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Hare_Geist
2008-10-28, 12:17
Cantor was a German mathematician who spent his life contemplating the nature of infinity. Last night, as I was dropping off to sleep, I very briefly wondered what the consequences of his theories are for Christian theology. However, Maths is a weak point in my education, so it is very possible that I do not understand him and misused his theories in my pure speculations, but that is why I have made this thread: hopefully someone with some knowledge of Cantor will be eager to correct me if I am wrong.

Now, to state his two major conclusions in non-mathematical terms, the first is that there are infinite sets of infinities and that they are all the same size. The second, which appears to contradict the first, is that there are infinities that are bigger than other infinities. A very poor way of conveying this -- and here I really show my ignorance -- would be to think of the numbers one and two. Two is larger than one, yet they are both infinitely divisible; they create two infinite sets, and they are both the same size (infinite), yet one set is larger than the other.

The first theological issue I thought this could be applied to was a particular refutation of God. A common line of reasoning is that if we can prove that the universe has always existed, that is to say, that it has existed for an infinite amount of time, then it quite naturally follows that God cannot exist; for he would have to exist prior to the universe by nature, yet there was no time before the universe, because the universe has always existed. However, this argument cannot survive if we apply Cantor's theories to it, because then it is possible for there to be two infinite existences, God and the universe, and for the former infinity to be larger than the latter infinity.

The second theological issue I thought Cantor’s theories could be applied to was the size of the physical universe, heaven and hell. People often think of heaven and hell in terms of extension and space, which, assuming their existence, is probably not the case. However, if we assume that it is the case that heaven and hell are spatial, then a problem arises. If they both need an infinite amount of space for an infinite amount of souls, then they cannot both exist, for they would impede upon one another and thus limit each other in terms of size. Yet if we accept Cantor’s conception of infinity, then it is possible that there is an infinite space in which there are three other infinite spaces (heaven, hell and the physical universe), because there can be infinities that are different sizes, while contradicting commonsense and remaining equivalent to one another in terms of size.

Now I really have to apologize for this thread before finishing up, because it is probably one of the dumbest things I have ever written. A major flaw I see in both arguments is that infinity as treated by our conceptual schema of numbers does not necessarily apply to infinity as space and time.

Hexadecimal
2008-10-29, 05:37
Maths are a strong point of my own studies. I'll look into this matter and dig this thread up again if/when I have gained some understanding.

Hexadecimal
2008-10-29, 07:00
His point presents a paradox of sorts: Infinity can be larger than another infinity, but is always equal to infinity. Consider a two inch line versus a one inch line. Both contain an infinite number of points, but one contains a (forgive the failure of my vocabulary here, I cannot find a better word) more infinite number of points. The two inch lines contains two subsets that are both equal to the one inch line in members.

p1=number of points on one inch line, p2=number of points on two inch line

where c(nx) is equal to the cardinality of p1+x.

You can define a function as such:

p1={n1,n2,n3,n4...}
p2={p1,c(n1),c(n2),c(n3),c(n4)...}

Both sets are uncountable, and this presents a twist: We cannot know precisely where in the set p2, the subset p1 ends, and c(nx) begins.

Fun Q here, just on the basis of this all: Where does space end and God begin?