We’ve discussed infinity before, but since the subject in inexhaustible, we’re discussing it again.
Infinity comes in sizes, as many readers know. What’s less appreciated is how these sizes relate to questions in epistemology, physics, even theology. We’ll explore some of these facets in future articles. For now, some basics.
The first and smallest infinity is the set of natural numbers. This infinity is not small. All together now: Just how big is it? We are be tempted to say it is incomprehensibly big, and there is some truth to that, but the sticking point is that difficult word comprehensible.
Let’s see if we can get a feel for this tiniest of infinities. Count 1, 2, 3, and keep going. Never stop. Eventually we get to 1010, which is 10 billion. How far is that from the end? Infinitely far away. Take that billion and make it the exponent, i.e. 10billion. How far is that one from the end? Same answer: an infinite distance.
Now that might seem like a lot of zeros after the 10, but it’s a pittance, not even a drop. Exponents are not very handy for counting really large numbers, so let’s work with tetrations. They look just like exponents, except the superscript is on the other side. So 210 = 10^10^10, and 310 = 10^10^10^10, and so on.
We’re getting really big here. Consider B = billion10, which is 10^10^… a billion times. Big number! Bigger than we will ever need for any counting of physical objects. But it’s still infinitely far from the end. Next try BB = B10, which is 10^10^… not just a billion times, but B times. This is so huge it can’t be well thought of. But it’s still tiny compared to the tiniest infinity.
Well, we can keep going, BBB = BB10, BBBB = BBB10, et cetera. Do that B times, then do it BB more times, then BBB more times, and then…you get the idea. You’ll eventually stop, and come to a finite number that is beyond anybody’s ability to grasp (if B isn’t already). But whatever this number is, it is still infinitely far away from the smallest infinity.
What I’m trying to imbue in you is an appreciation how mind-bogglingly big the first infinity is. The number is so large that numbers less than it are all we would ever need if we’re interested in counting (and, yes, many mathematicians call this infinity a number).
After this “simple” counting infinity probably comes what are called, in a playful fit of whimsy, real numbers. I say “probably” because nobody knows for sure if there is another kind of infinity between the counting kind and the so-called continuum, where the reals live. That there is no differently sized infinity between the counting numbers and the reals is called the continuum hypothesis, which most believe is true, but nobody knows how to prove.
Anyway, one way to think about the reals, is to take any two counting numbers, like 1 and 2, and imagine stuffing an infinite number of numbers in between. Count these “stuffing numbers” however you like. Then take any two of these next to one another, still inside 1 and 2, and stuff another infinity of numbers in between them. Like 1.1000000001 and 1.1000000002, and stuff an infinity between them. And keep doing this for any two successive numbers.
You can go on packing numbers into the gaps like this until you come to a point where you have formed a dense flood of numbers, the succession infinitesimally increasing.
The problem, which may be obvious to you, is that if you’re not careful, this infinity doesn’t seem as big as the counting infinity. That’s because it’s impossible, at least for me, to envision what an infinitely dense succession of numbers look like, when I can’t even tell you what BBBBBBBBB10 looks like. Best I can do is to tell you that the number of natural numbers between 1 and BBBBBBBBB10 is infinitely smaller than the number of reals between 1 and 2.
Strangely, counting big natural numbers is hard, yet working with reals is easy. That ease produces in mathematicians a sort of hubris, or rather, forgetfulness. We’re so used to calculating with reals that we forget just how impossibly large the continuum is. The forgetfulness arises when we try and apply real-number equations to things in existence. Are there any actual objects that correspond to the continuum? Depends on what you mean by “actual.”
Skip that question—for now—and think of this. As big as the natural counting infinity is, and as infinitely larger are the reals, there are more infinities larger still. They are comprehensible only in the sense we know they exist, and because we know minimal things about them, such as their ordering. But I don’t think anybody grasps what these numbers are really like. Not when we can’t even say what billion Bs tetrated to a billion BBBB10 is like.
So how many sized infinities are there? And what might this have to do with a proof of God’s existence?
Great question. We’ll do that another time. For those who are adept at math and want to read more, I recommend the paper “Infinite Sets and Infinite Sizes” by the very aptly named Gary Hardegree.
“So how many sized infinities are there? And what might this have to do with a proof of God’s existence?”
Quick question regarding this, insofar as it relates to infinite regression and the origin of the universe… am I right in thinking that it’s never possible to be the “infinitieth in line”? I ask this because as I understand it, the idea with an endless succession of universes is that we happen merely to exist in the latest or most recent universe in that endless succession, which to my mind means we would be the infinitieth – a nonsensical idea.
