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.
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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.
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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