[See part 5 here. All parts may be found here.]
The Intrigues of the Infinite are coming to play. Whether you are confused by the infinite or have some logical grasp of it, I’m going to take you on a ride through a few of those intrigues of the past. Hold on to your logical pants.
Few now appreciate the way religion was embedded in science, philosophy and yes, even mathematics, twelve decades back in the Western World. Certainly there were contrasting views between religion and science. Two examples suffice. The neo-Thomism of the latter half of the 19th century was a revival of St. Thomas’s Aristotelian theology. In neo-Thomism, God is seen as absolute perfection and distinct in substance from all other beings (which are “finite” – hehe). God’s knowledge includes all of physical and mental history, everything that will be, and everything that may be. The Thomists saw God as absolute infinity. Contrary to the free will postulates of the Thomists, determinism had a strong foothold in science and philosophy. But it (determinism) was seen by most of its adherents as companionate with Christianity and Judaism in particular. The point is that most thinkers of the time held their religion not just along side their disciplines, but saw it as deeply connected to them with a context that required due care. Moreover, this caution was usually evident whether the individual was a believer or not.
Georg Cantor, Catholic mother, Protestant father, Jewish heritage.
Into this era came G. F. L. P. Cantor. Georg Cantor was born in St. Petersburg, Russia roughly one year after the first King Follett discourse. Cantor would generate both an intellectual paradise and a thoughtful hell as he angered many of the big guns in religion, science, mathematics and philosophy. In other words, he caused fun. In the present world of “working” mathematics and science, much of Cantor’s thinking is a part of what we do. We take it for granted.
But now, drift back with me to the last quarter of the 19th century and the University of Halle (Germany). Here, Cantor became a full professor at the youthful age of 34. Cantor began his academic career in number theory[1] and his later ideas were connected to that field.
After Cantor finished his habilitation[2] he rolled out a theory that strained his relationship with his old Ph.D. advisor and brought accolades and criticism from all quarters. To understand what the fuss was about, we will look again at what it means to be infinite. After we get some sea legs I’ll come back to Cantor and his interesting psychological trajectory in a later post. We’ll see the Thomists again.[3]
If we are measuring size, it is a matter of counting and counting is just a matching exercise. Match numbers to the number of cows that pass the gate for example: one, two, three, . . . 25. 25 cows came through the gate. Our ordinary experience prepares us for such things. But when the number of objects becomes too large, the process becomes less meaningful. Scriptural accounts that suggest certain things are just too large to comprehend can be understood on several levels. Whether they entail the infinite is an open question – for example the question I’ve already mentioned: Is the universe infinite? But even questions like “How many moons does Jupiter have?” and “How many water molecules are in a cup of water?” are not just different in scope, they are different in meaning. Abstraction and approximation are the only ways to deal with the second question. (The “answer” is about 8 x 1024. Ten to the 24th power is so large that we can only deal with it as an abstraction. It is certainly finite, even though you will never “count” to it – try starting now. You should finish in about one quadrillion (US meaning) lifetimes – provided you can count in your sleep. Something like being in SteveP’s hell.)
In part 1, without being too explicit about it, we separated the concept of the infinite into two categories. We might call them the qualitative infinite and the quantitative infinite. We’ll eventually get back to thinking about qualitative usage but for now, let’s think quantitative. “Size” is the question we tackle first. I will continue that in the next couple of posts.
Can, or *should* one consider collections of things which are infinite in size? Partly the question is one of context. We may be in an infinite universe, but what of thinking about the infinite in an exterior way? Should we consider such a universe as an object? This seems a necessary feature of the Divine, does it not? But it led to challenges for some versions of classical theism.
We know from part 1 what it must mean for a collection of things to be infinite in size. An example would be the collection of “counting” numbers: {1, 2, 3, . . . }.[4] In this case the “. . .” is a comforting euphemism for “everything else after 1, 2, 3”! The idea that one should try to deal with non-finite collections of things is fraught with logical peril. But we shall be brave, or foolish, depending on how your philosophy has been or will be shaped by the infinite. The difficulty begins to rear its head when we try to push our understanding of size for finite collections or sets of things, to infinite sets or collections.
How big *is* {1, 2, 3, . . .}? We can just say infinite, and drop it at that. But measuring finite sets was a correspondence game. Counting the cows involved simply matching numbers with cows as they come through the gate:
1 <-> cow1
2 <-> cow2
.
.
.
25 <-> cow25
The problem with {1, 2, 3, . . . } appears when we try to count it:
1 <-> 1
2 <-> 2
3 <-> 3
.
.
.
