Replies to This Discussion

Permalink Reply by Olivia Kuo on May 3, 2010 at 5:51pm

I actually don't think I am right since infinite time wouldn't be countable, which undermines my whole idea. But this part: "an infinite being, subject to infinite suffering over a finite period of time would suffer more than a finite being, subject to infinite suffering over an infinite period of time" just doesn't sound right.

I'll look more into this problem after I'm done with exam season...

Permalink Reply by Michel-san on May 5, 2010 at 7:27am

In other words, a group of with an infinite number of sets that are finite is countably finite, and a group with an infinite number of sets that are infinite is uncountably infinite.

Some things to think about...
There are infinitely many prime numbers. For each prime number construct a set containing powers of that prime.
i.e.
[2]={2,4,8,16,32,64,128....}
[3]={3,9,27...}
[5]={5,25,125...}
[7]
[11]
etc...

All of these sets are infinite and disjoint (so no number is part of two sets), and there are an infinite number of them. But their union is clearly smaller than the natural numbers, so it's therefore countable. So it's not true that an infinite collection of infinite sets is uncountable.


The best way to define and deal with the cardinalities of sets is through looking at 1-1 (bijective) maps.

Finite:
There exists a 1-1 (bijective) map with a set of the form:
0 (the empty set)
or {0,1,2,...N} (the first N natural numbers)

Infinite:
Not finite.

Countable:
Finite or there exists a 1-1 (bijective) map with the set of natural numbers.
(This is why the term countable is used, the idea of counting them and giving them a numbering. It makes sense to be able to pick a member of the set and say "according to my numbering, this is the 2318th member".)

Countably Infinite:
Not finite, but countable, this leaves one possibility:
There exists a 1-1 (bijective) map with the set of natural numbers.

Uncountably Infinite:
Infinite but not countable.
Example: the real numbers.
(Think of anything uncountable as being part of the untamed wilds. There are many tools to deal with countable sets that won't work on uncountable ones. Also there are many uncountable infinities of different sizes.)


Finding explicit 1-1 (bijective) maps is difficult. An easier way is using a theorem (Cantor–Bernstein–Schroeder Theorem) which is basically the same as saying:
A less than or equal to B
B less than or equal to A
Therefore: A=B


For example: the rational numbers and the natural numbers.
The natural numbers are 'less than or equal to' the rational numbers. Since the rational numbers contain the natural numbers.
To see the rational numbers are 'less than or equal to' the natural numbers consider this:

For any rational number a/b where a and b are positive define:
n = (2^a)(3^b)
And where a is negative and b is positive define:
n = (2^a)(5^b)

Example:
5/3 --> 2^5 3^3 = 32*27 = 864
Or the other way around:
80 = 16 * 5 = 2^4 5^1 = -4/1


This way of 'hiding' rational numbers in the naturals shows that the set of natural numbers is very big! Don't be fooled by their way of being all spread out. Also consider the contents of your hard drive, or the contents of the entire internet can be represented by a single natural number. A hard drive contains nothing but a very long natural number written in binary.

Getting back on topic; this means the rational numbers are countable (not uncountable).

The real numbers are uncountable, however.


I like the picture, it's a great way of looking at it. Keep looking at problems like this (with pictures in your head), It's the best way. And yes, infinity is far more complicated than "god".

[Sigh... here I go writing again. On any other subject I struggle to say more than three sentences]

Permalink Reply by Wayne W on January 30, 2012 at 3:26pm

Countable times Countable is still Countable.  Countable sequences chosen from countable sets are uncountable; That's Cantor Diagonalization.  

However, what you are really trying to do is develop a measure theory and then integrate two different suffering functions.  Measure theory really doesn't care if the underlying sets are countable or uncountable, but it does like to satisfy  properties like countable subadditivity.  It also seems you may be confusing cardinality with values in the extended real numbers.    The limit of all the real numbers way down there at the end of the line is not a cardinality, it's a "value."     (Pascal's Wager would have made sense if he could have developed a proper measure theory. )

In the end there is no way to make a serious argument, because there is no reasonable way to get uncountable infinite as the value of a limit of finite integrals, because uncountable infinite is not a value.