Most paradoxes involves self-reference, the only exception known to me is Yablo's paradox, however it is still debated if it is really without self-reference. So, I was wondering, are there other known paradoxes that works without self-reference?

• Set set of all sets, which do not contain themselves? $$M := \{A : A \text{ is a set} \wedge A\notin A\}$$ (It can't be decided, wheter $M\in M$) – AlexR Aug 25 '13 at 14:39
• @AlexR I doubt that if you're able to define self-reference, your example won't be one such instance of self-reference. – Git Gud Aug 25 '13 at 14:47
• @AlexR I believe your example IS self-referential. – Trapszan Aug 25 '13 at 14:48
• I'd say $$\{ A : A\text{ is a set} \wedge A \notin A\} \in \{ A : A\text{ is a set} \wedge A \notin A\}$$ is "intuitively" not self-referencing, but yeah, maybe you're right. – AlexR Aug 25 '13 at 14:49
• @Trapszan, GitGud please show me the self-reference in this statement; I'm unable to find it. (I needn't even give it a Name to reference it by ^^) – AlexR Aug 25 '13 at 15:07

It is well known that the natural numbers have the property that if we have some non-empty set $A\subseteq\mathbb{N}$, then $A$ has a minimal element. Now let $$A = \{n\in\mathbb{N}|\,n\mathrm{\ can't\ be\ expressed\ with\ less\ than\ }100\mathrm{\ characters}\}$$ Let $a\in A$ be the smallest element in $A$. Then we can express $a$ as "$a$ is the smallest number that can't be expressed with less than $100$ characters". But this is an expression for $a$ containing less than $100$ characters, which would be a contradiction. We deduce that $A=\emptyset$. Thus every natural number can be expressed with less than $100$ characters. But there is only a limited number $n$ of characters, thus there are at most $n^{100}$ natural numbers.

There's the Achilles and the tortoise paradox:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

The Dichotomy paradox is in some sense a particular case of the above:
Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

• Those are more philosophical paradoxes than mathematical ones... – Daniel Robert-Nicoud Aug 25 '13 at 14:55
• Also this is not actually a paradox but a misunderstanding of infinite series. – ithisa Aug 25 '13 at 14:55
• @DanielRobert-Nicoud Agreed. – Git Gud Aug 25 '13 at 14:56
• @user54609 What is a paradox to you? Whatever meaning you give it it will be a 'proper' contradiction or it will be a contradiction that arises from the lack of rigour of the terms involved. Is there any reason why the former is more valuable than the latter in this context? Edit: In fact I argue that the latter in more valuable than the first, for a solution to a paradox lies exactly in defining everything in order to identify the issue. A 'proper' contradiction is just a contradiction. – Git Gud Aug 25 '13 at 14:57
• The liar paradox and also the "set of all sets" paradox seem to be true paradoxes, revealing incompleteness of certain systems of logic or set theory. – ithisa Aug 25 '13 at 14:58

A famous one was given by the Harvard logician WVO Quine (look up Quine's paradox for more information):

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

When I first saw it I had to take a minute to parse what it was saying and why it wasn't self-referencing.

• This seems self-referential to me. In the statement Yields falsehood when preceded by its quotation, what is the subject of the verb yields? If the answer is "This statement", then it is self-referencing. If the answer is "it is intentionally left ambiguous", then I would say that it is not a well-formed statement (is he threw the ball true or false if I don't say who "he" is?). – Zev Chonoles Aug 25 '13 at 16:05
• I agree with Zev; it seems to me that we have here an indirect self-reference. The sentence itself talks only about properties of some string of symbols; basically, that this string written two times, first time in quotation, will create a false statement. But this statement is our original sentence. – Trapszan Aug 25 '13 at 16:26
• The subject of "it" is the quotation, i.e. the subject of that sentence. The sentence is not the quotation, that's the point. If the subject had been "this sentence", e.g., then the sentence would have been self-referencing. This is an important distinction: not every use of the word it replaces "this sentence," including in Quine's famous example above. – josh Aug 26 '13 at 12:29