Questions tagged [paradoxes]

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

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Paradox of Fake Choice

This is a logic sort of question, so if it is better suited for a different stack exchange please let me know, not sure if this is the optimal location I thought of this interesting contradiction / ...
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Effect of squaring while finding roots of unity

Consider $$b=\frac{1}{b}\rightarrow b^2=1$$ Clearly $b=\pm1$   But if we square the above equation on both sides and then solve $$(b=\frac{1}{b})^2\rightarrow b^2=\frac{1}{b^2}\rightarrow b^4=1 $$ And ...
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A logic better adapted to quantum phenomena?

Our way of mathematical thinking is totally controlled by a simple two-valued logic $(\mathbf{false}, \mathbf{true})$. All deductions are due to this logic and we are unable to think otherwise. But ...
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Fruit Tree Paradox: Sum of disjoint probabilities not equalling $1$

This is a probability question that I came up with, and have noticed some things that do not seem to be right. Here's a description of the question: Suppose we have a fruit tree growing in our ...
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Paradox In the criteria for $a$ to be a removable singularity or a pole of $f$?

My complex analysis textbook stated the following proposition: Let $a$ be an isolated singularity of $f$ If $\lim_{z\to a}(z-a)f(z)=0$, then $a$ is a removable singularity If there exists ...
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Is Banach–Tarski paradox false without axiom of choice? [duplicate]

I know that you need axiom of choice to prove Banach–Tarski paradox. But what happens with paradox when we remove axiom of choice? Does theorem become false? Or is there just no proof of it without ...
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If $S=\{x:x\in x\}$, is $S\in S$ knowable? (Naive set theory)

Assuming the set theory we're working with allows self-containment, as well as arbitrary set building of the form $\{x:\Phi(x)\}$, if we define $S=\{x:x\in x\}$, is $S\in S$ knowable? As we see from ...
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Probability paradox: Is heads followed by heads or heads followed by tails more likely?

You repeatedly flip a coin until you either get a heads followed by a tails, or two heads in a row. Which is more likely to happen first? Solution 1: Both are equally likely, because each flip, there'...
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Removing the law of noncontradiction and paradoxes

Plenty of work has been done in intuitionistic logic, where we remove from classical logic the law of excluded middle: $\vdash P \lor \lnot P$. However, what if we instead removed the law of ...
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St Petersburg Paradox for a Risk Prone Subject

How can I show that maximising expected utility in the St Petersburg Paradox gives the same irrational behaviour that maximising expected monetary value does for a risk prone subject? I understand the ...
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What is the probability of three rabbits having the same sex? (Paradox)

My three children were given three rabbits from friends this easter. The rabbits were an "accident", and so that this doesn't happen again, we will have the male rabbits castrated. My daughter is very ...
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Proof that $\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error?

So, I proved that: $$\int f(\ln x)\ dx = x \sum_{n=0}^{\infty}(-1)^{n} f^{(n)}(\ln x) \ \ \ +\ \ C$$ where $f^{(n)}$ is the nth derivative of $f$. if we let $f(x) = e^{x}$ then $f^{(n)}(x) = e^x$ as ...
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On the boy-girl paradox and the TED-ed frog riddle

I'm aware this question has been asked many times - however, I feel I have a different take on the topic. First, let's take a classic scenario. Here, a doctor has 2 babies. He checks both of their ...
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Worm on the rubber band paradox problem

The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the "worm on the rubber band".Suppose that a worm crawls along an infinitely-elastic one-...
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Hilbert's Hotel Paradox: Guests moving to new room every day?

Suppose there are infinitely many coaches with infinitely many members in each coach. They stay at the hotel for infinitely many days. I know that guests can be accommodated using various methods like ...
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One paradox in limit situation [duplicate]

I am an undergraduate math major student. I found one paradox in the following situation. Consider a box, and say you put balls $b_1, b_2,..., b_{10}$ in the box at time $t=-1/2$, and simultaneously, ...
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How to resolve this paradox involving Cantor's diagonal argument?

