Let's assume for the sake of the argument that 'finite' is not included in the definition of 'person' (henceforth also called 'subject', so that it may also imply supernatural beings), so it does not figure a logical contradiction. If an infinite subject is that which encompasses the whole universe, it is metaphysically possible that this subject exists. If by infinite however we mean something that spans not only its world but all worlds, then it is not metaphysically possible because we know at least one world which he does not span: ours. However, I would say that by then, the definition of infinite is twisted to mean something that actually reflects "necessary (in all possible worlds)", after all. — Lionino
Infinity itself is a tricky concept. I once started a thread on the topic and got brutally pummeled by less than charitable mathematical folk for the terminology I employed. So I’ll here employ different terminology for it.
We often enough think of infinity in terms of quantity. Even Cantor’s Absolute Infinity, for example, is postulated to a number bigger than any other quantity. Going back to its roots, though, infinity signifies that which is devoid of limits and boundaries. In this sense then, to be bound, for example, to any aspect of space (distance), time (duration), or number (quantity) is to be finite, this rather than infinite. The Ancient Greek notion of Anaximander’s Apeiron gives one example of this latter notion of the infinite.
Most infinities we can think of are thereby in some respects bounded and only in some others unbounded. For example, Cantor’s Absolute Infinity is yet bounded to, and hence limited by, quantity. For the sake of convenience, I’ll here label these bounded infinities, or else the boundedly infinite—and contrast these to unbounded infinity, or else the boundlessly infinite. A line, a geometric plane in Euclidian space, and Cantor’s Absolute Infinity are all then bounded infinities; whereas the Apeiron is one example of that which is boundlessly infinite.
Given all that, what then does the phrasing of an “infinite subject” intend to imply?
I presume that “a subject” entails being a subject of awareness, which I further take to entail being aware of other. If so, all subjects are boundedly infinite in some regard—for one example, their/our awareness can, I think, be safely deemed infinitely divisible in principle. For instance, I see a rock + background: the rock has infinitely many aspects I could address in principle (variations in color, texture, curvature, etc.; relations between these; similarities and dissimilarities to other things I could be aware of; etc.)—given that I don’t get bored in so doing and that I would so address for all eternity. And this is not to yet get into the rock’s background.
We might by the just stated then affirm—in a rather bizarre formulation—that all subjects (of awareness) are thereby infinite at all times for as long as they occur. This on grounds that their awareness of other, by which they are principally defined, can only so be. (I might add, I can’t yet fathom of anything spatiotemporal that can’t be deemed to be boundedly infinite in at least one respect; as one example previously mentioned for an “infinite dog”, the mapping of anything physically spatial via geometric points leads to an infinite quantity of geometric points constituting whatever it is that is being addressed, thereby making the thing boundedly infinite. But if all items in a set of items are infinite in the same respect, this characteristic ceases to be any difference between them: an infinite apple then equals a plain old apple, so there then is no reason for the adjective of “infinite”.)
So no, when thus understood, an infinite subject would not be a logical contradiction—thereby allowing for metaphysical postulates wherein infinite subjects of awareness are interpreted to hold an awareness that encompass this world or that. (Don’t we as subjects of awareness in at least some ways encompass the very possible worlds we are aware of?)
But so construing all subjects of awareness to be infinite is intuitively odd. We typically by “infinite” want to convey something more. I venture that oftentimes (if not always) this something more will turn out to be logically contradictory: as I find to be the case with an omniscient, else omnipresent, else omnipotent subject (of awareness).
At any rate, the notion of “a boundlessly infinite subject” can only be a logical contradiction: at the same time and in the same respect that addressed is a) bounded to being a subject of awareness (entailed by being a subject) and b) is in no way bounded to being a subject of awareness (entailed by being boundlessly infinite). To use the same example I previously mentioned, the Apeiron cannot logically be a subject of awareness which thereby is cognizant of things which are other than itself—for, if for no other reason, this would then limit the Apeiron to so being a subject of awareness, which contradicts the Apeiron being perfectly limitless in all respects. (While I question Anaximander's notion of the Apeiron on logical grounds, another possible to conceive of boundless infinity is that of Nirvana without remainder, this among yet others, such as the Ein Sof; none of which are subjects of awareness that thereby dwell in a duality between self and other)
(In a bit of hurry at present so I’m not double checking the just written. If there’s glaring mistakes in it I’m sure someone will let me know.)