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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Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all seem to me like pathological, esoteric, ad-hoc…
BenW
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Can anything interesting be said about this fake proof?

The Facebook account called BestTheorems has posted the following. Can anything of interest be said about it that a casual reader might miss? Note that \begin{align} \small 2 & = \frac 2{3-2} = \cfrac 2 {3-\cfrac2 {3-2}} = \cfrac 2 {3 - \cfrac 2 {3…
Michael Hardy
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Zeno's Achilles & Tortoise - Where exactly is the proof wrong?

(For those who don't know what this paradox is see Wikipedia or the Stanford Encyclopedia of Philosophy.) Let us define $a_i$ and $b_i$ recursively $$ a_0 = 0\\ b_0 = 1\\ a_i = a_{i-1} + (b_{i-1} - a_{i-1})\\ b_i = b_{i-1} + (b_{i-1} -…
aman_cc
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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 you have seen it, the dealer offers you an…
Casebash
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Seems that I just proved $2=4$.

Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$. Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$. Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and $\sqrt 2^{\sqrt 2^{\sqrt…
JSCB
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Why does simplifying a function give it another limit

I'm asked: $$\lim_{x\to 1} \frac{x^3 - 1}{x^2 + 2x -3}$$ This does obviously not evaluate since the denominator equals $0$. The solution is to: $$\lim_{x\to 1} \frac{(x-1)(x^2+x+1)}{(x-1)(x+3)}$$ $$\lim_{x\to 1} \frac{x^2 + x + 1}{x +…
user347213
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Demonstration that 0 = 1

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e &&(^{1 + 2 \pi i n})\ \text{(raising both sides to the $2\pi in+1$…
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Why cannot a set be its own element?

When I study Topology, I met with a problem. On my book, it says 'we cannot admit that there exists a set whose members are all the topological spaces. That will lead to a logical contradiction, that there will be a set who is a member of itself.'…
Phil Wang
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Why can't Russell's Paradox be solved with references to sets instead of containment?

My background is in computer science, and I'm keeping the Java implementation in my mind as a model. Included in the Java language is the notion of sets. Now I understand that this is different from the model Russell and Whitehead had in their minds…
hawkeye
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"A two-envelopes puzzle"

This is Problem 1.25 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You are handed two envelopes, and you know that each contains a positive integer dollar amount and that the two amounts are different. The values of…
Spine Feast
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Russells Paradox and definition of a set in Terry Tao's Analysis I

In his book "Analysis 1", Terry Tao writes (check out page 39): To summarize so far, among all the objects studied in mathematics, some of the objects happen to be sets; and if $x$ is an object and $A$ is a set, then either $x\in A$ is true or…
HalluZerTuz
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What is wrong with the sum of these two series?

Could anyone help me to find the mistake in the following problem? Based on the formula of the sum of a geometric series: \begin{equation} 1 + x + x^{2} + \cdots + x^{n} + \cdots = \frac{1}{1 - x} \end{equation} \begin{equation} 1 + \frac{1}{x} +…
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For x < 5 what is the greatest value of x

It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?
Tom
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The set of all infinite binary sequences

Let's assume that there exists the set $S$ of all possible infinite binary sequences $s_i$: $$S=\{s_1,s_2,\ldots s_i \ldots\}$$ The sequences $s_i$ are such as $\{1,1,1,1,\ldots\}$, $\{0,0,0,0,\ldots\}$, $\{0,1,0,1,\ldots\}$ etc. Following the…
Andreas K.
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What's your explanation of the Raven Paradox?

The Raven Paradox starts with the following statement (1) All ravens are black. which is equivalent to the following statement (2) Everything that is not black is not a raven. In all the circumstances where statement (2) is true, (1) is also true.…
Dove
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