“That there is no differently sized infinity between the counting numbers and the reals is called the continuum hypothesis, which most believe is true, but nobody knows how to prove.”
This is not quite correct. The continuum hypothesis (CH for short) is independent of ZFC (results of Goedel and Cohen) and independent in a very strong sense. Roughly speaking, using Cohen’s forcing technique, starting from a model of ZFC + CH, one can always get a model of ZFC + not-CH. In fact, there are very few constraints that ZFC puts on the cardinality of the continuum, it can be pretty much anything and virtually every question on the cardinality of power sets (more generally, questions on cardinality exponentiation) is independent of ZFC. Some mathematicians have advanced axioms that decide the question one way or another (Woodin, Foreman, etc.), but they are far from self-evident and as I mentioned, the plurality and richness of set-theoretic model constructions makes the whole question extremely hard to decide and even dubious whether it is a sensible question.
As far as the opinion of most mathematicians, although I do not have any hard numbers, I am positive that it is *not* true that “most believe is true”; on the contrary, my impression is that Set-theory experts either believe it is false or that there is no fact of the matter on the issue, meaning that that there is no standard “true” model of Set theory (as opposed to arithmetic) which we could get hold of and answer the question.
Let us imagine a set, M, each element of which is an infinite set, N. So M{N} is a set of a single infinity, corresponding the the infinity of counting numbers. M{N,N} corresponds to the infinity of the reals. M{N,N,N} corresponds to the infinity of the complex numbers. It follows that there must be infinitely many subsets N in the set M{N,N,N,N,…}
Let us now imagine a set of sets, L, such that each subset is a variant of set M. So we have L{M}, L{M,M}, and so forth. This corresponds loosely to a two dimensional matrix of sets N.
Let us now imagine a set of sets, K, such that each subset is a variant of set L. So we have K{L}, K{L,L}, K{L,L,L}, and so forth. This corresponds loosely to a three dimensional matrix of sets N.
Continue ad infinitum, adding layers of nesting or matrix dimensions. Having reached infinite dimensions or layers of nesting, call this N’. Repeat the entire process ad infinitum, and so forth.
I don’t have any real comments on the body of text, but the title is fantastic.
the continuum hypothesis, which most believe is true, but nobody knows how to prove.
Ah, but we know how to prove that we cannot prove it!
Interestingly, if we use ZF+AC (Axiom of Choice), we cannot prove CH; but if we use ZF+CH, we can prove AC.
But Aquinas (and Aristotle) use infinity in a sense different from the numerical.
https://www.uibk.ac.at/philtheol/tapp/publ/tapp_a32_2016_infinityaquinas.pdf
Infinity in mathematics is an exercise in misleading by omission.
Let’s say some author claims that A = (1,2,3,4,…) is expressing an infinite set. The omission here is: proof. Having no proof is same as doing no maths.
Had they done their homework, it derived them B = ((1),(2),(3),(4),(…)). This is plain maths, the method of doing things correct and show that possibilities given by maths rules have been exhausted.
But B has finite number of elements, 5.
Contradiction. Q.E.D.
Chaeremon – I’m not sure if that’s an example of high wit or low intelligence. The internet needs a sarcasm font.
@McChuck, what surprises me [not] about the empty statement is the missing (omitted) reproach for: who was cited.
In the past I have bemused my tiny mind and amused my incredulous sheep by trying to explain the mystery of infinity. I eventually satisfied myself with the notion that there are two “types” of infinity (the sheep did not appear to be convinced though. Perhaps they were right).
First there is the transcendental perfection of everything; beyond which there is no more of anything. The perfect, indivisible essence of being.
Then there is the infinity of increments. (The infinity that enslaves the mathematicians.) The infinity of perfection can never be attained by any succession of increments (by addition, multiplication, or division). No matter how far you incrementally progress by observation, deduction, induction etc. there is always more. No matter how finely each increment is divided there is always more nuances within.
That is how, I contend, a being limited by time (successions of events) could spend all eternity getting to know the perfection of everything without ever being “bored” with nothing more to do.
That brings us around to the doctrine of the Trinity but I think it best left for another discussion.
@Oldavid, Re: “No matter how finely each increment is divided there is always more nuances within.” One is inclined to agree.
But there are cusps, some are easy & some are hard to find, e.g.: if all you can see around you is beautiful, the most beautiful of all is x.y.z. [put the cusp in here, should be easy].
I am contending, Chaeremon, that cusps are fractal.
@Oldavid, you might want to try easier, no need to restrict beautifulness to maths objects at the outset – they will be contained as all other objects are.
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