A Tristram Shandy argument proved invalid. -- Ben's Message Board

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Subject: A Tristram Shandy argument proved invalid.

Author:
Wade A. Tisthammer

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Date Posted: 03/13/04 10:11pm
In reply to: Wade A. Tisthammer 's message, "Proof against an infinite past?" on 01/ 9/04 5:47pm

First, I'd like to apologize for not bringing this to attention before.

>Can the paradox be resolved? It seems impossible for
>Tristram Shandy to finish his autobiography and yet
>the argument regarding an infinite past says
>otherwise. Let�s look at the part of the argument
>that claims an infinite past implies Shandy being able
>to finish his autobiography:
>

There is a one-to-one correspondence between years passed and days passed.
>
In each year Tristram Shandy records a
>different passed day.

>
>Therefore Tristram Shandy records all
>the days that have passed (follows from 1 and 2); he
>finishes his autobiography.
>
>Because the above is a deductive argument, it can fail
>in only one of two ways. Either the argument is
>invalid (i.e. the conclusion does not logically follow
>from the premises) or at least one of the premises is
>incorrect.

The argument is invalid. We can have two infinite sets with a one-to-one correspondence, with each element in one set recording a different element of the other, and still not have all the elements recorded. To demonstrate this a counterexample to the structure of the Tristram Shandy argument is useful. Suppose we have two sets, set A is a complete set of natural numbers (1, 2, 3...) and set B is a complete set of natural nubmers (1, 2, 3...). Suppose the first element in set A, signified by A(1), records the second element in set B, signified by B(2). From then we have the recurring pattern:

A(1) == B(2)
A(2) == B(3)
A(3) == B(4)
A(4) == B(5)

And so on out to infinity. And here we have it, two sets with a one-to-one correspondence (premise 1), and each element recording a different element of another set (premise 2), yet the conclusion (all elements in the second set are recorded) does not follow.

Thus this Tristram Shandy argument is invalid.

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Replies:

Subject	Author	Date
A Tristram Shandy argument proved invalid.	Wade A. Tisthammer	03/13/04 10:14pm

Sorry.	Wade A. Tisthammer	03/13/04 10:22pm

Everyone nearly said as much	Damoclese	03/14/04 4:01pm
I don't think they did. -- Wade A. Tisthammer, 03/15/04 12:07pm You're right. -- Damoclese, 03/15/04 4:55pm