EternityDate: 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. Novie 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 eternity." 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 numbers. 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! http://mathforum.org/dr.math/ Date: 01/01/98 at 10:51:03 From: Novie Subject: Re: Eternity Dr.Math, 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. Novie 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! http://mathforum.org/dr.math/ |
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