Questions tagged [paradoxes]

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

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Can Hilbert's Hotel be explained by a difference between ordinal numbers and cardinal numbers

In taking a philosophy of maths course I have been very curious about the notion of infinity, and whether or not it is paradoxical. One thing I have frequently thought is that "infinity" as ...
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Perimeter of Inscribed Square - Paradox?

Imagine a simple X / Y coordinate graph. A circle surrounds the point of origin. Let's say the radius = 3. We want to know how many points exist on the circumference of the circle through which a ...
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Is there an equivalent to Godel's theorem that looks like "This statement is provable."? [duplicate]

I've been thinking about Godel's thoerem and the liar's paradox. The liar's paradox, when flipped around, stops being a paradox and becomes valid logically whether the statement is true or not. "...
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Bertrand's Paradox: Is this proof valid?

Bertrand's paradox is a well known problem. In this I prove that "method 3" gives the correct answer. This is assuming that a good distribution should also look good. A distribution that ...
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Paradox: Derivative w.r.t. basis element

Let $z_1\in \mathbb{R}\setminus\{0\}$ and $z_2\in i\mathbb{R}\setminus\{0\}$. Then $\{z_1,z_2\}$ form a basis for $\mathbb{C}$. This means that any $z\in\mathbb{C}$ can be written as a linear ...
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Does Bertrand's Paradox depend on the Axiom of Choice?

This is the set up to Bertrand's Paradox: Randomly choose two points on a circle. Construct a line segment (circle chord) between them. Construct an inscribed equilateral triangle within the circle. ...
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Paradox of proof by induction

I'm having trouble understanding the following situation: Given an identity P(n) that is wrong (but I don't know whether it's right or wrong), I am trying to check whether this identity can be proven ...
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Axiom of replacement vs. Axiom of separation and Galileo paradox

Galileo's paradox says that on one side there are fewer square numbers (second powers) among natural numbers than all numbers because only some numbers are squares. On the other side, there are as ...
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2 votes
2 answers
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Two people Monty Hall paradox

We're on a game show, and we have to select between three doors, one of which has a Lamborghini behind it while the others have goats. After we've decided, the host opens one of the other two doors, ...
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A variation on the three prisoners problem

Three prisoners hear that one of them will be executed (the exact person who will be executed is determined upfront, and cannot be changed), while the other two will be released. Prisoner A asks the ...
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What is the real potential energy of an alternating q and -q infinity system?

We can create a model of an infinite one-dimensional ionic crystal. Considering a system of $N\gg1$ alternating point charges $Q$ and $-Q$, that are distributed as the distance between two neighboring ...
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The Engagement paradox

Firstly, I should say that I came up with this paradox after reading of the Grimm Reapers paradox, but I’m not quite sure how this should be resolved. Nevertheless here is the problem: Suppose a lady ...
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Dirac delta distribution in $1D$

We know that $$\frac{d^2}{dx^2}\left(\frac{1}{|x|}\right) =-4\pi\delta(x)$$ where $\delta(x)$ is Dirac delta distribution. $$\Rightarrow \lim_{x\to 0}\frac{d^2}{dx^2}\left(\frac{1}{|x|}\right) =-\...
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Question regarding Bertrand's paradox

The classic example of Bertrand paradox deals with the case where we count the uncountable set of chords in a circle in different ways and ends up getting different probability each time. The ...
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In the Staircase paradox, where does the limit function differ from the hypothenuse?

I am aware of the answers to the Staircase paradox here Now for the example of the unit square and the approximation of the hypothenuse by the staircase function, surely the limit function differs ...
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would a set of all countable sets have any paradoxical properties?

I recently talked with a friend about set theory and he mentioned "set of all countable sets". I think that such set does not exist (just like "set of all sets" does not exist) and ...
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Does the casino have an edge on a commission free 50/50 luck game?

I have an interesting discussion with someone on Youtube: Assume a commission free roulette (without the number 0) so you have a true 50/50 chance on black or red. Also assume that all players will ...
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Proving that the Yablo's paradox is $\omega$-inconsistent

Jeffery Ketland proved in his Yablo's paradox and $\omega$-inconsistency that the set of Yablo sentences, which leads to Yablo's paradox, is $\omega$-inconsistant, but I do not understand his proof. ...
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Which would be the "direct" formula for the birthday paradox , withouth subtracting to 1?

