Paradoxes are arguments which contradict logic or common sense, often by using false and implicit premises.

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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 ...
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Probability of being the millionth customer (What Would You Do?)

I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation. Here is the problem formulation: You are waiting ...
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Generalizing beyond proper classes

I noted that issues such as Russell's Paradox involving the set of all sets that don't contain themselves can be resolved by stating that the object that is all set that don't contain themselves is a ...
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How is this derivative paradox solved? [closed]

We have $$x^2=x+x+x+\cdots+x$$ with $x$ terms in the sum and $x\in\mathbb{Z}$. Taking the derivative of the above equation: $$2x=1+1+1+\cdots+1$$ again with $x$ terms. This implies $$2x=x$$ How ...
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Proofs that relied on paradoxical sentences

Graham Priest's Logic of Paradox is a modification of classical logic where the principle of explosion does not hold, so that there are inconsistent theories which are not automatically trivial. ...
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Bartlett's paradox in Bayesian evidence

I've come across Bartlett's "paradox" (not to be confused with Lindley's paradox, also known as the Lindley-Bartlett paradox) in Bayesian statistics. The paradox originates from Bartlett's 1957 paper, ...
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Does the Russell Set exist?

I am currently reading "Naive set Theory" by Paul Halmos. In the second chapter, on the axiom of specification we show that the Universal Set does not exist. The proof is the following: Lets ...
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G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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What's the relationship between half of a value x, and the midpoint of that value?

Say you have 4 apples: A A A A Where is the middle of this, I mean, on paper it's obvious, but what is the numerical value of this? Halving this value (or rather any value) is easy, 4/2 = 2, but is ...
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Simple question with a paradox

"I have three boxes, each with two compartments. One has two gold bars One has two silver bars One has one gold bar and one silver bar" You choose a box at random, then ...
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Are there fewer reals on $(0, 1)$ than on $(1,\infty)$?

I know that the cardinality of the sets of real numbers $(0, 1)$ and $(1, \infty)$ are equal. So what is the fallacy in this argument? For every real on $(0, 1)$, we can add any integer $n$ to it ...
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P(TT|T) in two coin tosses not 1/3?

So the question is, if two consequtive coin tosses occured and we know, in the aftermath, that at least of them resulted in a tails, what is the probability that they were both tails? The common ...
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What is the worst-case running time of this algorithm?

What's the worst case running time? while flip_coin() == heads: continue; In other words, I flip a fair coin until I get tails. Of course we knows the expected ...
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Self-application in Church's untyped lambda calculus

In "Proposition as Types" by Philip Wadler mentions the weaknesses of untyped lambda calculus and "Russell's logic" concerning self-application. Whereas self-application in Russell’s logic leads ...
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3 coins interpretation of the boy and girl paradox

Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? I think I have come up with a new interpretation for this problem. If children where ...
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Mirimanoff Paradox

The class $B$ is well-founded, if every descending chain of subclasses is finite. The class $B$ is not well-founded if there exists an infinite chain of classes $B_n $ such that $ \quad \dots \in ...
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Non WellFounded Set theories and Russell's Paradox

I am very puzzled by set theories which reject the axiom of regularity. If we reject the axiom of regularity, and allow a distinction to be drawn between wellfounded and non-wellfounded sets/classes, ...
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What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
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If the answer is “no” then “yes” and vice versa type of paradoxes. What are they?

I'm a complete layman, so my technical terms might be misleading. Sorry for the many small questions, it's just that I don't know how to formulate my question right. What is the deal with paradoxes ...
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Looking for explanation of Banach-Tarski Proof, preferably by visual methods “Video, Pictures, Diagrams…”

could someone please explain the four steps of Banach-Tarski? 1- Find a paradoxical decomposition of the free group in two generators. 2- Find a group of rotations in 3-d space isomorphic to the free ...
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The unexpected hanging paradox [duplicate]

monday ----- tuesday ----- wednesday I don't think you can apply this logic recursively. They're all different scenarios unrelated to each other, aren't they? Should I survive until the end of ...
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This sentence is false

"This sentence is false". Is this sentence true or false? My attempt: If this sentence were true, then what it says would be the truth , it implies that it is false which is a contradiction. if ...
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Undecidable definition of pure function

I am trying to come up with a formal definition for functional purity in a simple programming language (think JavaScript). What I've got so far is this: DEFINITION: A statement is impure if ...
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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 ...
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Soft question about logic and Banach-Tarski Paradox

I precise for the possible down voters that I'm not student in maths I'm learning chemistry, and my friend is learning litterature, and we were speaking about BT paradox, my friend discovers this ...
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Average Price Paradox

