preparing some lecture notes for next semester and going back to basics (set theory and proof strategies) I came along the following simple question which is about proving theorems in general ...
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How to ensure that you haven't run into a paradox proving a theorem e.g. by proof by contradiction? [duplicate]

While preparing some lecture notes for next semester and going back to basics (set theory and proof strategies) I came along the following simple question which is about proving theorems in general ...
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Paradoxes in Logic

What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about ...
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How can I understand is the picture $2D$ or $3D$

I can not understand is this picture 2D or 3D .what is the rule or condition to be a 2D or 3D picture.How can I understand that?please help me?
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On a scale of 1 to 10, how likely is it that this question is using binary? [closed]

I just read this interesting xkcd strip: At first I thought it was funny, but as I got to ruminate a little over it, I was surprised to be unable to find an answer. As Karolis Juodelė pointed out, ...
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Fingerprint match probability

I am trying to use the formula for the birthday paradox as a reference to figure out an equation that represents the probability of a fingerprint match. Here's the equation for probability of a ...
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Leibniz' Law and that good old riddle

There exists a Theory of Identity in mathematical logic. I've encountered it for the first time in Principia Mathematica by Alfred North Whitehead and Bertrand Russell (1910). Quote: "This definition ...
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Understanding: Axiom of Specification and Russell's Paradox: there is no universe?

Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those ...
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Notation in “proof sketch” of the Banach Tarski paradox on wikipedia

I'm trying to understand the proof sketch here. In step 3 of the proof sketch we have $A_{1} = S(a)M \cup M \cup B$. My understanding is that $S(a)$ and $M$ are both sets. I have failed to understand ...
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The set of all things. A thing itself?

If the universe is the set of all things. Does it contain itself? In other words is it a thing itself? I know its a stupid question, but it really grinds my gears. Thanks! Edit 8.12 Okey, someone ...
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Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized ...
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Why isn't $\int_0^1{1/x^2}~dx= 1$ + the integral from 1 to infinity.

This probably as nearly a stupid question as the one that isn't asked, but I've always thought the area under the curve is equal to the integral. So Given the function $\dfrac{1}{x^2}$ which is ...
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Mistake wikipedia article on St petersburg paradox?

I suspect that there is a mistake in the wikipedia article on the St petersburg paradox, and I would like to see if I am right before modifying the article. In the section "Solving the paradox", the ...
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Paradoxes in number theory

Does it exist any paradoxes within the field of number theory? Any examples? My thought is that since it is possible to find paradoxes in set theory, which in some sense more fundamental than number ...
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Variant on Russell's paradox: show $B = \varnothing$

Let $X$ be a set and $R$ a relationship on $X$. Define $N = \{x \in X\mid(x, x) \notin R\}$. Let $$B =\{b \in X\mid(\forall n \in N)(b\,R\,n) \land (\forall n \notin N)(\neg b\,R\,n)\}\;.$$ ...
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Simple tax puzzle

I recently saw some post on facebook whining about taxes. Simplifying it (and changing numbers, facts, etc.), this was saying: For each dollar an employer wants to pay you: 20% go in taxes that ...
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Is the answer to this question “no”? [closed]

The question in the title should be easy. So, yes or no?
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Why cannot a set be its own element?

When I study Topology, I met with a problem. On my book, it says 'we cannot admit that there exists a set whose members are all the topological spaces. That will lead to a logical contradiction, that ...
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Russell's Paradox

Many of you know such paradox... " $\exists y \forall x (x \in y \Longleftrightarrow \Phi(x)$" for any function $\Phi(x)$ substitute $x \notin x$ for $\Phi(x)$ Then by existential instantiation ...
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St. Petersburg Paradox

A fair coin will be tossed until a heads results. You will then be paid $2^{n-1}$ dollars where $n$ equals the number of flips. Now why is the expected pay out infinite? $$ \sum_{n \geq 1} ...
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Sleeping Beauty paradox - fair prior

I've been reading the Sleeping Beauty problem wiki. The contradicting answers, to me, appear to stem from frequentist and Bayesian interpretations: The "thirder" solution refers to a limit in an ...
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Paradoxes without self-reference?

