Geoffrey Klempner, moderator of Ask a Philosopher, has just posted this answer of mine:
(11) Dave asked:
I have a question regarding the existence of actual infinities. I've heard theists argue that an actual infinity cannot exist, yet claim that God is infinite. Some then say that an actual infinity cannot exist in 'the physical world' or in 'spacetime,' but outside of the physical world (but still in reality) actual infinities can exist. Isn't this an arbitrary distinction? Or are they using a different notion of infinity for God? The existence of God is such an obvious counterexample to their argument that I feel like I'm missing something. Thanks.
The contexts of metaphysics and mathematics are, as Dave knows, different, and therefore their use of a common symbol, e.g., 'infinite,' does not entail equivocation. Metaphysics explores the intelligibility of self-subsistent being, which finite beings allegedly participate (or, as modern syntax has it, participate 'in'). In the Thomistic tradition that has influenced subsequent philosophical theology to the present day, the symbol of 'subsistent being' is equivalent to 'God.' Mathematical infinity, by contrast, refers to the possibility of adding a member to a series: if it is always possible to add one more, then the series is infinite. The series of natural numbers (or of even numbers, or of prime numbers) is infinite in that sense. It is a metaphysical claim that no series of existents can correspond in a one-to-one fashion to the series of natural numbers, because such an actual or 'consummated' infinity would lead to absurdities.
For example, suppose an infinity of persons stands in a line to your left and each person has one coin. A superhuman being with magnetic powers causes the coin belonging to the person on your immediate left to travel instantaneously from his or her pocket to yours (so that now you have two coins);simultaneously, coins from the third and fourth persons wind up in the second's pocket; coins from numbers five and six go to number three; from seven and eight to four, etc. What such a transfer accomplishes is a doubling of the number of coins by the mere change of the location of existing coins, that is, without the production of new coins. That is metaphysically absurd, and that is why there cannot be a consummated infinity: it bears within it the possibility of absurdity, which is no possibility at all.
In the future, Dave will go further under his own steam to resolve more quickly, if not dispel, a problem if he clarifies who said exactly what (i.e., not settle for 'I've heard...', 'Some then say...'). I wish to assure Dave that what was 'obvious' to him has also been obvious to the intelligent writers who, he seems to have thought, missed the obvious and then contradicted themselves. Which philosopher forgot that he had said that God was infinite after declaring consummated infinities to be impossible? One should document such a self-damning performance before presenting it as an interesting case for the consideration of others. Having said that, I also want to assure Dave that I enjoyed answering his question and hope he will pursue his metaphysical studies.