I stumbled upon the birthday paradox, and I get it. However, all the explanations I see solve the probability by subtracting to 1 the probability of all people having different birthdays. What I am ...
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limit $\underbrace{\lim_{n\to\infty}\frac1n+\lim_{n\to\infty}\frac1n+\cdots+\lim_{n\to\infty}\frac1n}_{n\text{ times}}$

Basically this: 1 = 0 with limits \begin{align} 1&=\lim_{n\to\infty} 1\\ &=\lim_{n\to\infty} \frac n n\\ &=\lim_{n\to\infty} \left(\underbrace{\frac1n + \frac1n + \cdots +\frac1n}_{n\text{ ...
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Paradox in calculating Partial Derivative wrt xy.

I am trying to calculate $\frac{\partial (xy^2)}{\partial (xy)}$. To calculate this I am trying to substitute expressions in $\alpha, \beta$ instead of $x, y$ in two independent methods. Method-1: ...
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1 answer
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Is it possible to express and analyse Bertrand's paradox with terms and tools from set/space theory?

I recently watched Numberphile's excellent videos about Bertrand's paradox. https://www.youtube.com/watch?v=mZBwsm6B280&t=0s https://www.youtube.com/watch?v=pJyKM-7IgAU&t=0s See also https://...
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1 vote
2 answers
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Why this works for proving ZF axioms system is free of Russell's paradox?

In a book on mathematical logic, the author explains why ZF axioms avoid universal set like this: We may also now show that no universal set exists. Suppose $u$ is a set containing all sets. By ...
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Help me find the fallacy in these specific arguments re: the Boy-Girl Paradox [duplicate]

I'm wrestling with the classic problem called the Boy-Girl Paradox in the following simple form: Suppose you know that a given family has at least one girl. What is the probability that the other ...
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Banach Tarski, Axiom of Choice and non-measurable sets

I had a question about the link between Banach Tarski paradox, Axiom of Choice and non-measurable sets. I know that the construction of sphere decomposition in the Banach Tarski context is made by the ...
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linear regression simpson's paradox

I need advice on this problem. It is related to Simpson's Paradox. Consider three binary variables $X, Y, Z$ and all taking values in $\{0, 1\}.$ Consider the following inequalities: $P(X = 1) > P(...
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¬(A∨¬A)⊢¬B in paracomplete systems

I am considering a paracomplete logic where the principle ¬(A∨¬A)⊢¬B holds. What does it take for the principle to be explosive, such that we can infer any ¬B? If we have some statement that is ...
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Looking for the flaw in: Three-fourths of composite numbers are even.

Note: there is an older post How can all 3 of these be true? that is similar to this one, but I'm trying to find what's wrong. If we pick two positive integers at random, then there is a 50-50 chance ...
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Russel's antiparadox and Godel's incompleteness [duplicate]

In the famous paradox Russell tries to construct the set of all sets not containing itself. Then the question whether this whole set contains itself leads to a contradiction - if it does not contain ...
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To check proposition of Self-referential sentence

I want to prove two self-referential sentences $S, S_1$ are proposition or not. My approaches are given below. Suppose, There is a statement :- $S :$ This statement '$S$' is false Now, There are 2 ...
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4 votes
3 answers
221 views

A paradox in probability

I am trying to calculate the probability of picking perfect squares out of first $n$ positive integers. There are $\operatorname{floor}(\sqrt n)$ number of perfect squares less than $n$, if we assume ...
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1 answer
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Do the Zermelo-Fraenkel axioms prevent the existence of fractals?

It is well known that Zermelo-Fraenkel axioms solve the Russell's paradox - i.e. A set can not contain itself (singular set) without leading to logical contradictions- just saying that this kind of ...
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3 votes
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Solving Birthday Paradox with Triangular Number formula

I have an incorrect solution to the classic birthday paradox question, and while plugging in some values shows the formula is wrong, I can't see any intuitive reasoning as to why it is incorrect, or ...
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Why does a type of all types including itself create a paradox in Martin-Löf type theory?

