Theology and the Actually Infinite
One of the common claims which is utilized in arguments for the existence of God is that actual infinities cannot exist, implying that there cannot be an infinite regress of causal events in the history of the universe. If there cannot be such an infinite regress, then there must be some First Cause. Theologians then put forth other arguments attempting to show that this First Cause must be God. Blake Giunta, a Christian apologist, has constructed a very interesting and quite useful website cataloging common lines of argumentation from both sides of the debate (color coded Green for Christian arguments and Red for opposing arguments), along with citations and documentation for those claims, called BeliefMap.org. It does not take very long for a fairly cursory perusal of Belief Map to bring one to this exact claim regarding the actually infinite.
While I disagree with Mr. Giunta on many of his views, I have a great deal of respect for him and I think that his work with Belief Map is absolutely fantastic. He truly does attempt to give an irenic and charitable view to the positions of his opposition, and he does sincerely want to discuss the actual arguments being made, instead of being content to knock down Straw Men. To that end, I would like to help Mr. Giunta add to his encyclopedia of apologetics by addressing the manner in which one might answer the claims about actual infinities.
Under the heading, “Logically, prior events can’t number to infinity,” Belief Map separates the discussion into two further, green claims. The first of these is, “Infinity can’t exist in the real world,” which is further subdivided into three green categories and four red. Two of the reds are theological in basis, and not of much concern to me, but the other two are mathematical and interesting. Each of these red categories contains a minor rejoinder, so I’ll be addressing them as best I can, as well. The second claim under the “Logically, prior events can’t number to infinity” heading is that “Infinity can’t be formed by adding.” After discussing all of the “can’t exist” categories, I will then consider this one.
Infinity – Infinity = Contradictions
The problem with this argument is that it is not even cogent. Infinity is not a number. There are, classically, two ways in which “infinity” is discussed as a concept in the philosophy of mathematics (Katz, 45-50). The first is the “potential infinity,” which is the idea that an iterative process can be repeated without any apparent bound. In this case, “infinity” is a description of the manner in which a process is carried out, and certainly not a number. The second way in which the concept is discussed is the case of “actual infinity,” which is the idea that a completed set can contain a number of elements which is greater than any Natural number. In this case, “infinity” is not a number, but rather a quality of numbers– a number can be either “finite” or “infinite.” And just as there are a multitude of finite numbers, some of which are greater than others, there are similarly a multitude of infinite numbers, some of which are greater than others (Katz 795; Conway; Robinson).
Numerical operations can only be performed upon numbers. For this reason, the expression “Infinity – Infinity” is entirely incoherent. It is no mathematically different than saying “Red – Red” or “Delicious – Delicious” or “Blake Giunta – Blake Giunta.” These are not mathematical statements, and as such, we cannot draw mathematical conclusions from them.
If the argument is amended to discuss the subtraction of infinite numbers instead of the subtraction of infinity, it loses all weight. There exist systems of mathematics in which infinite numbers can be subtracted from infinite numbers perfectly consistently– for example, on Surreal numbers (Conway) or Hyperreal numbers (Robinson). These do not lead to the purported contradictions espoused by apologists.
Infinities yield metaphysically impossible scenarios
Belief Map offers three cases as examples of metaphysically impossible scenarios: an infinite tug of war, an infinite hotel, and an infinite popsicle.
The infinite tug of war is actually just a restatement of the question of subtracting one infinite number from another, which we’ve already discussed. It makes precisely the same mistake as before, treating “infinity” as a number and not recognizing that there are numerous infinite numbers, not all of which are equal. As such, it is therefore easily resolved by proper mathematics.
The infinite hotel illustrates a counter-intuitive property of actual infinities, but it does not illustrate a metaphysical impossibility or a contradiction. The only way one might legitimately claim that this is an absurdity would be to already reject the possibility of actual infinites. However, since this is being utilized as an argument in support of just such a rejection, to do so would simply be fallaciously circular question begging.
The infinite popsicle does present something of an absurdity. I’ll agree that Bernardete’s scenario is metaphysically impossible, but not for the reason which he suggests. Popsicles are composed of atoms. Atoms have a significantly non-infinitesimal volume. As such, one cannot create a popsicle with an infinite number of layers in 4 cubic inches (or any other finite volume) of space. This thought experiment doesn’t work, but not due to any metaphysical absurdity relating to infinity.
Proper parts always contain less than wholes
This bald assertion is an unfortunate bit of question begging. One of the primary definitions of an actually infinite set is a set which contains a proper subset of equal cardinality (Katz 792-795). Belief Map offers no good reason to accept the claim that proper parts always contain less than wholes. One might as well argue that actual infinities can’t exist because actual infinities can’t exist.
