Benardete’s Book Paradox

http://commons.wikimedia.org/wiki/File:Marinus_Claesz._van_Reymerswaele_002.jpg

Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.

— Patrick Hughes and George Brecht, Vicious Circles and Infinity, 1978

| Oddities · Science & Math

The Unrepentant Liar

ushenko russell liar paradox

“According to [Bertrand] Russell’s treatment the sentence within the rectangle of Fig. 1 is meaningless, and may be called a pseudo-statement, because it is a version of the liar-paradox. But Russell’s treatment is unsatisfactory because it resolves the original paradox at the price of a new one. For, if the sentence of Fig. 1 is meaningless we must admit, since we observe that there are no other sentences within the rectangle, that it is false that there is a genuine or meaningful statement within the rectangle of Fig. 1. And, if there is no statement within the rectangle of Fig. 1 then it is false that there is a true statement within the rectangle of Fig. 1. The italicized part of the preceding sentence will be recognized as identical with (even if a different token of) the sentence within the rectangle of Fig. 1. And since the italicized sentence is true, and therefore a meaningful statement, the sentence within the rectangle is not a pseudo-statement either. Thus, if the sentence in question is meaningless, then it is meaningful and vice versa.”

— A.P. Ushenko, “A Note on the Liar Paradox,” Mind, October 1955

| Science & Math

Faith No More

Kurt Gödel composed an ontological proof of God’s existence:

Axiom 1. A property is positive if and only if its negation is negative.

Axiom 2. A property is positive if it necessarily contains a positive property.

Theorem 1. A positive property is logically consistent (that is,
possibly it has an existence).

Definition. Something is God-like if and only if it possesses all positive properties.

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is logical and hence necessary.

Definition. A property P is the essence of x if and only if x has the property P and P is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. x necessarily exists if it has an essential property.

Axiom 5. Being necessarily existent is God-like.

Theorem 3. Necessarily there is some x such that x is God-like.

“I am convinced of the afterlife, independent of theology,” he once wrote. “If the world is rationally constructed, there must be an afterlife.”

| Religion · Science & Math

Hintikka’s Paradox

(1) If a thing can’t be done without something wrong being done, then the thing itself is wrong.

(2) If X is impossible and Y is wrong, then I can’t do both X and Y, and I can’t do X but not Y.

But if Y is wrong and doing X-but-not-Y is impossible, then by (1) it’s wrong to do X.

Hence if it’s impossible to do a thing, then it’s wrong to do it.

| Science & Math

Homework

One day while teaching a class at Yale, Shizuo Kakutani wrote a lemma on the blackboard and remarked that the proof was obvious. A student timidly raised his hand and said that it wasn’t obvious to him. Kakutani stared at the lemma for some moments and realized that he couldn’t prove it himself. He apologized and said he would report back at the next class meeting.

After class he went straight to his office and worked for some time further on the proof. Still unsuccessful, he skipped lunch, went to the library, and tracked down the original paper. It stated the lemma clearly but left the proof as an “exercise for the reader.”

The author was Shizuo Kakutani.

| Science & Math