In Per Martin-Löf (1998) "An Intuitionistic Theory of Types" in G Sambin and JM Smith (eds) Twenty-five years of constructive type theory Clarendon Press (original work written 1972 but ...
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-4 votes
3 answers
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Hilbert Hotel Paradox [closed]

In a video created by famous science blogger Veritasiuam(The Video), it explained the paradox, what to do if a finite number of guests come, what to do if even an infinite long bus of guests come, ...
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2 votes
1 answer
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Liar's Paradox & Tarksi — Does Tarski's theorem truly resolve Liar's Paradox (in Peano Arithmetic [and possibly outside of it])?

I was looking in the literature, and in my textbook, it was concluding Tarski's theorem after showing: $$\mathbf{PA} \vdash \varphi \; \longleftrightarrow \; \lnot \text{truth}(\ulcorner{\varphi}\...
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Probability Paradox - Converting Number Bases

What is wrong with this argument? Let $X$ be a random positive integer in base 2 The probability that the number of digits of X is even is $1/2$ (Since number of digits can be any positive integer) ...
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How to describe this (paradoxical) motion by the hidden geometric series?

High-schooler here trying to work through an interesting problem: A boy races an ant, giving the latter a head start. The motion of the two bodies can be decomposed according to the following steps: ...
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Wallet problem solution?

There is a problem proposed by Maurice Kraitchik that is supposedly unsolved(according to this VSauce2 video): Take two people and compare the amount of money that is in their wallets. The person who ...
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Analyzing a Gambling Race Paradox

Suppose a number of players are given $100$ points each, and repeatedly engage in a gamble having positive expected value, with the goals of being the first player to reach $100000$ points. Solving ...
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About the cardinality of reals, is there a paradox here?

Picking a real number from the unit interval with uniform distribution is having Khinchin's constant with probability $1$ as the geometric mean of its continued fraction's coefficients, not ...
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Is the mean of four integers equal to the mean of means?

It appears to me that the mean of four numbers is equivalent to the mean of the means of two pairs of those numbers: $\text{mean}(a,b,c,d) = \frac{a+b+c+d}{4}$ $\text{mean}(\text{mean}(a,b),\text{mean}...
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1 vote
1 answer
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System of logic where "this statement is false" can be encoded?

Is there a system of logic in which "this statement is false" can be encoded? I'm familiar with the incompleteness theorems, so I know "this statement cannot be proven" can be ...
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1 vote
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Mathematical theories where paradoxes are formed from the assumptions of the system about itself

There are in fact certain paradoxes involved in the notion of a system that predicts its own behaviour. These are reminiscent of Russell's Paradox in set theory and of the paradoxes that arise when ...
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5 votes
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Paradoxical homework

Let $z_1,z_2,z_3\in\mathbb{C}$ such that $|z_1|=|z_2|=|z_3|=1$. If $z_1+z_2+z_3\ne0$ and $z_1^2+z_2^2+z_3^2=0$ then prove $|z_1+z_2+z_3|=2$. What I did $z_1^2+z_2^2+z_3^2=0~~|\cdot(z_1+z_2+z_3)$ $\...
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Question on Russel's paradox [duplicate]

I am reading the first chapter of Aliprantis and Border's Hitchhiker guide and there is a passage on Russel's paradox that has me confused. It reads: Russell's Paradox is a clever argument devised by ...
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3 votes
1 answer
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Does permitting comprehension for all (and only) contingent formulas result in paradoxes?

Does permitting comprehension over all well-formed formulas that are neither contradictions nor tautologies result in paradoxes? I have a hunch that a simple extensional set theory with the "...
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2 votes
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Can we insist on having a universal category (with ZFC and a hierarchy of classes)?

I'm curious whether there is any harm on insisting on the existence of a universal category. I'm also curious if we can make one in an extremely naive way by stapling a hierarchy of classes to $\...
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3 votes
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Paradox of $-i$ seemingly equal to $1$ via the Wallis product for $\pi$ and the Euler sine product

Assuming $x$ is a real variable throughout,$$\frac{\sinh(ix)}{i}=\sin(x)$$ $$\frac{\sinh(\pi ix)}{\pi ix} = \frac{\sin(\pi x)}{\pi x}$$ $$\frac{e^{\pi ix}-e^{-\pi ix}}{2\pi ix}=\prod_{n=1}^{\infty}\...
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Using R, solve the birthday paradox

The probability that two students in a class have the same birthday is at least 75%. What is the minimum size of the class? I tried, ...
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