We thus distinguish two senses in which an actually infinite collection can be created through successive addition:
(A) An actually infinite collection is created particularly through successive addition if some act of addition creates an infinite collection out of a finite collection.
(B) An actually infinite collection is created procedurally through successive addition if the process of successive addition, carried through to completion, creates an infinite collection.
kuartus4 wrote: Now I dont expect to get an answer from Dr. Craig himself since I understand he's very busy and either way Im not even sure he participates in this open forum.
kuartus4 wrote: After having read the paper again, I believe that the only way out for Craig if he wants to maintain that the paradoxes he presents are truly absurd and agrue from them that the infinite is impossible is for him to deny mathematical legitimacy to the infinite and subscribe to mathematical finitism.
Dr. Craig is familiar with Dever’s argument and thinks it is a very powerful and even-handed critique, one of the best he’s seen. Dr. Craig’s claim is that an actual infinite is metaphysically impossible. So if mathematical objects really existed, his argument would, indeed, apply to them as Dever says. But do mathematical objects really exist? Only if Platonism is true. As Dr. Craig wrote in The Kalam Cosmological Argument, “For the nominalist, the conceptualist, and the formalist, the mathematical validity of the Cantorian system implies no commitment to the existence of the actual infinite in the real world. . . . Only for the Platonist-realist, who accepts the independent status of mathematical entities in the real world, do Cantor’s theories have ontological implications for the real world. This means that our argument against the real existence of the actual infinite would contradict Cantor’s work only if the Platonist-realist position . . . were proven to be . . . correct. . . , for our argument would be compatible with any of the other three.” (p. 89) If there were mathematical objects, there could be only a finite number of them, as Intuitionists believe. But Dr. Craig sees no reason to think there are such objects; hence there really are not two worlds, as Dever infers. One addendum: In Philosophical Foundations, Dr. Craig did use the language of the impossibility of an actually infinite number of physical things; but that was because his co-author is a Platonist and so he had to accommodate him! Obviously, if an actually infinite number of things, whether concrete or abstract, cannot exist, then an actually infinite number of physical things cannot exist, which is enough to prove the finitude of the past.
For God, all things are possible.