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

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What would be the consequences if ZFC proved its own inconsistency, nonconstructively? [on hold]

Let's say a nonconstructive proof was given in ZFC that ZFC was inconsistent. Note that this doesn't automatically make ZFC inconsistent. Given a consistent theory $X$, $X + \neg \text{Con}(X)$ is ...
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Why does simplifying a function give it another limit [duplicate]

I'm asked: $$\lim_{x\to 1} \frac{x^3 - 1}{x^2 + 2x -3}$$ This does obviously not evaluate since the denominator equals $0$. The solution is to: $$\lim_{x\to 1} \frac{(x-1)(x^2+x+1)}{(x-1)(x+3)}$$ ...
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Illusionary singularities in functions

If given the function $$f(x) = \frac{-x^3 + 1}{x^2 - 1}$$ one can clearly see that it is not defined when $x = \pm1$. We can rewrite the equation by factoring out $(x-1)$ in both the numerator and ...
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Balls and Urn Paradox

So, I came across the following paradox: At $1$ minute before noon, put in balls $1 \sim 10$ and take out ball number $1$. At $1/2$ minute before noon, put in balls $11\sim20$ and take out ball ...
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Second-order logic and Russell's Paradox

I know that in first-order logic the following holds [see e.g. George Tourlakis, Lectures in Logic and Set Theory. Volume 2: Set Theory (2003), page 121] : $\vdash \lnot ∃y \ ∀x \ [A(x,y) \...
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On logic vs information theory

If the statements All crows are black and All non black things are non crows are equal, then why is the former so much easier to communicate by giving examples? What implications does this ...
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Berry's paradox with Godel encoding

I thought this is so obvious that people would have asked this question before, but for some reasons I can't find it. So here go: We are working in PA. With Godel encoding, we can encode a FOL ...
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1answer
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Basic math paradox [duplicate]

$1 = 1$ $ 1^2 = 1$ and $(-1)^2 = 1$ Therefore, $1^2 = (-1)^2$ Square root both sides $\sqrt{1^2} = \sqrt{(-1)^2}$ Therefore, $-1 =1$ This is an obvious paradox, but I don't know how to approach ...
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1answer
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St Petersburg paradox, can't replicate max lottery price on wikipedia

I've tried to verify the below statements on wikipedia because I think they're mixed up: https://en.wikipedia.org/wiki/St._Petersburg_paradox#Expected_utility_theory "For example, with log utility a ...
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3answers
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Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
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39 views

Why can an event be independent and dependent simultaneously?

This question stems from the boy-girl paradox, but it is not a repeat question so please read on. The boy-girl paradox question is worded in many different ways, but in essence it says: "If a family ...
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27 views

How does this self referencing (circular reference) equation terminate (i.e. not create a paradox?)

I'm working with a financial equation which seems like it should result in a paradox but I'm told doesn't, however I haven't been told why it doesn't. (I don't work in the field I'm a programmer ...
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How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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122 views

Is it possible to prove that $\{\mathbb{N}\} = \{\}$?

A standard definition of the naturals is considered once again: $$ \begin{array}{l} 0 = \{\} \\ 1 = \{0\} = \{\{\}\} \\ 2 = \{0,1\} = \{ \{\} , \{\{\}\} \} \\ 3 = \{0,1,2\} = \{ \{\} , \{\{\}\} , \{ \{...
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Are physical/material/dimensional/temporal explanations of Banach-Tarski necessarily irrelevant?

I recently re-reviewed some of my undergrad analysis text and read the sketch of the proof of Banach-Tarski presented on Wikipedia, starting with a proof that the free group with two generating ...
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1answer
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Can the Monty Hall paradox be explained by Berkson's paradox?

I just learned about Berkson's paradox, which says that if $A$ and $B$ are independent, then $P(A\mid B,A\cup B) < P(A\mid A\cup B)$ (knowing that $A$ or $B$ occur creates a negative dependence on $...
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A solution for Russell's paradox

I think that the matter of the paradox is that it causes explosion and trivializes the formal system, and if this explosion can be prevented, there is no problem. I'm an unprofessional person but ...
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Terry Tao, Russells Paradox, definition of a set

In his book "Analysis 1", Terry Tao writes (check out page 39): To summarize so far, among all the objects studied in mathe- matics, some of the objects happen to be sets; and if x is an object ...
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3answers
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“This statement is false.” [duplicate]

In propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition. ...
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3answers
136 views

Gasoline Paradox: Car Can't Run out of Gas?

I have heard of a statement like this: A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time. Is this possible (mathematics wise)?
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Just got confused with what my friend asked (paradox and fake proofs). [duplicate]

Take $x^2=x+x+x+\cdots$ ($x$ times). Now differentiating both sides wrt $x$, we get: $$2x=x.$$ This means $x=0$ or $2=1$. How? Where did I go wrong?
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1answer
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Exchange paradox: PDF over the positive reals such that f(2x) = f(x/2)?

I was reading about the exchange paradox (two envelopes, one with $m$ dollars and another with $2m$ dollars, you get one at random, would you switch before looking?). The paradox is that if you get $X$...
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The Liars's paradox: Is it a paradox? Is it a truly a simplified version of the Bertrand's Paradox?

I must state that I consider myself inadept at using proper math language. Therefore, I must state my thoughts in word statements. Let us take a statement. We will call this statement 'This'. I now ...
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1answer
142 views

The three-coin-flip riddle

Is the following true (It seems obvious to me that it's not... but... a PhD in physics, Derek Abbott, seems to think others explanation at end of post): Someone flips 3 coins on the table, they are ...
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Why does the Axiom of Selection solve Russell's Paradox in Set Theory?

I am a beginner in mathematics and I was reading a text on Set Theory that talked about how Zermelo's Axiom of Selection "solves" Russel's Paradox. I understand that the the axiom does not allow ...
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1answer
51 views

Is there a paradox that doesn't require self-reference and negation? [duplicate]

Is there a paradox that doesn't require self-reference and negation? The set of all sets that do not contain themselves uses both, for example. Same with ...
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1answer
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De Moivre's paradox

According to Bernoulli's law of large numbers, in the coin tossing game the probability that the number of heads equal to the number of tails tends to $1$ as number of tosses increases. In other ...
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1answer
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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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2answers
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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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1answer
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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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3answers
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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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1answer
57 views

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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1answer
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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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1answer
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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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1answer
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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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Strange problem with the imaginary unit [duplicate]

In class while messing with fractions and complex numbers I found this "paradox" $$ \sqrt{-1}=\sqrt{-1} $$ $$ \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}} $$ $$ \frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\...