We never run out of things to count. On the other hand, does everything in {1, 2, 3, . . .} get counted by this matching? The answer is clearly, YES. You can’t name a number that fails to be matched in this scheme. We apparently had to use the whole of {1, 2, 3, . . .} to get a complete match (with itself).
So that I don’t have to keep writing {1, 2, 3, . . .} I will just use the symbol N for it. Saves ink.
Next, observe that we can match the whole of N with other things. Such as the numbers that are multiples of 5: {5, 10, 15, 20, 25, . . .} Consider:
1 <-> 5
2 <-> 10
3 <-> 15
4 <-> 20
5 <-> 25
etc.
This matching rule is obvious, and everything gets matched. 1024 for example gets matched with 5 x 1024. Think what this means: the multiples of 5 form an infinite collection, *exactly the same size as N.* But of course, the collection of numbers which are multiples of 5 is completely and *properly* contained in N. There is clearly more stuff in N. That’s odd. But for things of infinite size, the principle of measure seems to work a bit differently than it does for sets of finite size. Things that appear to be larger in one sense, may fail to be so in the “counting” sense.[5]
Next up: *really* infinite stuff and what it might mean. (Part 7 is here.)
——————
Sophie Germain, courtesy Archives de l'Acadmie des Sciences.
[1] You may have heard of Princeton Professor Andrew Wiles and his colleague Richard Taylor’s 1995 proof of Fermat’s Last Theorem. This is a result in number theory and the solution to a 358 year old problem. It made the mainstream media, an unusual thing. (Fermat’s theorem seems to have the distinction of being the result with the greatest number of incorrect proofs in print – and a rather tragic association with prejudice: a woman, Sophie Germain, had to pretend to be male so she could work on the problem 200 years ago).
[2] In Europe up until a few years ago, advanced education was still not complete with the Ph.D. – a second thesis consisting of independent research from the Ph.D. thesis (and advisor) had to be submitted to obtain the final degree, allowing one to supervise Ph.D. students. This system seems unwieldy and is probably heading toward a US-UK type thing.
[3] The Catholics weighed in on both sides of Cantor’s paradise. Some saw it as proof of transcendence. Others as proof that God was not, could not be, top dog.
[4] It has been argued that we should regard the counting numbers as only “potentially” infinite. You just make larger numbers when you need them. The platonists don’t care for this. This “finitism” may be satisfactory for some purposes, but may be inadequate to discuss Mormon theology for example. We’ll come back to this Aristotelian position later. The fact that the universe may be infinite has interesting consequences for these holdouts. I chuckle, I chuckle.
[5] The size of N, measured by using N itself(!) is usually designated by the Hebrew letter alef, with subscript 0: אo. Observe then that the size of {5, 10, 15, 20 . . .} is also אo. There are אo things in N. Ht: Galileo for observing that N had properties like this. Cantor would make sense of it all. Sort of.
By Common Consent, a Mormon Blog 
Very interesting. As there are an infinite number of irrational numbers between any two rational numbers, it would seem that the “size” of infinity associated with irrational numbers is greater than the “size” of infinity associated with the rational numbers…? And what, then, would be the source of the biggest “size” of infinity, if that is a meaningful question?
Mogs
Cantor actually proved that there are an infinite number of infinities above Aleph null (the number of rational numbers) and Aleph prime (the number of irrational numbers). And thy curtains are stretched out still…
Also, there was an interesting BYU Studies article looking at the LDS theological implications of some of this. Bessey, Kent A. “To Journey Beyond Infinity” 43:4.
Cantor is well known for his diagonal proof. Depending on how one wants to look at it though he can either be a genius or a less calculated magician. In theory it may be possible to think about infinite numbers but when you start adding words such as “sets” to numbers they must take on a more finite proposition. A “set” is something “whole” or “known”. To say that there is a complete set of numbers can only include a finite and known amount. This is where Cantor became a clever magician over being a true mathmatician. In his diagonal proof he sets out to show that he can make an infinite list of numbers and then one by one go down that list and formulate a new number that isn’t in the list of infinite numbers.
