Paradoxes are arguments, which are contradicting with logic or common sense, mostly using a false and implicit premises.

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Infinity Paradox: Is $\infty + 1 > \infty$ [on hold]

For Instance: If I say a person has infinite number of oranges and by him, is a person who has infinite number of oranges also and assume that I provide one more orange to the first person. Can I say ...
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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$
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What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
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$x=yx$. Can this statement be true when we don't know that $y=1$?

I am dealing with an equation which is saying that $yx=x$. On the other hand it is telling us that $\frac{x}{x}=1$ which connotes that $x=x$. Is it not absurd to say that $x=x=yx$ when we don't know ...
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Ice cream issue in Lem's 'Extraordinary Hotel'

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?
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How to solve this equality? [duplicate]

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF How 1+2+3+ ... = - 1/12 ? Why? Shouldn't the result be infinite?
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Is $B = \{x:x \notin B\}$ a valid paradox in Naive Set Theory?

The version of Cantor's notion of sets that I've come across goes something like this: "...collection of well defined, distinguishable objects of our intuition or of our thought to be conceived ...
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Is Russell's paradox really about sets as such?

It seems to me that Russell's paradox rather is a "paradox" concerning relations. Suppose we want to construct a graph (with finite or infinite number of nodes) and want some node to be adjacent ...
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Is the following derivation of Gödel's Paradox generalizable to an intuitionistic one using provability?

I will follow Goedel's original proof of Goedel's paradox located on pg. 264 of Hao Wang's A Logical Journey: From Gödel to Philosophy. With this in sight, and to also avoid confusion, I will also ...
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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} ...
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What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
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Clarification regarding Drinker's paradox [duplicate]

This is the informal proof of Drinker's paradox The proof begins by recognising it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one ...
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How can this paradox be resolved?

I came up with a (probably unoriginal) paradox today, and was wondering how it might be resolved. Its approach to reasoning seems to resemble basic game theory techniques. Suppose a casino game has ...
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How is the Liar Paradox a paradox?

In the Liar Paradox, someone says "I am a liar.", which we assume means "Everything I say is false." (although even that's not correct, a liar is defined as someone who says lies, not someone who only ...
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Is it a “paradox”, or a flaw in the question?

(Clearly not a pardox per-se but I would like to hear what you think) The basic riddle (not a very interesting one even) goes as follows: A first client comes into a barber shop, takes a hair cut ...
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Coin flips and prediction - Is this a paradox?

Let's say a coin is given to you which is shown to have two sides (head and tail). I threw the coin 10 times and I got the sequence HHHHHHHHHH (all heads). Now, I am about to throw it the eleventh ...
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How can a set contain itself?

In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this ...
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resolving expected utility of st. petersburg paradox with logarithmic utility

St. Petersburg paradox is a game where you toss a fair coin repeatedly and if it lands heads on the $k$th trial you get $2^n$ dollars. Expected utility of game is: $E(U) = \sum_{k=1}^{\infty}[0.5*0 + ...
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mathematical limit for a ouroboros torus

The other day i was watching an episode of Tom and Jerry in which a similar situation was present toms head comes out of his own mouth. My head hurts when i think how is that even possible so i ...
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Pythagorean “Paradox” (right-angled triangle). [duplicate]

Consider an isosceles right-angled triangle as shown in the figure (top left). The length of its hypotenuse is $c$. The figure distinguishes both legs of the triangle, however, from now on let's ...
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Probability Paradoxes that Puzzle Professors.

There is a class of probability puzzles that includes Monty Hall/Three Prisoners, Three Cards/Pancakes, Two Children/Boy or Girl, their common antecedent Bertrand's Box Paradox, and (a more ...
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Explanation of two boxes problem? [duplicate]

Two piece of gold are contained in two same-looking black boxes respectively. It is known that one piece weights twice as the other, but do not know which is which. Two persons, say A and B, ...
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The sum of all the odd numbers to infinity [duplicate]

We have this sequence: S1: 1+2+3+4+5+6.. (to infinity) It has been demonstrated, that S1 = -1/12. Now, what happens if i multiply by a factor of 2? S2: 2+4+6+8+10+12.... (to infinity). I have ...
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Fallacy - where is the mistake?

Could anyone help me to find the mistake in this fallacy? Because the actual result for $I$ is $\pi/2$ \begin{equation} I = \int_{0}^{\pi} \cos^{2} x \; \textrm{d}x \end{equation} \begin{equation} I ...
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How would you explain this graph illustration of Simpson's paradox?

I need your help for understanding WHAT in the graph you find in the following link proves Simpson's Paradox. For those who don't know about Simpson's paradox, it is the inversion of the inequalities ...
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Points and real intervals [closed]

The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, ...
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If you have two envelopes, and …

Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch ...
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Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
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Why is Banach–Tarski's paradox so interesting?

Here is how I understand the Banach–Tarski paradox, based on the Wikipedia article : with a clever partitioning, one can decompose a solid ball into two solid balls, each identical to the first one. ...
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Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
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Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. ...