How can "owning 10 shares of ABC bought for average price of 2 per share" be an equivalent position to "owning 10 shares of ABC bought for average price of 6 per share"? Scenario A A1. Cash = 120, 0 ...
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Limiting the set of “constructible” properties, and loosening comprehension axiom

My historical understanding (which may very well be wrong) is that initially there was naive comprehension for set construction, which required no superset. Russell's Paradox came along and blew that ...
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Boy and girl paradox is driving me crazy

I know this question is asked over and over, but I still can't understand anything. Say I'm introduced to a random father of two and I want to know what's the probability that both his children are ...
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The domain of a function as a function: the “domain-function”

The domain of a function $f:X\to Y$ is normally defined as $\operatorname{dom}f\equiv X$, but I would like the domain-function $\operatorname{dom}$ to be a funtion itself, i.e. I would like to define ...
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Extending Banach-Tarski paradox?

I've learned the Banach-Tarski paradox as following: The points on the sphere (but not the fixed points) are drawn as a square grid, form each point there are three new directions plus the direction ...
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Does Russel's paradox preclude us from using the power set to generate every possible set?

Suppose I have the set of all things $\{a, b, c,... \}$. It seems to me that $ \mathcal P \{a, b, c,... \} $ would be the set of all sets, which sounds like it includes the set of all sets that do ...
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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 ...
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How many $9$'s are there in $0.999\dots$ in the first $n$ places after the decimal point? [closed]

I am counting the number of $9$'s that will come in the first $n$ digits after the decimal point in $0.999\dots$ How many $9$'s will be there? For example, if I take $0.333\dots$ we will have $n$ ...
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How much math was “Broken” by Russell's Paradox?

As you know, the phrase "the set of all sets that don't contain themselves" caused a paradox that "broke" (made trivial) Naive set theory. How much mathematics had to be redone because of this? Most ...
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Why “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” is paradoxical?

In a paper(see here) by Adam Brandenburger and H. Jerome Keisler, they give a game-theoretic impossibility theorem akin to Russell’s Paradox: Ann believes that Bob assumes that Ann believes that ...
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Existence only as a result of its presupposition?

Is there an analogy in logic to the paradox that a concept comes into existence only by presupposing it as already existing?
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What does it mean by Proving false

With respect to the recent finding of a bug in a Coq theorem prover in which false was proved, I'm asking this question. As a hobbyist studying maths, I'm ...
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can someone explain the proof of russels paradox (barber)?

So I understand Russels paradox (barber) but I do not understand the proof, I've looked everywhere online and youtube videos but it doesn't seem to make sense. Please note, I have compensated ...
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Can we avoid an axiomatic theory of sets by never formulating paradoxes?

We know that ZFC was formulated to avoid some paradoxes inherent to Cantor's naive set theory, such as Russell's paradox, which inquires about the truth of the existence of the set of all sets. The ...
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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 ...
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How do sets of language used to formulate ZFC axioms escape Russell's paradox?

We formulate sets using ZFC. Though, to write its axioms we already use the notion of sets. For instance, in formulating the Axiom of Extensionality, we write the following concatenation of symbols: ...
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Apples and Infinity

I am taking a proof writing and discrete mathematics course and we are learning about infinity. My TA asked me the following question and I'm wondering if my solution is correct? Question:Suppose ...
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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 ...
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If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one ...
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Birthday Attacks and Stacked Decks

So this may be a common question, I have no idea. Similar to the birthday paradox, I was wondering how many times one would need to shuffle a 52 card deck (assuming instant shuffles, no set starting ...
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Why is the Berry Paradox a paradox at all?

The Berry Paradox arises from first assigning every combination of 12 english words to an integer and then asking for "the smallest positive integer not definable in fewer than twelve words". This ...
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Should consistency exclude paradox for keeping reduction to absurdity hold? [duplicate]

Consistent means no contradiction only. It doesn't exclude paradox. So, either paradox deducts contradiction (in this case, paradox is excluded also) or a consistent logic system can contain paradox. ...
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Puzzle : Truant List of Statements

I was working my way through some puzzles in Discrete Maths by Rosen, when I came across the following question: The $n^{th}$ statement in a list of 100 statements is : "Exactly $n$ of the ...
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Russell's Paradox for the zero set and a set with the zero set.

So I have a question: Let: Allow set B = {x: x $\notin$ x}. Then, B $\in$ B $\iff$ B $\notin$ B ? Does this apply for the zero set? Because I'm a bit confused. The definition is a zero set is always ...
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How can I solve this para Paradox? [duplicate]

How can I solve this para Paradox? $ -1={(-1)}^{1/2} {(-1)}^{1/2}={[(-1)(-1)]}^{1/2}=1$