Actual infinities are consistent in Set Theory
Belief Map cites this as an objection to the claim that actual infinities cannot exist, and it is absolutely correct to do so. For more than 100 years, mathematicians have been developing and utilizing a valid and consistent framework for math which deals perfectly well with actual infinities.
However, Belief Map offers a rejoinder to this: “But so what? A concept’s being logically possible (free of formal contradictions) doesn’t entail that it is actually/metaphysically possible.” Certainly, I agree– though, I must say, I find a little bit of irony in this position being raised here, since I quite often see the exact same sentiment brought up by atheists in regards to God’s possibility.
That said, at best this argument is merely inconclusive. It does not say that actual infinities aren’t metaphysically possible, but only that they might not be. In 1925, David Hilbert addressed this precise line of argumentation:
Also old objections which we supposed long abandoned still reappear in different forms. For example, the following recently appeared: Although it may be possible to introduce a concept without risk, i.e., without getting contradictions, and even though one can prove that its introduction causes no contradictions to arise, still the introduction of the concept is not thereby justified. Is not this exactly the same objection which was once brought against complex-imaginary numbers when it was said: “True, their use doesn’t lead to contradictions. Nevertheless their introduction is unwarranted, for imaginary magnitudes do not exist”? If, apart from proving consistency, the question of the justification of a measure is to have any meaning, it can consist only in ascertaining whether the measure is accompanied by commensurate success. Such success is in fact essential, for in mathematics as elsewhere success is the supreme court to whose decisions everyone submits. (Hilbert)
The mathematics of the infinite has been successful– inordinately successful, in fact. It forms the basis upon which mathematics has been securely founded. Given that the previous arguments on Belief Map aren’t very convincing– or even coherent, in cases– I see no reason to think that actual infinities are not metaphysically possible.
Intervals contain infinite subintervals
This is another good objection to the impossibility of actual infinities. If we are discussing an interval with cardinality of at least 
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Belief Map’s rejoinder to this is to claim that such intervals can only be potentially infinitely divided, and are not actually infinitely divided. However, this seems to very clearly not be the case. All of the subintervals on the interval 
Now, perhaps Mr. Giunta might respond that he has already granted that such mathematical intervals may reasonably be consistent, but that he is arguing against physical intervals, and that these are only potentially infinitely divisible. However, this seems to be yet another bit of question begging, and is only a reasonable assumption if one already denies the metaphysical possibility of actual infinities. After all, if (for the sake of thought experiment) we adopt the assumption that actual infinities are metaphysically possible, then all of the infinite subintervals of a given, physical interval would be coextant with that interval.
Infinity can’t be formed by adding
Since adding a finite number to a finite number always results in a finite number, Belief Map argues that an infinite collection cannot be formed by the sequential addition of finite elements. However, this seems to be just another circular attempt to reject actual infinities by rejecting actual infinities. No one is suggesting that adding a finite number to a finite number will yield an infinite number. We are suggesting that adding a finite number to an infinite number yields an infinite number, and similarly that adding an infinite number to a finite number yields an infinite number.
To be fair, Belief Map does note that a possible objection to this claim is that past events may “have always been infinite in number.” Of course, if it is the case that past events are infinite in number, then it must be the case that past events have always been infinite in number. This is a necessary consequence of the infinitude of past events. It would therefore seem that the claim that “infinity can’t be formed by adding” is entirely irrelevant to the situation under discussion.
From time immemorial, the infinite has stirred men’s emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, no other concept needs clarification more than it does. (Hilbert)
Belief Map, unfortunately, has some ill-formed views regarding the nature and mathematics of infinity. This is owed, at least in part, to the fact that Mr. Giunta borrows heavily from William Lane Craig’s work in the discussion of this subject. However, as I have discussed before, William Lane Craig has a gross misunderstanding of the concept of infinity (Part 1 and Part 2). Hopefully, the information which I have presented here can help Mr. Giunta to improve his wonderful work and correct some of the misconceptions which Belief Map’s arguments present in regards to the actually infinite.
WORKS CITED
Conway, J. H. On numbers and games. A.K. Peters, 2006.
Hilbert, David. “On the Infinite.” 1925. URL: https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html
Katz, Victor J. A History of Mathematics: An Introduction. Pearson, 2018.
Robinson, Abraham. Non-Standard analysis. North-Holland Pub., 1974.
Boxing Pythagoras 
“The second way in which the concept is discussed is the case of “actual infinity,” which is the idea that a completed set can contain a number of elements which is greater than any Natural number. In this case, “infinity” is not a number, but rather a quality of numbers– a number can be either “finite” or “infinite.” And just as there are a multitude of finite numbers, some of which are greater than others, there are similarly a multitude of infinite numbers, some of which are greater than others (Katz 795; Conway; Robinson).”