Several problems here abound though and in essence it becomes just a clever magic or number trick on the mind and nothing more. As he goes down his list he makes his new number with each decimal place in succession a different number than the one found in the list. The proof seemingly checks out with everything he goes over. But with infinities (that is what we are trying to deal with here) we cannot be so sure. You see, Cantor is just playing the mind always keeping cleverly ahead in the number game by only showing at any given time that his new finite number is different than all of the previous finite numbers. But surely there must be a number identical to his in the infinite list below up to any point thus far, right? Absolutely! All Cantor is thus showing is that he can keep ahead of all the previous possibiliteies because he has the advantage of the possibilities always showing greater each decimal place he moves in every turn. For instance, in this sequence- .00 to .99 there are a 100 possible different sequences. So, when cantor makes his number he is just picking a number not in the previous numbers already counted (a possibility) and then in the next turn he moves over another decimal point (example- .000) changing the odds to 1000 possibilities. So, as we can see, Cantor is always cleverly ahead of all the possible choices in his “finite” list he is making. But with logic, we know that for every new decimal place in any infinite set (if there were such a thing) there will already be every possible known possibility. So, in truth, Cantors diagonal argument is flawed and only shows that no matter how big a finite number one can make, he can always make one the next number bigger. Well, no duh! that is just like adding one more number to an ever increasing finite number forever- anyone can do that! He is not allowing for each decimal place to be checked with the entire list ( I know- seemingly contradictions of words) for that possibility which of course would always already exist for any finite number he already came up with.
Math would be wise to remove the notion of something infinite from its base. Math is only figured with real known finite numbers. There is no proof for infinite sets. Cantor was a magician.
Love this. Takes me back to the time my diminutive, stuttering mission companion explained to me how there were more numbers between 0 and 1 than there were counting numbers. A whole new perspective was opened to my eyes. That kid had a mind to comprehend the universe.
It actually can’t be proven there are more numbers between 0-1 than all counting numbers. It’s ilogical that you could have a set of something infinite bigger than another set of something infinite. This is a fallacy of human thinking and not logical with real true math.
“real true math”? What did my companion get his doctorate in, then, I wonder? Fake lying pseudo-math?
Just check your math- that’s all. Anyone can make a claim with the infinite but to actually check- check your math. Show how he was right- that is all. I am betting he is or was wrong and I have math to show why.
Don’t worry, Rob, I’m sure you are right and set theory is wrong.
Care to link me to your published paper where you disprove it?
Mogget, there are an infinite number of rational numbers between any two irrationals too. The question as to which collection is “larger” in the counting sense was actually first resolved by Cantor himself. I’ll mention that in the next post.
WVS, thanks for this series. If you take requests, I have a couple.
First, complexity. You’ve brought up how even finite things can be unimagineably large. Even with reasonably small sets, though, the possible interactions of members of a set can snowball beyond comprehension. An expanding universe can even seem like a relief to the madness, putting most of everything beyond our concern. Some people wonder about Heavenly Grandfathers, and Heavenly Great-grandfathers. It’s the aunts and cousins that scare me.
Second, as an aside in a number theory class, Jack Lamoreaux once mentioned that any set that can be described with a finite number of statements is finite, so in some sense the reals and the natural numbers are finite sets. This apparently was a standard, well-known bit of mathematics. Could you elaborate?
John, something about that doesn’t sound right unless there are severe restrictions upon what sorts of statements are permitted. I have a feeling that was originally a commentary on the intuitionist and constructivist theories of mathematical foundations that perhaps got a bit mangled in the telling.
John, I am going to consider some theological stuff. As far as Lamoreaux (R.I.P.) is concerned, I don’t think he was an intuitionist, but I don’t really know. Perhaps he was talking of direct representation.
This is one of the interesting things I’ve seen when dealing with infinities – how there can be different sizes of infinites, but at the same time, also not. For example, x<2x, unless x is infinity.
Ah Frank, ordinals. I different ballgame. But important to our discussion.
Maybe I shouldn’t have rambled off theologically. In this post, the number of water molecules in a cup is mentioned as a finite number that is too large to comprehend. We get numbers like that when considering things as small as molecules or as big as galaxies, but we can also get them from something like the number of ways to group the students in a classroom. With ten students there 115,975 possibilities, and with thirty there are about as many as an ounce of water molecules.
On the other thing, I’ll have to read up sometime on what intuitionist and direct representation mean. It gives me a direction to go.
It actually can’t be proven there are more numbers between 0-1 than all counting numbers. It’s ilogical that you could have a set of something infinite bigger than another set of something infinite. This is a fallacy of human thinking and not logical with real true math.
Rob Osborn:
In regards to your reference to the continuum hypothesis (first sentence), it’s true that this has not been demonstrated (although Cantor claimed that God directly revealed its truth to him, but unsurpisingly that didn’t hold much weight). The second statement is very well established by virtually all of set theory since Cantor. This argument is about two hundred years out of date. “No one shall expel us from the paradise Cantor created.”-David Hilbert.
Nevermind that last reference to the continuum hypothesis, I misunderstood what you were saying. Yes, I think his proof is solidly accepted by the math community.
