Brian Bosse
"The Brain"
Hello T.E.,
I love this topic!
If this interests you, we should probably start another thread. But I will make some comments here.
To make this proof go through, one must assume an actual infinity.
Mathematicians are partly responsible for the confusion between the number of elements in a set and the set's cardinality. Even though they have carefully defined 'cardinality', when speaking informally they often speak like "Some infinities are larger than other infinities" without qualifications. You seem to make the same point when you say...
I would agree with you that when strictly speaking greater cardinality does not correspond to more elements. Of course, this only matters when sets are infinite. Otherwise cardinality corresponds precisely to number of elements. It also hints at the problems inherent in trying to have an actual infinity encompass a potential infinity.
It is not quite question begging. I have already provided an argument for why there cannot be an actual infinity earlier in this thread. I have a reason that is more than just pure conjecture or some crass assumption on my part.
I am open to the idea that philosophical conceptions of God may be incorrect. Our philosophy of God, so to speak, must be Biblically informed. However, I am fairly confident that one can overcome any objection rooted in actual infinity. In other words, I am not sure I agree with you that traditional formulations do assume an actual infinite. Maybe they do? I have been wrong before.
If am not mistaken, Craig's view of God and time is not traditional. Me, on the other hand, hold to the more traditional view. Interesting stuff...much of which is beyond me.
Sincerely,
Brian
I love this topic!
But what about infinite sets? The natural numbers vs. the real numbers.
If this interests you, we should probably start another thread. But I will make some comments here.
You cannot order the real numbers by the natural numbers, as Cantor's diagonal proof shows.
To make this proof go through, one must assume an actual infinity.
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?
Mathematicians are partly responsible for the confusion between the number of elements in a set and the set's cardinality. Even though they have carefully defined 'cardinality', when speaking informally they often speak like "Some infinities are larger than other infinities" without qualifications. You seem to make the same point when you say...
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.
I would agree with you that when strictly speaking greater cardinality does not correspond to more elements. Of course, this only matters when sets are infinite. Otherwise cardinality corresponds precisely to number of elements. It also hints at the problems inherent in trying to have an actual infinity encompass a potential infinity.
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.
It is not quite question begging. I have already provided an argument for why there cannot be an actual infinity earlier in this thread. I have a reason that is more than just pure conjecture or some crass assumption on my part.
We are talking about what philosophical theology says about God, and has said since ancient times. God is simple, unchanging etc.
I am open to the idea that philosophical conceptions of God may be incorrect. Our philosophy of God, so to speak, must be Biblically informed. However, I am fairly confident that one can overcome any objection rooted in actual infinity. In other words, I am not sure I agree with you that traditional formulations do assume an actual infinite. Maybe they do? I have been wrong before.
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.
If am not mistaken, Craig's view of God and time is not traditional. Me, on the other hand, hold to the more traditional view. Interesting stuff...much of which is beyond me.
Sincerely,
Brian