This is really confusing to me. On the one hand, you say that an actual infinite is an IDEA. On the other hand, you’re saying that it’s the idea that a complete set contains a NUMBER of elements greater than any natural number. So, I guess you’re saying it’s an idea, not a number —- even though it’s an idea about a number?
THEN, you’re saying that it’s a QUALITY of numbers, NOT a number itself. But then you say a NUMBER can be INFINITE? Huh? And that there’s a MULTITUDE of infinite NUMBERS. I’m so lost. How on earth can an actual infinite NOT be a number, and yet there can be (1) a NUMBER that is INFINITE, (2) a MULTITUDE of INFINITE NUMBERS, and (3) a NUMBER of elements greater than any natural number????
Please be more precise with your language because it sounds to a laymen like me that you’re just contradicting yourself.
Great questions, Mr. Scollo! I’ll do my best to answer them clearly.
I like to use the concept of Even-ness to be a direct parallel of what I mean when I say that “infinity is a quality of numbers.” A number can be even, but a number cannot be Even-ness. Similarly, a number can be infinite, but a number cannot be infinity.
I think that you and I will agree that Even-ness is not a number; however, a number CAN be even and there exist a multitude of even numbers. I could quite easily say that I have two numbers, x and y, such that x<y despite the fact that both x and y are even. None of this sounds unreasonable, right?
In exactly the same way, I might have some numbers, X and Y, such that X<Y despite the fact that both X and Y are infinite.
As to precisely what is meant by an "infinite number" depends on the particular number system being used. However, in general, a number which is infinite has an absolute value which is greater than any Natural number.
Imagine, for a moment, I told you that I have some number, x, which is positive. Without having to do any calculations and without knowing anything more about the exact value of x, you immediately know that x is greater than every negative number. In exactly the same way, if I have some positive infinite number, X, I do not need to know anything more about the exact value of X to know that it is greater than any Natural number.
I hope this helps. If it's still not completely clear, I'll be happy to address any further questions!
“I like to use the concept of Even-ness to be a direct parallel of what I mean when I say that “infinity is a quality of numbers.” A number can be even, but a number cannot be Even-ness. Similarly, a number can be infinite, but a number cannot be infinity.”
Okay, but I’m pretty sure that’s what he meant. He’s talking about collections that are actually infinite. So, that seems nit-picky. But I follow you know. So, an actual infinite is a number. A transfinite number. A countable number, in some cases, I assume.
“I think that you and I will agree that Even-ness is not a number; however, a number CAN be even and there exist a multitude of even numbers. I could quite easily say that I have two numbers, x and y, such that x<y despite the fact that both x and y are even. None of this sounds unreasonable, right?"
Right.
"In exactly the same way, I might have some numbers, X and Y, such that X<Y despite the fact that both X and Y are infinite."
I mean, I guess it depends. I don't know what "<" means. Cardinality? Size? Quantity?
"As to precisely what is meant by an "infinite number" depends on the particular number system being used. However, in general, a number which is infinite has an absolute value which is greater than any Natural number."
Yea, that's why it's a transfinite number.
"Imagine, for a moment, I told you that I have some number, x, which is positive. Without having to do any calculations and without knowing anything more about the exact value of x, you immediately know that x is greater than every negative number. In exactly the same way, if I have some positive infinite number, X, I do not need to know anything more about the exact value of X to know that it is greater than any Natural number."
Yep.
"I hope this helps. If it's still not completely clear, I'll be happy to address any further questions!"
Okay. I just don't know why his argument is bad. I don't care if the mathematics has been inordinately successful. That's not his point. Craig agrees with this. Mathematically, it's fine. How does this have anything to do with metaphysical possibility? The 'imaginary number' point doesn't do anything for me. Use imaginary numbers all you want. In mathematics, that's great! I have no idea what it means to some 'real-world' counterpart to an imaginary number. But even if there were, I don't see the analogy between imaginary numbers and actual infinites. I can imagine some world where there's an actual infinite collection of stuff; I can't imagine a 'real-world' counterpart for imaginary numbers. The analogy just has to do with the logical/metaphysical possibility distinction. That's all.
"There exist systems of mathematics in which infinite numbers can be subtracted from infinite numbers perfectly consistently– for example, on Surreal numbers (Conway) or Hyperreal numbers (Robinson). These do not lead to the purported contradictions espoused by apologists."
I don't see the relevance of this at all. Great. Surreal and Hyperreal numbers can be subtracted and divided. Awesome. I think everyone agrees you can do this in the world of mathematics. This just ignores the metaphysical-possibility point. I really can't see how you avoid contradictions either way. Maybe I'm dumb. I admit. The only difference between Set Theory and these other numbers is that Set Theory prohibits subtraction and division of infinite sets. The contradictions come from actually subtracting/dividing in the real world. So, if the Hyperreals and the Surreals permit this, all the better for bringing about real-world contradictions/absurdities.