Let's define the elements of a countable infinite set of numbers as follows: $$s_1 = 0.0000000...$$ $$s_2 = 0.1000000...$$ $$s_3 = 0.1100000...$$ $$s_4 = 0.1110000...$$ $$s_5 = 0.1111000...$$ $$...$$ ...
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Equivalence of “The Unexpected Hanging” and “The Two Generals”

The "Unexpected Hanging Paradox" is a situation in which a prediction is successfully made, even though a logical proof indicates that it couldn't possibly happen. A judge tells a condemned ...
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Simpson's Paradox: Is the data discriminatory?

I have data depicting college admission statistics in a combined two-way table and a three-way table (with colleges C1&C2) with the following probabilities: Overall acceptance: $72.37\%$ Overall ...
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Real 1 not equal to complex 1 [duplicate]

We have the set-theoretic definition for pairs as: (x , y) = {{x}, {x, y}} Also we have the definition: complex 1 = (1, 0) So if real 1 = complex 1 we would have: 1 = (1, 0) = {{1}, {1, 0}} Which ...
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I think I found a mathematical paradox…

As we know the expansion of $\ln(1+x)$ is as follows: $\ln (1+x) = x - x^2/2 + x^3/3 - x^4/4+\cdots$ Let $S_1 = 1 + 1/3 + 1/5 + 1/7+\cdots$ Let $S_2 = 1/2 + 1/4 + 1/6 + 1/8+\cdots$ $S_1 - S_2 = ...
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Trading Interview - Wealth Maximization Question (St. Petersburg variation?)

Recently got asked this question and am not sure how to answer it. I'm not sure there is a singular concrete answer but any general thoughts would be appreciated. -You are allowed to play ONE round ...
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Set of possibilities for Simpson paradox

Italy is playing the U.S.A. in a football World Cup match. A successful pass is when a player on one team kicks the ball to a player on their team and it is not intercepted by the opposition. Is it ...
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The case of the missing ninth of a $2$€ coin

In answering Expected value of the number of bills, I came across a phenomenon the likes of which I don't think I've encountered before, and I'd like to know more about it. You draw coins, each coin ...
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How exactly Arnauld's Paradox is solved in modern mathematics?

Wells, David Graham, The Penguin dictionary of curious and interesting geometry, New York, NY: Penguin Books. xiv, 285 p. (1991). ZBL0856.00005. A friend of Pascal, Antoine Arnauld, argued that if ...
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Banach-Tarski Paradox and Infinite Sample Spaces

Recently, I was reading a paper on Stochastic Processes and I saw a footnote on page 9 of the PDF that stated: When $Ω$ is infinite, not all of its subsets can be considered events, due to very ...
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Apparent paradox in Peano arithmetic

In trying to understand Godel’s incompleteness theorem, I have come across an apparent paradox. There must be a mistake somewhere but I cannot find it and would be very grateful for anyone to point ...
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Braess's paradox

Let's consider the Braess's paradox. I have question how were computed the Nash Equilibrium points with values $80$ and $85$ before an after adding the route with the value $0$ from $C$ to $D$, ...
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What does $\ln{z} - \ln{z}$ equal, given $z \in \mathbb{C}$?

On first glance, the expression $$ \ln{z} - \ln{z} $$ where $z$ is complex and of form $a + bi$ should always evaluate to zero. Subtracting something by itself should be zero. However, when one takes ...
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Probability of a picking a given number from (0,1) = 0. Is this paradoxical? [duplicate]

If I pick an arbitrary interval like (0,1), the probability of picking any given number (like 0.5) = $\lim_{x \to \infty}\frac{1}{x}=0$ I at first thought that this was a paradox, and questioning ...
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What exactly is the paradox in St. Petersburg paradox?

Currently I am reading St. Petersberg paradox. However, I do not see the paradox. Can someone explain to me why is this considered a paradox?
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Hilbert's Hotel: why does infinite nesting break everything?

