# HELP: Did I butcher mathematics by attempting to count infinity?

I don't know if you guys are following that huge discussion. I am kind of accompanying it, but not really posting because I feel like most of the other people who are responding basically express what I feel. I did, however, come across this post, that stood out:

"When you sin against an eternal god what kind of punishment should you receive? Naturally an infinite punishment because your crimes are against a higher being. Your [sic] committing an infinite crime. Christ being fully god is an infinite being. So he took on our infinite punishment because he was the only one who could so we wouldn't have to spend eternity in hell. An infinite being would experience more pain in 3 days that a human would over the course of an eternity. Christ suffered more pain than anyone could possibly suffer..."

David claims that 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. In chart, this would look like:

___________Nature of being ____ Nature of suffering _____ Time period of suffering

God_______ Infinite ____________ Infinite _________________ Finite (3 days)

A human __Finite ______________ Infinite _________________ Infinite (eternity)

Before I start my point I'd like to ask "when you sin against a benevolent god, do you receive punishment?" say that infinity, like god, is just a concept. Also, as a disclaimer, I would like to explain that I've only taken one semester of abstract algebra and acquired a very limited understanding of set theory and the concept of infinity. Moving on...

Now, infinity can divided in two categories - countable and uncountable - according to my understanding they can be defined as following:

Countable infinity [ex: set of naturals, integers] can be defined as an infinite set, each set containing a finite amount of values. The natural set would be countably infinite because you can divide it into infinite intervals containing a finite quantity of numbers; ex: with naturals you have { [0,1) , [1,2) , [2,3) ... } The interval [0,1) contains a finite quantity of real numbers (one number, which is zero), so does the next interval, but there's an infinite number of intervals.
Uncountable infinity [ex: set of reals, rationals] can be defined as an infinite set, each set containing infinite quantity of numbers. For example, with the reals, the interval [0,1) contains infinite quantity of real numbers; even if you were to divide it into smaller intervals such as [0,0.00000001), the new interval would still contain infinite amount of values. Thus, { ... [-1,0) , [0,1) , [1,2) , ... } set of real numbers is uncountably infinite.
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.

Back to the context of David's post, the finite nature of human over an infinite period of time would seem to generate a countably infinite human existence in hell. On the other hand, the infinite nature of human punishment over an infinite period of time would seem to generate a uncountably infinite human suffering. Thus, because a one-to-one mapping (for every a, there is one and only, b) involving a countable infinity can only exist when a countable infinity maps to another countable infinity; in the case of human suffering, for every quantity of human existence in hell, there exists more than one quantity of human suffering. Or, in other words, the ratio of (in hell) human suffering/human existence > 1.

I don't know if the nature of god or of god's suffering would be qualified as countable or uncountable infinity. However, if we assume that god is the "supreme being" and he is greater than anything else, then god's suffering < god's existence. Thus, god's suffering/god's existence < 1.

Therefore, HS/HE > 1 > GS/GE. The ratio of suffering a human being faces in hell is greater than the suffering Christ faced at the cross. Proportionally, one human sent to hell suffers more than god.

I hope I have not killed math or logic in the process. I apologize for any existing errors, please correct me. In my defense, infinity is a difficult concept to wrap one's mind around, I would daresay even more intricate than the god concept.

This is a pictorial of what popped up on my mind as I read David's post.

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### Replies to This Discussion

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...
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]

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.

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