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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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Paradox: Is $1 \in (0,1)$?

Consider the set of numbers such that $x \in (0,1)$. Their decimal expansion is $0.b_0b_1b_2\ldots$, with $b_n \in \{0,1,2,3,4,5,6,7,8,9\}$, and they are not all zero (or else $x = 0$). Then choose all $b_n = 9$, we have $0.999\ldots = 1$. But…
MT_
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Two envelopes problem, what is the problem with my solution?

This question is regarding the two envelope problem. http://en.wikipedia.org/wiki/Two_envelopes_problem It seems to me the very simple solution to this problem is to make the sum of all the money in the envelopes (SUM) = X. No matter how you…
Question3
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How does the barber's paradox apply to the halting problem but not similar solvable problems

I am trying to understand Turing's halting problem proof by applying the same paradox to a similar problem where, instead of determining if a given code will halt, you instead determine if it will return True or False (assuming it will always return…
Alex Breeze
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Russell's paradox explanation

The Russell's paradox deals with the set of all sets that do not contain themselves. So I want a example of a set that do not contain themselves. I got a examples of set of turtles.It will contain turtles, but I want to understand how should I…
math student
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Negative equal positive paradox

I was just bored and started practicing even more the exponentiations, and as I was working, I went ahead and did this: $(-1)^{2} = ((-1)^{\sqrt{2}})^{\sqrt{2}}$ So, I entered in my calculator of what is negative one to the power of the square root…
serkan
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The Rolling disk paradox

The rolling disk is the cornerstone to nonholonomic analysis. Its nonholonomic property constitute on equal translational and rotational velocity on the contact point. However, if the speeds are equal in modulus, but opposite in direction, the…
Bruno Peixoto
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Real 1 not equal to complex 1

We have the set-theoretic definition for pairs as: (x , y) = {{x}, {x, y}} Also we have the definition: complex 1 = (1, 0) So if real 1 = complex 1 we would have: 1 = (1, 0) = {{1}, {1, 0}} Which seems paradoxical. Am I missing a point?
hadizadeh.ali
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Are there other shapes like the Koch snowflake, with infinite perimeter but finite area?

Are there other known paradoxes in which a shape has infinite perimeter but finite area like the Koch snowflake paradox?
Binh Ho
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Banach-Tarski paradox in a countable set?

Let $M$ be a countable set. I can take a finite set $F$ of n-ary operations and construct a minimal set $M'$ for which: $M \subset M'$ For each $f \in F$ and $m_1,m_2…m_n \in M$ where $n$ is the arity of $f$: $f(m_1,m_2…m_n) \in M$ $M'$ is a set…
Hume2
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Why is $((-8)^2)^{1/6} > 0 \text{ and } -2 = (-8)^{1/3}$?

Why is $((-8)^2)^{1/6} > 0 \text{ and } -2 = (-8)^{1/3}$? Doesn't this contradict the exponentiation rule (power of power)?
mavavilj
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Trouble understanding Banach-Tarski paradox proof on wikipedia

When I read https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#A_sketch_of_the_proof I do not understand this part of the proof: $$F_2 = \{e\}\cup S(a) \cup S(a^{-1}) \cup S(b) \cup S(b^{-1}) =aS(a^{-1})\cup S(a)$$ where S(a) is the set of…
Emil
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Changing odds paradox

A paradox of changing odds I read about - doing my head in, but it must be easy to explain why. Three sets of two playing cards: AA KK AK With the cards turned face down, you task is to pick the AK pair. Odds are 3 to 1 you pick the correct pair.…
Linicks
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Gasoline Paradox: Car Can't Run out of Gas?

I have heard of a statement like this: A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time. Is this possible (mathematics wise)?
SaltMeter
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About the solution of $2^x-x^x=0$

Obviously $x=2$ is root of this equation $$2^x=x^x$$ if you plot it by some graphing software ,you will see x=0 is another root. and now,my question: is it true that $x=0$ is the solution of equation ?
Khosrotash
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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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