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

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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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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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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 ...