Can actual infinities exist in the real world?

  • Clarifying the question

    A hotel extends to the clouds. It has no bottom.

    While actual infinities may have some mathematical legitimacy (or even mathematical “existence”),1 is it metaphysically possible for an actual infinity to be instantiated in the mind-independent world?

    • Actual infinity (ℵ0) = def. An actual/completed infinity. A collection of definite and discrete members whose number is greater than any natural number 0, 1,2,3. (In set theory, a set with an infinite number of members: e.g. {0, 1, 2, 3…} or {2, 4, 8, 16…}.)
    • Exist = def. To be instantiated in the mind-independent world; “realized” or “actualized.”2
    • Mathematical existence/legitimacy = To be a legitimate mathematical notion (whether or not the entity has any non-mathematical existence).3
      • Edward Kasner & James Newman: “‘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 Lane Craig & James Sinclair: “Cantor called the potential infinite a “variable finite” and attached the sign ∞ (called a lemniscate) to it; this signified that it was an “improper infi- nite” (Cantor 1915, pp. 55–6). The actual infinite he pronounced the “true infinite” and assigned the symbol א0 (aleph zero) to it. This represented the number of all the numbers in the series 1, 2, 3, . . . and was the first infinite or transfinite number, coming after all the finite numbers. According to Cantor, a collection or set is infinite when a part of it is equivalent to the whole (Cantor 1915, p. 108). Utilizing this notion of the actual infinite, Cantor was able to develop a whole system of transfinite arithmetic.” [The Blackwell Companion to Natural Theology (Blackwell, 2009), 104.]
      • William Lane Craig & James Sinclair: “Historically, certain mathematical concepts have been viewed with suspicion and, therefore, initially denied legitimacy in mathematics. Most famous of these are the complex numbers, which as multiples of √–1, weredubbed“imaginary”numbers. ... [Mathematically legitimate objects are] “as 'real' as the real numbers” [The Blackwell Companion to Natural Theology (Blackwell, 2009), 104.]
  • Many experts say "no"

    • Edward Kasner, James Newman: “At the beginning of the twentieth century it was generally conceded that Cantor's work had clarified the concept of the infinite so that it could be talked of and treated like any other respectable mathematical concept. The controversy, which arises wherever mathematical philosophers meet, on paper, or in persons, shows this was a mistaken view. In its simplest terms this controversy, so far as it concerns the infinite, centers about the questions: Does the infinite exist? Is there a thing as an infinite class? Such question can have little meaning unless the term mathematical "existence" is first explained.” [Mathematics and the Imagination (Courier Corporation, 2013), 61.]
    • David Hilbert: “Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought.… The role that remains for the infinite to play is solely that of an idea.” [Philosophy of mathematics 2nd ed., Eds. Benacarraf and Putnam (Cambridge): Online.]
“No, after all…
  • Infinities yield contradictions

    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

    An optical illusion square with an aleph null inside of it.

    If infinities existed, they would entail the existence of metaphysically impossible scenarios (absurdities)

    For example, absurdities like…

    • The infinity tug-of-war demonstrates how we can pit infinities against each other, manipulating tuggers so-as to preserve the infinity on both the red team and the blue team but which would nevertheless obviously result in one side winning (e.g. imagine every other member of the blue team suddently letting go), and yet we are forced to say that in fact neither side gets an advantage at any point regardless of our drastic manipulations.
    • The infinite-room hotel, where a hotel with no vacancies can mysteriously obtain vacancies by performing contrived mathematical manipulations, and where we can subtract an infinity of guests in contrived ways so-as to have 1 guest remaining, or 2 guests, or any number we want, such that infinity minus infinity can be anything.
    • The infinite rainbow popsicle, where we could look at the top of a popsicle (with an inifinity of colored layers), and literally have no possible answer as to what the top color would be. Reality itself would not have an answer.

    These apparently metaphysical absurdities help us think about the topic, because if an infinity could be instantiated in the real world, then so too could these impossible scenarios. It is arguably easier to admit these scenarios can’t be instantiated, and so therefore 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
      • 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.]
      • 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]
  • Parts always contain less than wholes

    For any multitude in the real mind-independent world, and for any part of that multitude, the part of the multitude is necessarily smaller than the whole.1 That is to say, in the real world, if M’ is a submultitude of M, then there are more things in M than M’.

    After all, this is one of the stronger intuitions that humans cross-culturally and cross-generationally have had, with no good reason to change. Most would in fact call it common sense, to the point that any mathematical frame-work (e.g. the standard Zermelo–Fraenkel axiomized set theory) which assumes its falsity could not, in the end, be interpreted as a realistic representation of reality. (Note: unrealistic mathematical frameworks can still be interesting and useful, even if some features cannot correspond to reality itself.)

    The necessity of parts being less than wholes matters because, if infinities are possibly instantiated in the real world, then this simple truth about parts and wholes in the actual world has to be false. This is because actual infinities in set theory are such that a submultitude of a given infinity can be put in a 1-to-1 correspondence with its parent.2

    1. We of course mean a "proper" part, where a proper part is a mereological part that is not the whole. Some philosophers may want to say the dog itself is part of the dog, and so to be clear other philosophers to accomodate this focus on proper parts of the dog, e.g. the dog's tail but not the dog as a whole.
    2. For example, the natural numbers {1, 2, 3, 4...} can be put in a 1-to-1 correspondence the set of even numbers: {2, 4, 6, 8...}, where each member of the natural numbers is multipled by two. Despite the even numbers being a submultitude, both sets have the same number (cardinality) of members.
“Yes, after all…
  • Infinity is consistent in Set Theory

    An umbrella with an aleph-null below it.

    “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)—especially within an ad hoc system of rules designed to prevent contradiction, like subtraction's being disallowed in set theory—doesn't entail that the concept is actually/metaphysically possible.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…

    • The interval is not comprised of an infinity of point-parts or divisions (i.e. there is not infinitieth cut. Instead, the cuts depend on the pre-existing interval.)1
    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.
      • 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.]
    2. In fact, the Grim Reaper paradox may prove that time and space cannot be mere constructions out of an actually infinite number of points. If such an infinity were possible, the following impossibles scenario involving an infinity of Grim Reapers seems like it would have to be possible: Let “Reaper x is set to kill you” mean “If you are not already dead, Grim Reaper x will swing his scythe instantly resulting in your death, where x is that Reaper's number in the infinte series of reapers." Now...
      • Reaper 1 is set to kill you in 60 seconds.
      • Repear 2 is set to kill you in 30 seconds.
      • Reaper 3 is set to kill you in 15 seconds.
      • Reaper 4 is set to kill you in 7.5 seconds.
      • …on and on ad infinitum (i.e. there is no highest numbered reaper and therefore no first or earliest reaper). A contradiction results from the fact that you must die within 60 seconds, and yet you cannot die because there is no first reaper. (And yes, you need a first reaper, because otherwise a higher numbered reaper would have already killed you.)
  • 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”. (That is to say, couldn't it represent the world the way a map does, which is not inherently propositional in form, but which can nevertheless be described with a potential infinity of propositions [e.g. city A is 3 miles from city B; one can get there via route 1 or 2, or 3, etc., ad infinitum]).2

      • 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.]
      • 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] 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.”