Paradoxes are arguments, which are contradicting with logic or common sense, mostly using a false and implicit premises.
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Infinity = -1 paradox
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:
Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
12
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2answers
802 views
Card doubling paradox
Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value x) and after ...
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3answers
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difference between class, set , family and collection
In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
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4answers
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How can a structure have infinite length and infinite surface area, but have finite volume?
Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = ...
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11answers
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Paradox: increasing sequence that goes to $0$?
It is $10$ o'clock, and I have a box. Inside the box is a ball marked $1$.
At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$. At $10$:$45$, I will remove the ...
12
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1answer
433 views
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 ...
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8answers
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Why is “the set of all sets” a paradox?
I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand.
Why is ...
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3answers
622 views
Explain why calculating this series could cause paradox?
$$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + ...
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votes
3answers
347 views
Reference about the Banach-Tarski paradox
I think the title says it all. I am planning on giving a talk in a few weeks about the Banach-Tarski paradox and I have some pdfs found online which describe the paradox a little but I am looking for ...
2
votes
2answers
238 views
An unwanted property of the set $T=\{x,\{x\} \}$
Let $T$ be $T=\{x,\{x\},y \}$ and let $f:A\rightarrow T, \ f(a):=x$, where $A=\{a\}$. Define $B=\{x,y \}$. Now a weird thing happens: We should have that $x \in f(f^{-1}(B))$ by construciton of $f$, ...
2
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1answer
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Statistics: Bertrand's Box Paradox [duplicate]
Possible Duplicate:
Probability problem
This is the Bertrand's Box Paradox I read about on Wikipedia:
Assume there is three boxes:
a box containing two gold coins,
a boxwith ...
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6answers
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A variant of the Monty Hall problem
Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
9
votes
4answers
695 views
Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]
Possible Duplicate:
Is value of $\pi = 4$?
Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
2
votes
2answers
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Can one come to prove Cantor's theorem (existence of higher degree of infinities) FROM Russell's paradox?
I have been thinking about this: One can arrive at Russell's paradox from Cantor's argument, but can we go the other way round, i.e., can we prove Cantor's diagonal argument(often referred to as ...
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vote
3answers
348 views
Achilles and the tortoise paradox?
Let's say we decide to race on a track 1000km long. You are a 100 times faster than me, meaning if we both start at the beginning you obviously win. To make things more fair you give me a head start ...
8
votes
3answers
333 views
Is $SO_2$ an amenable group?
In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows:
[amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined ...
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votes
3answers
214 views
How can any statements be proven undecidable?
As I understand it, undecidability means that there exists no proofs or contradictions of a statement.
So if you've proved $X$ is undecidable then there are no contradictions to $X$, so $X$ always ...
2
votes
1answer
64 views
Conditional Probability and Division by Zero
Suppose we are picking points uniformly at random from the surface of the Earth. I want to compute the probability that I pick a point in the Western hemisphere, given that I pick a point on the ...
2
votes
2answers
146 views
Probabilistic paradox: Making a scratch in a dice changes the probability?
For dices that we cannot distinguish we have learned in class, that the correct sample space is $\Omega _1 = \{ \{a,b\}|a,b\in \{1,\ldots,6\} \}$, whereas for dices that we can distinguish we have ...
