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

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Hard time on understanding real analysis. [closed]

I am learning real analysis now but I really dislike the notion of limit, infinity... They seems to generate lots of paradoxes and unreasonable results. For example, when I am reading the uniform ...
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How to resolve the apparent paradox resulting from two different proofs?

Definition of Open Ball Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set, $B_d(x, r) := \{y \in X : d(x, y) < r\}$ will be said to be the open ball of radius $r$ ...
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Russell's paradox question

Tao's analysis book uses following example for Russell's paradox: $$P(x) \Longrightarrow `` x\text{ is a set, and }x \notin x"\\ \Omega := \{x : P(x)\text{ is true} \} = \{x : x\text{ is a set and }x ...
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What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
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Paradoxes that remain paradoxical even when you understand the underlying theory

It strikes me that the Banach-Tarski paradox (rearranging ball partitions) is not dispelled even when you understand the underlying mathematics. Perhaps Parrondo's paradox in Game Theory (sawtooth ...
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A paradox with the additivity axiom of probability theory

Suppose F is a finite set of propositions such that, for every proposition A in F and every proposition B in F such that A is distinct from B, P(A) = P(B) and A is inconsistent with B. By using the ...
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Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
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Expanding a randomized Two Envelope Problem

There is a previous problem called the Envelope Paradox with a detailed explanation and solution given here. In short, the problem involves two envelopes with random (on some probability distribution ...
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Calculate Inclusion-Exclusion in Birthday Paradox

Follow-on from this post I was trying Birthday Paradox for 5-day calendar, with 3 people The probability that NONE of them have matching birthday is $5/5 * 4/5 * 3/5 = 0.48$ The probability there ...
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Wikipedia's explanation of the lambda-calclulus form of Curry's paradox makes no sense

Wikipedia gives multiple explanations of Curry's paradox, one of which is expressed via lambda calculus. However, the explanation doesn't look like any lambda calculus I've ever seen, and there's an ...
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Zero equals one?

By using inverse trig identities, it can be shown that $sec(x)*cos(x)=1$. However, when $x=π/2$, the resulting function would be $$(1/0)*(0/1)=1,$$ and cross multiplying yields $$1*0=1.$$ I'm not sure ...
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Impossible events that actually happened [duplicate]

I'm looking for the name of a paradox I found on the internet some time ago. I'm not entirely sure about the formulation which involved a few mathematical expressions, but this is how I recall it in ...
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2answers
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Can you fix uncountably many holes in a circle using one rotation?

Let's say you have a circle with a single hole in it (it is exactly the same as a circle, except there is exactly one point a complete circle has that this figure does not). There is a way to fix this ...
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1answer
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How would you solve this paradox?

Three friends walk into a bar, sit down, and when waiter comes to them, they ask him how much is the beer. The waiter tells them that the beer is $10$ dollars. They order three beers and each of them ...
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Thomson's Lamp Question

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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A mathematical explanation of the Simpson's Paradox?

In general, Simpson's Paradox occurs because situation such as following occurs for some arbitrary events $A,B,$ and $C$: $P( A | B , C) < P(A| B^c,C)$ $P( A | B , C^c ) < P(A| B^c,C^c)$ But, ...
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Is the axiom schema of specification sufficient for solving Russell's paradox? If so, why?

This is basically a two part question, as the title indicates. I understand why unrestricted comprehension will produce paradoxes like the Russell set, but I'm less clear on the question how the axiom ...
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The Banach–Tarski Paradox [duplicate]

Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? What is the true nature of this paradox ? I don't really understand this ?
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Difference between fractions at group level have different sign than difference between fractions in aggregate

I have obtained a result (perhaps incorrectly; we shall find out) that appears paradoxical. Suppose I am interested in comparing fractions between 'groups' (not in the strict mathematical sense) ...
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How is it possible that if I have $2$ choices and $1$ of them is taken away, I have $0$ choices? [closed]

I'm a simple man living his life and enjoying mathematics. Today while thinking about choices I realized this paradox: If I have $2$ choices and $1$ of them is taken away, I have $0$ choices. How is ...
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Bertrand paradox solutions

Wikipedia states the problem as follows: "The Bertrand paradox goes as follows: Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the ...
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Given max force due to friction and change in mass, calculating angle of inclination. Paradox?

