Hello T.E.,
This may be the case. His appeals to things like the infinite hotel, etc... do point to non-intuitive issues that arise within Set Theory. However, I do not yet see how any set theory gets around these issues. For instance, all set theories will argue that the cardinality of the nuatural numbers equals the cardinality of the even numbers. It seems you are saying that there is a set theory where this is not the case. This is news to me, but I am no expert.
Sure. But transfinite arithmetic is done with ordinal numbers, not cardinal numbers.
What's more, there is a reason for that. Cardinality is defined as relative ordering. Two sets have the same cardinality if the members can be mapped one to one onto each other. This is the same as saying that each set can order the other (just so long as at least one can generate an order). For finite sets, this is the case when there are
just as many members in each set. But if set A is mapped onto set B one to one, and set A still has members left over when B runs out, then A has a greater cardinality, and it is also the case that it has more members, is bigger than B.
But what about infinite sets? The natural numbers vs. the real numbers. You cannot order the real numbers by the natural numbers, as Cantor's diagonal proof shows. But does that mean, in the case of infinite sets, that one has
more members or is
bigger? Or in the case of infinite sets is a difference of cardinality merely a matter of relative orderability and does not imply the informal concept of
more?
If you look at the standard texts: Kleene, Church, etc. the all assert without argument or even raising the question of equivalency that greater cardinality means "more", but I maintain that this is a unjustified metaphysical interpretation of set theory. Cantor just assumes this also, which I suppose is where the problem started.
This is why in the transfinite realm, cardinal numbers fail. Cardinality, in my view, is not a fully arithmetic concept, and in the transfinite realm this becomes apparent.
With all of that said, I proposed an argument that does not presuppose a particular set theory. Rather, it denies the actual infinite in terms of the real world. It does not deny it in strict theoretical constructs. Most set theorists would not find this problematic. They will admit that the transfinite may not have a real world application and remain content with their construct. I love to study formal systems just for the sake of the logic behind formal systems (set theories are in this catagory).
Brian
In this form it is question begging. You don't believe and actual infinite can exist in the real world, because you think that in the real would it is impossible.
For the argument to not apply against the infinity of God, Craig has to argue that God is not composed, either in his being or in his duration.
I disagree. The argument as stated only argues against an actual real world interval that is infinite as opposed to an interval that is finite divided into infinite parts. Also, when you speak of the infinity of God, I suspect you are equivocating on the term 'infinity'. I do not think the Bible uses the term in such a technical sense as mathematicians have defined it. A discussion regarding this would be interesting.
Sincerely,
Brian
Well, we aren't talking about what the Bible says about God. We are talking about what philosophical theology says about God, and has said since ancient times. God is simple, unchanging etc.
There are plenty of people who reject this sort of traditional metaphysics, but as far as I know only process theology people have an developed answer about what they would put in its place.
Craig has a section in at least one of this books (I remember reading two or three) where he talks about the simplicity of God and how that exempts God from the argument, and he has a discussion trying to show that time must be thought of as consisting of moments or events and is therefore composed.
Of course this gets Craig (if his argument is valid) only to the God of the philosophers. He then has to call on supplementary arguments as to why the God that his argument proves to exist would be the God of the Bible. Craig is well aware of that.
