The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math


Date: 12/30/97 at 21:32:59
From: Novie
Subject: Eternity

Hello Dr. Math,

If you must choose from two alternatives:

    1. eternity
    2. half of eternity

which is the best?

Can you match the eternity and the measurement?

Thank you very much.


Date: 12/31/97 at 17:25:03
From: Doctor Jerry
Subject: Re: Eternity

Hi Novie,

To answer such questions, one should have agreed upon the rules in 
advance, but neither you nor I has the time to go through the theory 
of cardinal numbers.

However, maybe I can answer by analogy. Would you rather have $1 for 
each counting number (1,2,3,4,...) or for each even number 
(2,4,6,8,...)?  The second of these is something like "half of 

I think the answer to the above question is that I don't care, since 
there are the same number of counting numbers as even counting 

This may be surprising to you. Here is the standard way of arguing: 
Put the counting numbers in wastebasket 1 and the even counting 
numbers in wastebasket 2. We're going to draw one object from each 
wastebasket, tape them together, put them aside, and then repeat this 
process. We're going to match them this way: I'll use (a,b) to mean 
that a comes from wastebasket 1, b from wastebasket 2, and we've tied 
them together.

   (1,2),(2,4),(3,6),(4,8), and so on

You can see that we will have tied everything from both baskets 
together, with nothing left in either basket. In this case, we say 
that the two baskets have the same number of objects.

So, I'd say about the two eternities, it doesn't matter. There are the 
same number of seconds in them.

-Doctor Jerry,  The Math Forum
 Check out our web site!   

Date: 01/01/98 at 10:51:03
From: Novie
Subject: Re: Eternity


Thank you for replying to my question. I have a conclusion about your 
answer: Eternity and half of eternity are the same.

But why did you start with a number for eternity (not half of 
eternity), presented by counting number (1,2,3 ...)?

We know that eternity has no beginning or end. In the case of half of 
eternity we can say that wherever and whenever we are standing now is 
half or 1/3, 1/4, 1/5 and so on of eternity. To our left is one part 
of eternity, and the other is to our right.

Imagine we have two eternities and we fold one of them into two parts. 
Because of the fold, an edge separates the left and right sides. The 
other edge is still infinite (the edges for the left and right parts).   
On the left we now have the ending number that reaches the folding 
part, and on the right we have the starting number that begins with 
the right folding part.

The other eternity is a whole without a beginning number. So there is 
a space between eternity and half of eternity (or eternity divided by 
any number).

So my theory of eternity is that if the eternity is a whole, it has no 
beginning or end, but if we divide it by any number (like folding it) 
we can have an end number on one side (the left side of the fold) and 
a starting number on the other side (the right side of the fold).

My conclusion is that eternity and eternity divided by such a number 
are not the same. Do you agree?

Thank you very much.  I look forward to your answer.


Date: 01/01/98 at 16:08:49
From: Doctor Jerry
Subject: Re: Eternity

Hello again Novie,

My argument was an attempt to give an idea about how mathematicians 
count or compare infinite sets. They follow rules which are stated 
ahead of time. Almost never do they enter into philosophical arguments 
or discussions based on "common sense."  They know from experience 
that there is no way of settling such questions.

I will make one more observation, one that somewhat fits with your 
opinion that infinity has neither a beginning nor an end. The set R 
of real numbers, including all negative numbers and all positive 
numbers, is without beginning or end. Yet the set R has the same 
cardinality (the same number of elements) as the set of real numbers 
between -pi/2 and pi/2. This follows from the fact that the function

   f(x )= arctan(x), -pi/2<x<pi/2

maps (-pi/2,pi/2) onto the set R.

-Doctor Jerry,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Analysis

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.