Given an object resting on an inclined plane, we have the following equations to determine the forces acting on said object: $$\sum f_x = mg\sin(\theta) - f_s = 0 $$ Which says that the sum of the ...
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1answer
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Strange problem with the imaginary unit [duplicate]

In class while messing with fractions and complex numbers I found this "paradox" $$ \sqrt{-1}=\sqrt{-1} $$ $$ \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}} $$ $$ ...
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Intuitionistic response to Russel's Paradox

I'm having a look at intuitionistic approach to mathematics, and stumbled upon a derivation of Russell's Paradox that doesn't use the LEM. (Why did mathematicians take Russell's paradox ...
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$2=1$ Paradoxes repository

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one ...
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Why is that sometimes eigenvalue decomposition produces a rotational matrix, other times it does not

Suppose we want to perform eigenvalue decomposition on this matrix: $$P = \begin{bmatrix} 1.5 && -0.5\\ -0.5 && 1.5 \end{bmatrix}$$ I obtain the corresponding eigenvectors as ...
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How to show that the event that a prisoner does no go free is not measurable

I was reading this webpage a few months ago about the following problem- A countable infinite number of prisoners are placed on the natural numbers, facing in the positive direction (ie, everyone ...
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ZFC,unprovability of existence of a countable model,Skolem construction and paradox

The well-known Skolem construction yields a countable model of ZFC,elemetarily equivalent to the universe of sets $V$.Why this construction is not a proof of existence of models of ZFC,as such proofs ...
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About the solution of $2^x-x^x=0$

Obviously $x=2$ is root of this equation $$2^x=x^x$$ if you plot it by some graphing software ,you will see x=0 is another root. and now,my question: is it true that $x=0$ is the solution of ...
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Probability of being the millionth customer (What Would You Do?)

I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation. Here is the problem formulation: You are waiting ...
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Generalizing beyond proper classes

I noted that issues such as Russell's Paradox involving the set of all sets that don't contain themselves can be resolved by stating that the object that is all set that don't contain themselves is a ...
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How is this derivative paradox solved? [closed]

We have $$x^2=x+x+x+\cdots+x$$ with $x$ terms in the sum and $x\in\mathbb{Z}$. Taking the derivative of the above equation: $$2x=1+1+1+\cdots+1$$ again with $x$ terms. This implies $$2x=x$$ How ...
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Proofs that relied on paradoxical sentences

Graham Priest's Logic of Paradox is a modification of classical logic where the principle of explosion does not hold, so that there are inconsistent theories which are not automatically trivial. ...
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Bartlett's paradox in Bayesian evidence

I've come across Bartlett's "paradox" (not to be confused with Lindley's paradox, also known as the Lindley-Bartlett paradox) in Bayesian statistics. The paradox originates from Bartlett's 1957 paper, ...
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Does the Russell Set exist?

I am currently reading "Naive set Theory" by Paul Halmos. In the second chapter, on the axiom of specification we show that the Universal Set does not exist. The proof is the following: Lets ...
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G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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What's the relationship between half of a value x, and the midpoint of that value?

Say you have 4 apples: A A A A Where is the middle of this, I mean, on paper it's obvious, but what is the numerical value of this? Halving this value (or rather any value) is easy, 4/2 = 2, but is ...
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Simple question with a paradox

"I have three boxes, each with two compartments. One has two gold bars One has two silver bars One has one gold bar and one silver bar" You choose a box at random, then ...
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Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
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P(TT|T) in two coin tosses not 1/3?

So the question is, if two consequtive coin tosses occured and we know, in the aftermath, that at least of them resulted in a tails, what is the probability that they were both tails? The common ...
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What is the worst-case running time of this algorithm?

What's the worst case running time? while flip_coin() == heads: continue; In other words, I flip a fair coin until I get tails. Of course we knows the expected ...
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Self-application in Church's untyped lambda calculus

In "Proposition as Types" by Philip Wadler mentions the weaknesses of untyped lambda calculus and "Russell's logic" concerning self-application. Whereas self-application in Russell’s logic leads ...
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2answers
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3 coins interpretation of the boy and girl paradox

Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? I think I have come up with a new interpretation for this problem. If children where ...
2
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1answer
106 views

Mirimanoff Paradox

The class $B$ is well-founded, if every descending chain of subclasses is finite. The class $B$ is not well-founded if there exists an infinite chain of classes $B_n $ such that $ \quad \dots \in ...
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Non WellFounded Set theories and Russell's Paradox

I am very puzzled by set theories which reject the axiom of regularity. If we reject the axiom of regularity, and allow a distinction to be drawn between wellfounded and non-wellfounded sets/classes, ...
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What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
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If the answer is “no” then “yes” and vice versa type of paradoxes. What are they?

I'm a complete layman, so my technical terms might be misleading. Sorry for the many small questions, it's just that I don't know how to formulate my question right. What is the deal with paradoxes ...
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Looking for explanation of Banach-Tarski Proof, preferably by visual methods “Video, Pictures, Diagrams…”

could someone please explain the four steps of Banach-Tarski? 1- Find a paradoxical decomposition of the free group in two generators. 2- Find a group of rotations in 3-d space isomorphic to the free ...
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The unexpected hanging paradox [duplicate]

monday ----- tuesday ----- wednesday I don't think you can apply this logic recursively. They're all different scenarios unrelated to each other, aren't they? Should I survive until the end of ...
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This sentence is false

"This sentence is false". Is this sentence true or false? My attempt: If this sentence were true, then what it says would be the truth , it implies that it is false which is a contradiction. if ...