In a recent thread, the Kalam Cosmological argument family was challenged on the issue: can an actual infinity exist? If not (presumably due to Hilbert’s Hotel-like absurdities), then God could not be an infinite being as such is impossible of being.
A thread of discussion developed, and I thought a summary intervention may be helpful. On further thought, perhaps it should be headlined:
KF, 12: >> I think several themes are worth highlighting.
It can be discussed that non-being, true nothingness cannot be a causal source. Were there ever utter nothing, such would therefore forever obtain. There would be no world.But, manifestly, there is a world.
So, we must ponder the logic of being, at least in a nutshell.
Candidates to being may be such that core characteristics central to identity stand in mutual contradiction such as those of a square circle. Such are impossible of being. And, we see principles of distinct identity necessarily embedded from the outset, especially that truths must be all so together so X and Y where Y = ~ X is not a possible state of affairs. If something could exist in a possible world were it actualised as a state of affairs, it is a possible being.
Of these, some Z can be in at least one possible world Wi, but not in another “neighbouring” one, Wj — contingent beings. The difference Wi – Wj will contain some unmet on-off enabling circumstance for Z . . . a necessary causal condition or factor for Z, say C. If Z is a fire, it requires heat, fuel, oxidiser and an uninterfered-with combustion chain reaction (cf. how Halon extinguishers work).
By contrast, we can see a being N that has no dependence on any such C, which will be in any and all possible worlds. That is, N is a necessary being and will be part of the common framework for any world W to exist. For example, distinct identity (A vs ~A) entails that two-ness and so also the endless set of naturals, must exist. [Beyond which lie the transfinites and the surreals as illustrated.] And, without necessary causal factors C, such has no beginning or end. Given that a world exists, at least one being N must be necessary. (Theists, classically hold that things like numbers are eternally contemplated by God, for instance.)
Any given case Ni is eternal, causeless, framework for any possible world, enabling and structuring it in some way. Notice, eternality not infinity, has been asserted, on the strength that for some W to be, some N must be as key to its framework. We readily see this for two-ness etc.
Such is strange to our ears, maybe, but that is a fault of our education not the logic.
Now too, our world is one of finite stage causal succession as we can see from succession of generations. But it is dubious for such to have existed to the infinitely — endlessly — remote past, as to succeed from some stage s_k to s_k+1, s_k+2 etc is equivalent to a counting succession 0,1,2 . . . which succeeds without limit but in an instance will be such that some later s_p to s_p+1, s_p+2 etc can again be matched 1:1 with 0,1,2 . . . thus showing that a transfinite span, credibly, cannot be traversed in finite-stage successive, cumulative steps. Thus if we are at a now, no S_k is transfinitely remote, even beyond say a big bang at 13.8 BYA. Our world W_a is credibly not some Ni, and has a beginning. It has a cause, a capable, sufficient one.
Where, we exist therein as responsibly and rationally free, morally governed creatures. This constrains the N_a that is at the world-root. For, post Hume et al, only at that level can the IS-OUGHT gap be soundly bridged. And we all know that after centuries of debate, only one serious candidate stands — just put up a viable alternative if you think you can. Good luck with that: ______ (Predictably, a fail.)
This is: the inherently good creator God, a necessary and maximally great being; worthy of loyalty and of the reasonable, responsible service of doing the good in accord with our evident nature.
But, what about, God is infinite?
I suggest, this first means that God is not externally limited or weakened so that he can be defeated or utterly frustrated in his purposes. Which, is among other things a way of saying that God is not evil, that being the privation, frustration or perversion of good capabilities out of their proper end.
God is also eternal and indestructible, as he has no dependence on external, on/off enabling causal factors. Thus, his being is without beginning or end, endless. This, being a characteristic of necessary being, which is required once a world is.
So, I think we need to reflect on what sense is meant when it is suggested that an actual infinity is impossible of being, and what are its strengths and limitations. For sure, an endless past of finite stage causal succession seems impossible and a physical, materially based infinite quantity is also dubious. But, the transfinite set of natural numbers and beyond the surreals great and small all seem necessary — framework to any world. Which in turn suggests mind capable of such a contemplation.
Food for thought. END