Most paradoxes involves self-reference, the only exception known to me is Yablo's paradox, however it is still debated if it is really without self-reference. So, I was wondering, are there other ...
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Proof by contradiction and division by $0$

Proof by contradiction is based on the fact that if, as a consequence of a statement's truth, we reach a contradiction, then that statement must be false, since contradictions do not exist in ...
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Achilles without tortoise (II)

I am Achilles II and, on a straight line, I start running really fast: The first $1$ meter I cover in $1$ second. The next $\frac{1}{2}$ meters in $\frac{1}{5}$ seconds. The next $\frac{1}{2^2}$ ...
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Achilles without tortoise

I am Achilles, and, with no tortoise in front of me, I start running on a straight line. The $1$st metre, I cover in $1/2$ second. The $2$nd metre in $1/4$ second. The $3$rd metre in $1/8$ second. ...
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Thomson's Lamp and the possibility of supertasks

The Thomson's Lamp paradox: A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a ...
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Problems with using logic to study logic

Does using logic as a tool to study logic itself lead to problems/paradoxes? Similarly to how self-referencing sentences sometimes make no sense, e.g. This statement is false. When we try to study ...
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Intuition about whether to switch in box problem

I ran across an apparent paradox which I then located in the paper The Box Problem: To Switch or Not to Switch as such: Imagine that you are shown two identical boxes. You know that one of them ...
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For x < 5 what is the greatest value of x

It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?
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Seems that I just proved $2=4$.

Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$. Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$. Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and ...
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Proof that there is no Banach-Tarski paradox in $\Bbb R^2$ using finitely additive invariant set functions?

I am wondering if anyone is familiar with the above topic? I have found a proof that it is possible to define a finitely additive invariant set function in $\mathbb{R}^2$ on the circle in Lax's book ...
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How to formalize this paradox?

A friend gave me this problem (in the "blue box") An interesting fact about the number $2$. How many times the number $2$ appears in this text? It appears $2$ times. Well I see the ...
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What is the problem? [duplicate]

Where has the one block gone in lower image,after we rearrange the triangles? (Found it on G+ as a post.)
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Paradox of Infinity? [duplicate]

If a series such as '$a$' below adds to infinity: $a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$ Multiplying '$a$' by $2$ yields: $2a = 2 + 4 + 8 + 16 + \cdots\to \infty$ However when I subtract ...
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The set of all infinite binary sequences

Suppose that we have the set $S$ of all possible infinite binary sequences $s_i$ (a sequence is simply an ordered set): $$S=\{s_1,s_2,s_3,\ldots \}$$ where the sequences $s_i$ are like ...
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The condemned philosopher paradox. Can someone explain it to me?

the following paradox is a variation of the Barber Paradox, I don't quite understand why this is a paradox so I'd like to hear you tell why, please. There was a philosopher who had committed a ...
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8 x 8 = 13 x 5 ???

I'm sure some (if not most) have seen this by now and since I'm fairly new to real deep mathematical explorations I'm stumped as to why this would be true. For class I was asked to cut out an 8" x ...
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Why does a Penrose Stair seem to be correct?

Penrose Stairs seem to be a locally valid but globally inconsistent contraption. I have a couple of questions: Is it physically realizable? In other words, is it possible to build a 3-D structure of ...
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Random Point on Infinite Line Paradox

I've invented a paradox, or at least I think I have. Here is how it goes: On an infinite line, a point is placed at random. You start at point 0 on the line, and your job is to find the point, but ...
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Does a complement contain itself?

I think it is safe to assume many sets do not contain their complements. {1, 2} for example. Now, by the definition of complement, that would mean the complement would have to contain itself. This ...
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What is a paradox in mathematics?

I was reading some of these "mathematical paradoxes", and trying to understand why the list presents only counterintuitive mathematical results. Is there room in mathematics for logical paradoxes?
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Why is the hypergame not simply well founded?

According to Cameron, the hypergame paradox proceeds as follows: A game is considered as well founded if ANY play of the game ends in a finite number of moves. A hypergame is where the first player ...