According to Wikipedia: Although a room can be found for any finite number of nested infinities of people, the same is not always true for an infinite number of layers, even if a finite number of ...
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Area of a circle smaller than one?

If a circle has radius less than one, does it mean that the circumference of the circle is bigger than the area? How can that even be possible? For example, if a circle has radius $0.5$, then the ...
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Hilbert's hotel: why can't I repeat it infinitely many times?

I was wondering about the following: Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest ...
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Are there other shapes like the Koch snowflake, with infinite perimeter but finite area?

Are there other known paradoxes in which a shape has infinite perimeter but finite area like the Koch snowflake paradox?
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Is this Simpson’s Paradox?

In January, there were 2,700 new sign ups, and 3,500 who opt out. As at end January, there are 60,000 customers in our database. In February, there were 3,400 new sign ups and 4,300 who opt out. As ...
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Is this a solution to Russell's paradox?

Wikipedia says "Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, ...
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Defining sets in intuitionistic logic

I'm somewhat familiar with the school of intuitionistic logic. I know that an intuitionistic logician thinks of infinity as constructive as apposed to complete. Thus a intuitionistic logician cannot ...
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Paradox with probability becoming $1$ from $\frac{1}{2}$

There are two players $A$ and $B$ that play the following game. Each of them is given a random positive integer number by a fair judge, and the player with the biggest number wins. The judge tells ...
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Mathematical explanation for rim point moving backwards.

The paradox is given in the chap. 1 of the book titled : Mathematical Fallacies and Paradoxes, by Bryan Bunch; as given here. The book explanation has no mathematical formulation, say if states the ...
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Are there any disasters with Projective Determinacy?

It is well-known that the Axiom of Choice entails a number of counter-intuitive results, Banach-Tarski paradox is only one example. In this MSE question, Martin Sleziak asked about similar undesirable ...
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Banach-Tarski paradox in a countable set?

Let $M$ be a countable set. I can take a finite set $F$ of n-ary operations and construct a minimal set $M'$ for which: $M \subset M'$ For each $f \in F$ and $m_1,m_2…m_n \in M$ where $n$ is the ...
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Why does the comparison test fail here?

Someone recently stumped me here. Consider the series $\sum_3^{\infty} \dfrac{1}{x^{1.0001}}$ and the series $\sum_3^{\infty} \dfrac{1}{x\ln{x}}$. The first series in convergent, and the second ...
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Contraposition fallacy in raven paradox?

I do not understand how the contraposition "all non-black objects are not ravens" is logically identical to "all ravens are black." I see that IF all ravens are black, THEN all non-black objects are ...
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Prove the sequence $\{ 2^n \}$ is unbounded using Russell's paradox

I am really stuck on how to make use of Russel's Paradox. This is where I started. Proof by contradiction. Suppose $\{ 2^n \}$ converges then it is bounded. Let $M$ be an element of the reals and a ...
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Does $\int_{0}^{\infty} \frac{1}{x} dx$ Converge?

I'm sure countless people has tried to prove this but this is how I came across this, although I do remember seeing this in the past. Background https://www.youtube.com/watch?v=vQ0himyDR2E I was ...
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Brainteaser question/solution which reasonment is correct and why?

I am struggling to find some explanation to this: here is my problem: "A cube of ice melts without changing shape at uniform rate 4cm$^3$/min. Find the rate of change of the surface area of the cube ...
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The paradox of random occurrence the family problem

we have 100 families: 10 families have no children, 40 families have 1 child for each one, 30 families have 2 children for each one, 10 families have 3 children for each one and 10 families have 4 ...
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Mathematical proof of Olber's paradox

Is there a mathematical proof regarding Olber's Paradox? For me this feels a false paradox as I think that it IS possible to have an infinite (in space and in time) Universe with an infinite number ...
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Minimal example of Simpson's paradox

Let's say that a finite probability space $(\Omega,\mathscr P(\Omega),P)$ has Simpson's property if you can find events $A,B,C\in\mathscr P(\Omega)$ such that $P(C) \in (0,1)$. $A$ and $B$ are ...

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