"This bald assertion is an unfortunate bit of question begging. One of the primary definitions of an actually infinite set is a set which contains a proper subset of equal cardinality (Katz 792-795). Belief Map offers no good reason to accept the claim that proper parts always contain less than wholes. One might as well argue that actual infinities can’t exist because actual infinities can’t exist."
Bald assertion? I'm sorry, but this is the intuitive position here. Yes, the part and the whole have equal cardinality. That's not the issue. It's size/quantity. You're being unclear here again. Did he mention this in terms of cardinality? If so, show me where. And further. The 'equal cardinality' thing is what leads to absurd results in the real world. I don't care what happens in the mathematical world. Do all you want there if the rules of mathematics permit it. That's not the argument. It's the metaphysical possibility point that's the argument. Do you know the difference between the two? If you do, I can't tell by what you've written.
"The infinite tug of war is actually just a restatement of the question of subtracting one infinite number from another, which we’ve already discussed. It makes precisely the same mistake as before, treating “infinity” as a number and not recognizing that there are numerous infinite numbers, not all of which are equal. As such, it is therefore easily resolved by proper mathematics."
Ha, ha. I just find this uncharitable. He means an actual infinite! And again, you say 'not equal'. Right! Spell out what you mean. Quantity/size, correct? Resolved by proper mathematics . . . right. That's not the point. You don't seem to understand mathematics possibility and metaphysical possibility.
"The infinite hotel illustrates a counter-intuitive property of actual infinities, but it does not illustrate a metaphysical impossibility or a contradiction. The only way one might legitimately claim that this is an absurdity would be to already reject the possibility of actual infinites. However, since this is being utilized as an argument in support of just such a rejection, to do so would simply be fallaciously circular question begging."
I want you to define 'metaphysical possibility' for me. Because it makes no sense to say I need to reject the possibility of actual infinites because I reject the metaphysical possibility of actual infinites. You're being imprecise again. What do you mean by "reject the possibility"? What do you mean by possibility? Please be more clear. I can reject it in one sense, and affirm it in the mathematical sense. Are you a mathematical formalist? If you are, that needs argument, or you're begging some questions.
"The infinite popsicle does present something of an absurdity. I’ll agree that Bernardete’s scenario is metaphysically impossible, but not for the reason which he suggests. Popsicles are composed of atoms. Atoms have a significantly non-infinitesimal volume. As such, one cannot create a popsicle with an infinite number of layers in 4 cubic inches (or any other finite volume) of space. This thought experiment doesn’t work, but not due to any metaphysical absurdity relating to infinity."
So, here you agree this is metaphysically impossible. I want you to define this please. You appeal to atoms. This isn't strictly metaphysical possibility, though it involves it. This is physical impossibility. It's a physical fact that atoms have the volume they do. So, please define metaphysical possibility for me.
My apologies! I honestly don’t know how I missed this reply. If you’ll forgive the many-months’ lapse, here is my reply.
No, an “actual infinite” is a collection. I understand that the numerous different uses of “infinite” can be a bit ambiguous, though.
A number is a means of describing properties of a collection. An infinite number can be used to describe properties of an actual infinite, a collection of more items than can be contained in any finite collection.
Though I’m sure we could construct a mathematical system in which there exists only one infinite number, I’m sure you’ll agree that we can discuss a cogent X<Y regardless of whether we are considering cardinals, ordinals, Hyperreals, Surreals, or any of a host of other consistent mathematical systems.
Usually “transfinite” refers particularly to the ordinals of set theory. I wouldn’t generally use the term to refer to, say, an infinite Hyperreal number.
The argument to which I was responding claimed that actual infinites are a metaphysical impossibility because they lead to absurdities. Since we have consistent mathematical systems for describing actual infinites, it is not the case that these concepts necessarily lead to absurdities. It has therefore not been demonstrated that actual infinites are metaphysically impossible.
Which contradictions do you think persist when considered by means of Surreals or Hyperreals?
If you are saying that the “size” or “quantity” of a set differs from the “cardinality” of that set, then what do you mean by “size” or “quantity?” How do you measure these properties?
At no point did I state or imply otherwise. Indeed, that is the prime assumption of my response, so I am more than a bit unclear as to what your objection actually is, here.
Again, the argument which was proffered for the metaphysical impossibility of actual infinites was the absurdities implicit in mathematics utilizing them. If there are no absurdities in the mathematics, then the argument fails to show that actual infinites are metaphysically impossible.
If I am understanding Belief Map and Dr. Craig correctly, “metaphysical possibility” means “it could be the case that such a thing exists in reality.” Both of these sources argue that actual infinites are NOT metaphysically possible. I see no reason to think that this is the case, and the arguments which they offer in that regard fall far short of demonstrating it.