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Different Infinities


Date: 02/19/97 at 21:21:20
From: Brian Hicks
Subject: Infinity and the number line

Hello!  

My friends and I were debating this issue.  Let's say you have a 
number line. We were wondering how many different types of infinity 
there are there.  I say that there are two: one that goes from 0 to 
the negatives forever (negative infinity) and another that goes from 0 
to the positives forever (positive infinity).  Now, is there a 
positive infinity and a negative infinity?  Or is there just infinity?  
What form would the number line actually have if there were just one 
infinity?  Wouldn't it be a circle shape or oval?   Since there would 
be just one infinity, if you went negative, you would get infinity, 
and if you went positive you would get infinity. Would those two 
infinities be the same? 

Could you clarify a bit more on an uncountable infinity?  How can we 
say there is an uncountable infinity (which has more numbers than 
there actually are, correct?).  How can that be true?  If we have an 
infinite number of natural numbers, how can there be an uncountable 
infinity?  

We say that infinity is a concept, because we have proven it's not a 
number. But does infinity = infinity?  If there is only one type of 
infinity this should be true?  

Thanks for your help.  :)
Brian


Date: 02/23/97 at 17:47:19
From: Doctor Ceeks
Subject: Re: Infinity and the number line

Hi,

Well, you've asked some excellent questions!

To really answer them completely would require quite a bit of writing, 
so I hope for now, you will be content with a few statements of 
relevance.  Perhaps, these statement will give you enough food for
thought and you can come to your own answers about your questions.
Otherwise, please ask more questions!

In your message, you use the word "infinity" in two different ways: 
one to express size, or "cardinality", of a set, and another way,
to express limiting notions.

Let's separate the two concepts now.

1. As a limiting notion.

This refers to your discussion about going on to the right along the
number line, or going on to the left along the number line.

Because I want to get some sort of response to you quickly, let me
just make two observations:

Observation 1: You can map the real line to the set of points between 
-1 and 1 by sending the real number x to the real number 
2Arctan(x)/ Pi. (There are many other possible maps.)  Notice that 
there are points between -1 and 1 which can be chosen closer and 
closer to 1, and similarly, there are points between -1 and 1 which 
can be chosen closer and closer to -1.

Observation 2: You can map the real line to all but one point of a 
circle too. One way to see this is to draw the unit circle in the 
plane with center at the origin.  Now send a point on the x-axis (the 
real line) to a point on the circle by noting where the line which 
goes through the point on the x-axis and the point (0,1) intersects 
the circle at a place other than at the point (0,1).  Now, as you move 
along the number line to the right, you approach the point (0,1) on 
the circle under this mapping.  But you also approach the point (0,1) 
on the circle as you move along the number line to the left.  Contrast 
this with Observation 1.

2. Infinity as the cardinality of a set.

We can say two sets S and T have the same size or cardinality if and 
only if there exists a pairing between elements of S and elements of T 
so that every element of S gets paired with exactly one element of T, 
and every element of T gets paired with exactly one element of S.

We say S is finite if S has the same cardinality as some set of the
form {1,2,3,4,...,N}, where N is a positive integer.

Cantor (refer to "Cantor's Diagonalization arugment") observed that
there is no pairing between the set of real numbers and the set of
integers.  The integers are a subset of the reals, but the reals
cannot be imbedded in the integers.  Consequently, the set of reals is 
considered to have a larger cardinality than the set of integers.

Anything with the same cardinality as the integers is said to be
"countable".  Because of Cantor's observation, we say the reals are
"uncountable".

I'm sorry this answer is so terse and more like a puzzle than an
answer.  Feel free to let us know what you make of it.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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