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. ...
1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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: ...
1 vote
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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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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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### 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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### 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 "...