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 Banach-Tarski paradox realistic? Why is Volume not an invariant?

Banach-Tarski says that given a glass ball, we can break it into two glass balls of equal volume to the original (plus other generalizations). The explanations I have found for this paradoxical notion is that volume is not an invariant when we do…
MCT
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Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$

Possible Duplicate: Is value of $\pi = 4$? Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
glebovg
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The class of all classes not containing themselves

In ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox manifest itself when discussing the class of all…
Mozibur Ullah
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Is there an absolute notion of the infinite?

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be rich enough to talk about a bijection that a…
Elchanan Solomon
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Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. A lion would rather stay a lion than be eaten by…
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Spheres cause contradictions in dimensions $10$ and more?

According to this Numberphile video, if you tightly pack hyper-spheres into a hyper-box and then find the radius of the largest hyper-sphere that could possibly fit in the remaining space, the resulting hyper-sphere would somehow exceed the confines…
Lorry Laurence mcLarry
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Logic nonsense/paradox

I'm not sure if this is a paradox or a nonsense or neither of both. Anyway this is the "problem" if we can call it like that: A: B is True B: A is False How can you solve it?
gyosko
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If you have two envelopes, and ...

Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch to the other one? Intuitively, you don't know…
terrace
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$2=1$ Paradoxes repository

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one week, I explain the solution in the blackboard…
Kikolo
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Why is this inclusion of dual of Banach spaces wrong?

Ive been struggling the last days on this paradox, please I need help! Let $$E\subset F$$ be two Banach spaces equipped with the same norm. Some people told me that $$F^* \subset E^*$$ with $E^*$ denoting the dual space of $E$ (space of continuous…
Vintarel
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Problems with using logic to study logic

Does using logic as a tool to study logic itself lead to problems/paradoxes? Similarly to how self-referencing sentences sometimes make no sense, e.g. This statement is false. When we try to study logic we use logic to arrive at any conclusions…
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Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations,…
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What exactly is the paradox in Zeno's paradox?

I have known about Zeno's paradox for some time now, but I have never really understood what exactly the paradox is. People always seem to have different explainations. From wikipedia: In the paradox of Achilles and the Tortoise, Achilles is in a…
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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 independently being a $1$€ coin or a $2$€ coin…
joriki
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Paradoxical models of $\sf ZF$ without choice

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold in the same model). Paradoxical means that the…
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