# How to formalize this paradox?

A friend gave me this problem (in the "blue box")

An interesting fact about the number $2$.

How many times the number $2$ appears in this text?

It appears $2$ times.

Well I see the number two $3$ times but if i put $3$ in "It appears ... times." it becomes false ...

$(1)$How we can rewrite this paradox in the classical logic (cutting useless parts, can this paradox be reduced to a simple form?)?

$(2)$What is its deep meaning and where comes from?

The weird thing I notice, that maybe is the cause, is that the 3rd phrase try to state something about itslef and the system..even if it is inside the system...

My intepretation

My attempt to find a similar "paradox" inside naive sets theory's concept is :

lets define two sets $A=\{2\}$ and $B=\{\delta \}$ , $2$ and $\delta$ are finite ordinals

then lets define a set $X=A\cup B$

We have that $|A|=|B|=1$ and we know that $|X|=|A|+ |B|-|A\cap B|$

now we say that $|X|=\delta$ that can be

$\delta=1$ (if $A\cap B=\{2\}$ ) or

$\delta=2$ (if $A\cap B=\varnothing$ )

then $|A\cup B|=\delta=1 \rightarrow A\cap B=\{2\}$

but $A\cap B=\{2\} \rightarrow ((2\in A) \land (2\in B))$ and we know that in $B$ ther is only one element so $2=\delta$

If this "paradox" is of the same kind of the first (inside the "blue" box) now I can see better that there is something circular. Anyways I'm not able to continue and find a simple (and correct) formula in the language of classical logic that shows this problem, so my questions $(1)$ and $(2)$ are still valid for me.

UPDATE

As the user Charles noticed my interpretation I can't redefine define the two objects.

In fact I wanted to obtain the paradox with that:

$i)$ $A:=\{2\}$

$ii)$ $B=\{|X|\}$

$iii)$ $X = A \cup B$

these tree definition aren't ordered, are given at the same moment, but the definiton is circular because I use $ii)$ in $iii)$ and viceversa.It is maybe this the origin of the paradox?

But what I am really interested in is:

$1)'$ Can be my friend's text reduced at this? How can I express this with a formal formula? (Like the Russel's paradox for example)

that is the question 1).

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"the number $2$" appears only twice. –  Arthur Fischer May 2 '13 at 13:03
I'm not sure this is a paradox. It's just false. It's interesting that if you were to attempt to correct it, and write "it appears 3 times", it would still be false. But it's still not a paradox. There's no contradiction, it's simply that it is impossible to write a text that correctly states how many times it contains the number 2. –  Jack M May 2 '13 at 13:05
Fill in the blank with any word to make a correct sentence: "This sentence does not contain the word ______ ". –  Carl Mummert May 2 '13 at 13:08
I see that someone has voted to close this question. To improve the question, you can include additional information on where you encountered the problem, and your thoughts on the matter. Separately, some contributors here have a distaste for "paradox" questions, because the site could be swamped with them, so the close vote could be related to that. It's advisable to make questions about paradoxes as informative and high-quality as possible. –  Carl Mummert May 2 '13 at 13:10
@Gerry For example, one can follow the same recipe I suggested above, replacing the phrase "this text" with "the text in the gray box at the URL …". One shouldn't need to do that, since that is most likely what everyone understood by "this text" anyway, but the fact that such an easy solution is available should make clear the poverty of the argument from quibbling over the boundaries of the self-reference. –  MJD May 2 '13 at 13:24

This is an attempt to answer the new question.

I don't think this is a paradox. You've already defined $X$ and $\delta$ so it's not legitimate to redefine them with

now we say that $|X|=\delta$ that can be

If this is instead treated as a condition

For what $\delta$ does $|X|=\delta$?

then the answer is simple: there are no such $\delta.$

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... if you put it in this way..this is the right solution, but it is obvious to me that my interpretation changed the real nature of the text that my friend gave me..In the first version of the question, I did't added my interpretation for this reason. The purpose of my question ( my two qestions ara the same) is to obtain a "pure logic interpretation" of the paradox. Really I don't know how to rerwrite the questions. but +1 because it is related with my interpretation (attempt) –  MphLee May 2 '13 at 19:45
I wanted to obtain the paradox with that $A:=\{2\}$, $B=\{|X|\}$ and $X = A \cup B$ these tree definition aren't ordered, are given at the same moment, but the definiton is circular.. is maybe that the origin of the paradox? But what I am really interested in is: can be my friend's text reduced at this? How can I express this with a formal formula? (Like the Russel's paradox for example) –  MphLee May 2 '13 at 19:57
An interesting fact about the number 2:
How many times does the number 2 appears in this text?
It appears 2+1 times.

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flagged, because it doens't help me, and it has not a description. –  MphLee May 2 '13 at 15:12
@MphLee: And to think I voted for your question, and to reopen it! –  Charles May 2 '13 at 16:18
I can say you thanks, anyways your answer make more confusion in my mind, is indeed interesting... but not helpfull sorry –  MphLee May 2 '13 at 18:26
@MphLee The digit 2 appears three times in this snippet. Three is equal to two plus one. –  Arkamis May 2 '13 at 18:29
@Akamis Yea I got that... –  MphLee May 2 '13 at 18:30