Questions tagged [paradoxes]

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

A paradox is an argument that produces an inconsistency, typically within logic or common sense. Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. However some have revealed errors in logic itself and have caused the rules of logic to be rewritten. (e.g. Russell's paradox)

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Is the liars paradox actually a paradox at all? I don't think it is.

Note - I am not versed in logic notation or whatever the particular notation is for this kind of thing. Sorry in advance. Edit - This is to address the argument about taking the statement as a whole rather than in parts. If you take the statement as…
Rex.zip
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How to disprove Russell's paradox in ZF?

How to avoid a Russell-style paradox in ZF? The Zermelo-Fraenkel system, by not using a theory of types, does not prevent us from posing Russell's paradoxical scheme. Only when the argument is replaced by the set being defined, the resulting…
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Another take at a Debt 'paradox'

I use quotes around paradox because this is certainly not a mathematical paradox but only used in common usage. The situation goes as follows A tourist $\beta$ visits hotel $\nabla$ in a poverty-ridden town. He wants to first check the hotel rooms.…
sato
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Does Godel 1st theorem make sense?

It seems to me that there is 2 family of statements that I call "logically undecidable" (to distinguish with computationally undecidable which is define by turing machines), i.e. of statements for which it doesn't exist any proof that the statement…
François
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Significance of Banach Paradox

What makes Banach paradox important? Could we not just take the set of points defining a sphere and break them into two sets, one of rational and one of irrational points, so won't we again have two spheres with the same area?
Sadikov
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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 there are signs seemingly suggesting that this…
Lehs
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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 a number $m \in \mathbb{N}$ such that: $\lim_{z\to…
Eduardo Magalhães
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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 negative numbers exist, then $$\frac{-1}{1} =…
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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 circumference is $2 \pi \cdot 0.5 = \pi$, and the…
arsene stein
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Why does Turing-computing (being an inconsistent formalism) has undecidable problems?

I'd like to apply Church-Turing thesis to Kleene-Rosser paradox: Since untyped lambda-calculus is an inconsistent formalism AND Turing machines are equal in decisive power to lambda-calculus SO We have the same Kleene-Rosser paradox in Turing…
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Friday analysis of the unexpected hanging paradox

The judge told me: A1. You will be hanged on day X. (X is some day from Monday to Friday) B1. You can't deduce what X is. It's Friday morning and I'm still alive. My first deduction is (please tell me if it's not sound): A2: I will be hanged…
Asmani
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Find a perfect strategy algorithm for finding another person in a shop

'There is a row of 9 consecutive shops, John will visit a shop for 14 consecutive days. John moves venues daily to a shop directly left or directly right (end of row means forced move). John moves completely randomly. I want to find John in a shop…
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Paradox? What's wrong with this thinking?

Let $x = 2 \times 2 \times 2 \times 2 \times 2 \times \cdots$ Thus, $x = 2x \Rightarrow x = 0$ So, $2 \times 2 \times 2 \times 2 \times 2 \times \cdots = 0$ This clearly doesn't make any sense. What's wrong then? Thank you in advance.
Gooble
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What's wrong with this argument about fractional exponents?

I am sure that this has been discussed many times and the answer may be trivial. But I was confronted with the following issue and could not give a satisfactory answer: $$1=(-1^{6})^{\frac{1}{2}}=-1^{6\cdot\frac{1}{2}} = -1^{\frac{1}{2}\cdot…
Florian
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How do these logical contradictions differ in status?

Firstly contradiction 1: $s_1$="There exist distinct integers $a,b,c$ such that $a^n+b^n=c^n$ for some $n>2$" With this contradiction, we first suppose that it is true. Then we find that doing so introduces a contradiction, so we reject our…
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