# Can actual infinities exist in the real world?

“No, after all…

Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.1 This is relevant because if an infinity could be instantiated in the real world, then so too could these contradictions. (But in fact, contradictions can’t be instantiated, and so modus tollens infinities can’t either.)

1. William Lane Craig & James Sinclair: “For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic – a mere stipulation which has no force in the nonmathematical realm.” [The Blackwell Companion to Natural Theology (Blackwell, 2009), 112.]
• Infinities yield metaphysical absurdities

Infinities yield metaphysically impossible scenarios (absurdities)

For example, absurdities like… • …the infinity tug-of-war.
• …the infinite-room hotel.
• …the infinite rainbow popsicle.
This is relevant, because if an infinity could be instantiated in the real world, then so too could these impossible scenarios. (But in fact, these scenarios can’t be instantiated, and so modus tollens infinities can’t either.)

No, after all…
• …actual infinities are mathematically legitimate (in set theory).1 [See below]
So? Plausibly…
• …these only show certain kinds of infinity are impossible.2
1. Michael Martin: “…a priori arguments… show at most that actual infinities have odd properties … This latter fact is well known, however, and shows nothing about whether it is logically impossible to have actual infinities in the real world.” [Atheism: A Philosophical Justification (Temple University Press, 1992) 104-105.]
2. Graham Oppy: “At most, it seems that one might suppose that these puzzles show that there cannot be certain kinds of actual infinities; but one could hardly suppose that these puzzles show that there cannot be actual infinities of any kind.” [Arguing about Gods (Cambridge, 2006), 140]
• Proper parts always contain less than wholes

A part of the whole contains less than the whole (i.e. In reality, if M’ is a submultitude of M, then intuitively there are more things in M than M’). This is relevant because if infinities are possibly instantiated in the real world, then this intuitive proposition about the world is actually false.

• “No, after all…
• Infinity is consistent in Set Theory

“The notion of an 'actual infinity' is logically consistent/possible within Axiomatized Set Theory.”

But, so what? A concept's being logically possible (free of formal contradictions) doesn't entail that it is actually/metaphysically possible.1

1. A. W. Moore: “[Cantor] was adament throughout his life that the whole idea of an infinitesimal was demonstrably inconsistent,” [The Infinite (Routledge, 2001), (As cited by Craig)]
José Benardete: “Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face” [Infinity: An Essay in Metaphysics (Oxford, 1964), 238.]
Kasner, E & Newman, J.: “‘Existence’ in the mathematical sense is wholly different from the existence of objects in the physical world … the infinite certainly does not exist in the same sense that we say, ‘There are fish in the sea’” [Mathematics and the Imagination (Simon & Schuster, 1940),61]
William Craig & James Sinclair: “…mathematical legitimacy of certain notions does not imply an ontological commitment to the reality of various objects. … Cantor’s system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments. … On antirealist views of mathematical objects such as Balaguer’s ﬁctionalism (Balaguer 1998, pt. II; 2001, pp. 87–114; Stanford Encyclopedia of Philosophy 2004b), Yablo’s ﬁguralism (Yablo 2000, pp. 275–312; 2001, pp. 72–102; 2005, pp. 88–115), Chihara’s constructibilism (Chihara 1990, 2004; 2005, pp. 483–514), or Hellman’s Modal structuralism (Hellman 1989; 2001, pp. 129–57; 2005, pp. 536–62), mathematical discourse is not in any way abridged, but there are, notwithstanding, no mathematical objects at all, let alone an inﬁnite number of them. The abundance of nominalist (not to speak of conceptualist) alternatives to Platonism renders the issue of the ontological status of mathematical entities at least a moot question. The Realist, then, if he is to maintain that mathematical objects furnish a decisive counterexample to the denial of the existence of the actual inﬁnite, must provide some overriding argument for the reality of mathematical objects, as well as rebutting defeaters of all the alternatives consistent with classical mathematics – a task whose prospects for success are dim, indeed. It is therefore open to the _mutakallim _to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world.” [The Blackwell Companion to Natural Theology (Blackwell, 2009), 107-108.]
• Intervals have an infinity of subintervals

Any interval contains an infinity of subintervals (e.g. a meter and minute can both be divided in half an infinity of times).1, 2 This is relevant because if there are an infinity of sub-intervals inside any interval, then an actual infinity of subintervals must exist.

But plausibly... intervals can only potentially infinitely be divided (i.e., divided, then divided again, then again, with no end), so it is not an actual (completed) infinity. That is to say, the interval is not comprised of an infinity of point-parts or divisions; there is not infinitieth cut. Instead, the cuts depend on the interval (i.e. the interval is logically prior to any potential infinity of divisions/points that we continually impose on it with our conceptual dividings).
1. I.e., allegedly, a) Between any two numbers, there are an infinity of other numbers. b) Between any interval of time, there are an infinity of instants. c) Between any interval of space, there are an infinity of points. It's worth noting that the latter two require that space/time be continuous (such that it could be divided infinitely many times), which is controversial.
2. Walter Sinnott-Armstrong: “When your hand moves a foot…, it goes through an infinite number of intervening segments: half, then half of that… and so on.… These areas of space and periods of time really exist, regardless of our limitations and actions.” [God? A Debate between a Christian and an Atheist (Oxford, 2003), 43.]
• God is infinite

God is an infinite being. This is relevant because if God is an infinite being, then God's existence entails the existence of an actual infinity.

But wait, when we say God is infinite, we are ascribing a qualitative attribute to God (e.g. God is maximally good and powerful). We are not ascribing a quantitative infinity.
• God's omniscience requires infinity

The extent of God's knowledge would be infinite (given God exists and is omniscient). So if actual infinities are impossible so is God.1

But so what if the extent is infinite? Couldn't it be that God's knowledge is metaphysically "simple", akin to a map, which isn't inherently propositional in form, but which can nevertheless be _represented _propositonally (e.g. city A is 3 miles from city B;one can get there via route xyz, etc.).2
1. Graham Oppy: “we cannot then say either that an orthodoxly conceived monotheistic god is, or that an orthodoxly conceived monotheistic god’s attributes are, actually inﬁnite.” [Arguing about Gods (Cambridge, 2009), 139.]
2. William Lane Craig & J. P. Moreland: “…some thinkers such as William Alston, while rejecting complete [divine] simplicity, have advocated that God's knowledge be construed as simple. On Alston's view God has a simple intuition of all of reality, which we human cognizers represent to ourselves propositionally. Such a view is in line with Aquinas's…  contending that God does not, strictly speaking, have a plurality of Divine Ideas but rather an undifferentiated knowledge of truth. We finite knowers break up God's undivided intuition into separate ideas. Similarly, Alston maintains that God's knowledge is strictly non-propositional, though we represent it to ourselves as knowledge of distinct propositions. Thus, we say, for example, that God knows that Mars has two moons, and He does indeed, know that, but the representation of His knowing this proposition is a merely human way of stating what God knows in a non-propositional manner. Such a conception of divine knowledge has the advantage that it enables us to embrace conceptualism without committing us to an actual infinite of divine cognitions or Divine Ideas.” [Philosophical Foundations for a Christian Worldview (IVP, 2003), 526.] Elswhere Craig writes “when we say that God knows an infinite number of propositions, we are speaking of the _extent _of His knowledge, not the mode of His knowledge” [“Does God Know an Actually Infinite Number of Things?” Online at ReasonableFaith.org] and cites a previous publication where he elaborates on his view further: “…propositions are the byproduct of human intellection and so merely potentially infinite in number, as we come to express propositionally what God knows in a non-propositional way.”