The Paradox of Tristram Shandy

The Paradox of Tristram Shandy

Bijections

between different

infinite sets

can lead to

paradoxes

like that of Tristram Shandy that Adolf Fraenkel tells us in order to explain this fundamental feature of

set theory

(A.A. Fraenkel: ”Einleitung in die Mengenlehre” 3. Aufl., Springer, Berlin 1928, p. 24): ”Well known is the story of Tristram Shandy who is going to write his biography so pedantically that he needs one year to

complete

a single day of his life. Of course his biography will never get ready, when continuing in this manner. But if he lived infinitely long (a

countable

infinity

of years, say) then his biography would get ’ready’, every day of his life, how late ever, finally would get a description.” Tristram Shandy borrows time in order to catch up with time already spent. (Similarities with modern finance schemes are unmistakable.)

But we will show that this method leads to a

contradiction

. For that sake we use the less spectacular but better understandable ratio of 2 to 1. We fill successively pairs of consecutively enumerated marbles into an urn. Between every two steps of filling we remove the marble with the lowest number from the urn. If the numbers

contained

in the urn are separated by a point from those removed already, this yields the following

sequence

21., 2.1, 432.1, 43.21, 6543.21, 654.321, …

which however gets confusing when multi-digit numbers are involved. The picture gets clearer, when the

even numbers

are denoted by 0 and the odd numbers by 1.

01., 0.1, 010.1, 01.01, 0101.01, 010.101, …

After infinitely many exchanges we have a contradiction between set theory and mathematics: Set theory yields as the limit the empty urn because for every marble the step can be determined, when it is removed.

But this very sequence can also be understood as a sequence of real numbers r. Mathematics shows that the (improper) limit is

infinite

, proved by the proper limit 0 of the sequence of the

reciprocals

1/r.

Since the

logarithm

is a

strictly increasing function

of its

argument

and gives the number of digits of x by [log x] + 1, we have a contradiction in that, according to mathematics, there will be a set of infinitely many indexed digits on the

left hand side

of the point in the limit.

As one and the same sequence of sets of digits leads to different results in set theory and in mathematics, set theory can no longer be accepted as the basis of mathematics. In particular the results of a scientific theory must not depend on the choice of symbols. But even that

occurs in

set theory. If we let the above thought-experiment run materially exactly as before, but remove always the largest number instead of the smallest, then the urn contains infinitely many marbles in the limit. The question what might happen when an accidental number is removed remains without answer.

How could such a contradiction appear? The true reason is the identification of all and each that is based upon and suggested by the use of the same symbol in logic. A bijection between

natural numbers

and a

countable set

enumerates

each

element

like a street number is fixed on a house. But that means only for

finite sets

that all elements are enumerated. In infinite sets there is for each enumerated element an infinity of not enumerated elements - and this

relation

does never change. Unfortunately in set theory ”never” is mistaken for a time that can be arrived at. Just like every infinity can be completed, so can

eternity

as an infinite set of seconds.

The basic error of set theory is that the correct meaning of potential infinity has been reversed to actual infinity.

Potential infinity: For every initial segment (1, 2, 3, …, n) of natural numbers, there is a larger initial segment (1, 2, 3, …, n, n+1).

Actual infinity: There is an initial segment (1, 2, 3, …), namely the sequence of all natural numbers, that is larger than every other initial segment (1, 2, 3, …, n).

That means, that a

quantifier

exchange has occured. The correct

sentence

”for every initial segment (1, 2, 3, …, n) of the natural numbers there is a larger one” has been inverted to ”there is an initial segment of the natural numbers (namely the

complete set

of all natural numbers) that is larger than every other initial segment”. Actually mathematicians,

strictly

committed to logic, should have refused to be taken for a ride. Who would invert the sentence ”every US citizen lives in America” to ”every American lives in the US”?

Title

The Paradox of Tristram Shandy

Canonical

name

TheParadoxOfTristramShandy

Date of creation

2013-05-06 7:43:07

Last modified on

2013-05-06 7:43:07

Owner

WM (16977)

Last modified by

WM (16977)

Numerical id

1

Author

WM (16977)

Entry type

Definition

Classification

msc 03Exx

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