The argument in a nutshell: the reality of any sort of real thing presupposes, prerequires, and supervenes the reality of the proper formal ultimate of its sort. E.g., 5 presupposes, prerequires, and supervenes infinity. No infinity → no 5. The reality of its proper formal ultimate is logically implicit in any real. There are reals. So there is a real Ultimate.
Along every dimension of formal configuration spaces there is some ultimate; some value that is its outmost bound, and that helps to define it as a formal dimension. Were it otherwise, the dimension would be inadequately specified, and so could not operate as a real formal dimension. For, if a formal dimension is not itself limited, it cannot limit; so, it cannot form. In that case, it is not a form to begin with. It is rather nothing at all.
Consider for example a dimension consisting of the set of natural numbers from 0 to n. Is 5 a member of that set? No way to tell, unless n is specified. Thus “the set of natural numbers from 0 to n” is not a definite set at all. It is a string of characters with no definite denotation, and thus with no meaning. It is noise.
Excursus: Implicit in this consideration is that the Logos has a complete and coherent logical structure, along all dimensions; or else, he is not a logos in the first place, let alone the Logos himself; but rather just a mish mash; a chaos; not a he; not even an it; not at all.
But obviously there must be a Logos, for we see formed things all about us, and we are ourselves formed things.
If there is no form of things, then there are no things; for, to be a thing is to be in a certain specific way; to be is to be formed.
There are things. So is there is a form of things.
There being a form of things, ergo there must be a former of things, a conditioner; a creator. For, the former of things is implicit in the form of things. No former, no formation. A form that does not form is not a form. Ideas don’t have themselves; so likewise forms don’t themselves form. No former, no form.
But I’m getting ahead of myself.
In respect to the most tractable, familiar sorts of formal dimensions, such as those that are characterized by maxima (e.g., the possible interior angles of a triangle), it is easy to see that for each formal dimension there must be some ultimate (viz., you can’t adequately specify triangularity without implicitly specifying the limit of the interior angles of triangles). There seems to be a definite maximum of velocity within our cosmos, e.g.; which is to say, a maximum rate of causation (this maximum being implicit in the minimum rate of causation (in a coherent cosmos, could it be otherwise?)). But even when that outmost bound of a formal dimension is itself boundless, as is the case with mathematical infinity, there must be such an outmost bound. And infinity is just such a bound. Boundlessness then is a precise specification of the outmost bound of some sorts of form.
Excursus: How big can a given sort of thing be? In the absence of other constraints, such as the bearing capacity of bone per pound thereof (a limit on the size of terrestrial endoskeletal animals), there would seem to be no such limit. A thing might be infinitely extensive. Indeed, never mind this or that thing in this or that extensive cosmos; there might be an infinitely extensive system of extensive systems, e.g.; secula seculorum might not be finite. This is to say that the thing constituted of the created order in toto might not be finite.
If the creator is infinite, it is rather hard to see how his act of creation might possibly be finite.
But, again, I’m getting ahead of myself.
Given any thing, the set of which it is a member must be real – not itself concretely actual qua set, mind you, but itself real, at least formally – a real set, that is to say – or there could be no actual or formal instance of it, nor therefore of any of its members. No x, no instance of x.
Here, then, is the argument.
Along every properly specified formal dimension, there is necessarily a thing than which there is no greater along that dimension. Infinity along all dimensions capable thereof, e.g., is then the indispensable forecondition of any lesser value of any such dimension: ∞ ∋ x; ∴ ¬∞ → ¬x. By extension, the reality of the maximum of x along any formal dimension (whether it is capable of infinity or not) is the forecondition of the reality of any x along that dimension.
It does not matter whether we are arguing about actual infinity or potential infinity. Either sort of infinity might be real. It is that reality we are concerned to demonstrate.
- ∞ ∋ x; ∴ ¬∞ → ¬x
So likewise for all sorts of ultimacy.
The concrete actuality of any subultimate entails the concrete actuality of its proper formal ultimate. Likewise, the merely formal potentiality of any subultimate entails the merely formal potentiality of the ultimate. But, NB: a merely formal ultimate would be subultimate to its concrete actual. For, since actuals must all have forms, actuality is ipso facto also formality; whereas potentiality is only formality. So, actuality is greater than potentiality.
The actual infinite is ultimate to the merely formal infinite; so the merely formal infinite presupposes, prerequires and supervenes the actual infinite.
Thus a merely formal ultimate could not be obtained except insofar as it had been actualized concretely. Then even to obtain the mere form of 5, e.g., and whether or not there are actually five of anything, the proper formal ultimate of numbers such as 5 – infinity – must be concretely actual.
If there is anything at all, then, the Ultimate than which no greater can be conceived is therefore necessarily real and actual.