Actually, Cantor’s arguments of infinity are disputed by many including the the math itself. For starters one can’t prove exactly what numbers would all be included in an infinite set, so to state that one could make a new number from a collection of unknowns is ludicrous as Cantor suggested.
All I say is show me the math showing how one infinity couldn’t include any number not already in that set. Cantor’s diagonal argument only checks numbers in a finite set not an infinite set as he thinks.
Rob, given that set theory is the generally accepted mainstream view, I’m still waiting for your link to your published paper where you refute it. When you are attempting to discredit the consensus view, the onus is on you. Show us the math.
You may think that the number of molecules in a cup are uncountable? I assert that they are indeed countable. People are trying hard to define mass in absolute terms as a specific large number of atoms of a particular isotope, likely, of silicon. If we, humans, are working the problem, it can not be too hard.
I am not too worried about mathematical infinities. Being a modeling language with rules, large countings are part of the rules.
I have no doubt that there are real existential infinities in terms of time and space and many other quantities. The question is how does a being, existing within the infinities, interact? Since we are under some commandment of knowing God, this is not necessarily an idle speculation for Mormons.
TUBES,
You are aware of Cantor’s diagonal argument?
I don’t want to debate if you aren’t aware of it.
RW: That’s why I’m surprised that more Mormons aren’t set theorists. I think all Mormons should have at least a basic understanding of set theory; It’s quite pertinent to our quest to know God.
Rob – indeed I am. But much has come since then, particularly with Cohen’s work in the 60’s. And again, as set theory is the mainstream position, the burden is on you to refute it. Were you in fact correct, I’d imagine that your work would easily find a home in Acta or the Journal of the AMS. You have published such an article, I presume?
Oh, I guess not. Apparently you don’t in fact “have the math”, as you claim.
tubes,
I amnot going to attempt to disprove set theory of the which I am comfortable with, at least as far as I understand it. My beef is with “infinity”, especailly the claim that one infinity can be of a different size (in count) than another infinity. It is an absurd statement to make such a claim. Cantor tried to prove this with his diagonal argument but as I have said above, its just a jedi mind trick- magic at best!
Cantor uses the numbers to the right of the decimal placement. But, it can be showed in a mirror using the same numbers on both sides that the argument works on both sides of the decimal placement for a true direct one on one correspondence. The problem trying to compare infinite sets is that the feeble mind only calculates and thinks in the abstract of finite things. We have no way to truly grasp the infinite, partially due to the fact that there is no such thing as an actual number having an infinite count of numbers in it. That is paradoxial in and of itself. No matter how big we make a number it always has a finite property to it. It must because it would be impossible to actually have one so big that one end was of infinite length away from one in the middle or end…wait- in an infinite number there can be no middle, no end, no start, in fact- there can be no part of it that has any significant value when compared to its entire number. Its meaningless- it doesn’t nor cannot exist.
Let’s relate this to something physical- give something some value here rather than just an abstract.- (next post)
(continued)
Suppose you were to take our universe and let’s say for the sake of things that it seemingly goes on forever in every direction. We may say it has this “infinite property” or potential about it, but there is no way to know for sure nor to place it into correspondence with any abstract number to see which one is bigger. If I were to place an actual marker somewhere in the universe and then go to another place and add another marker there will and “must” always be a spatial relationship between the two markers, meaning- I will always be able to count how much distance between any and all markers I place anywhere in the universe some greater and some lesser. What I could never do though is place a marker that was an infinite distance from any other marker.
Now of course we are just dealing with the abstract of one infinite set, so to speak. So what happens if we were to try to compare one infinite set with another? This is where it gets confusing and contradictory though because as I have shown with Cantor’s argument, he is never concluding that he has actually shown he can make a number of infinite size and have it not be in a list of numbers which have no end in every direction. Lest’s apply this to something non-abstract. Go back to our universe-
Sppose that the infinite list of his numbers were the placement markers in correspondence and that each different number was in accordance with it’s spatial relationship- kind of like on an infinite grid. As this applys, it is like saying that he can place a marker somewhere in the universe where there isn’t already a marker. But, this is a contradiction because every place (because it is infinite) already would have that placement taken. This simple illustration shows the ridiclousness of infinities and especially in trying to compare two different sets of them. There is no such thing as an infinite number in length just as it is also impossible that there is some place in the universe that is an infinite amount of distance away from us in our galaxy.
You’re phrasing it wrong, which is probably why you are getting hung up here. Properly stated, the principle is that one infinite set can have a cardinality that is different from another infinite set.
“Infinity” as used with respect to real numbers (i.e., as a limit of an integral in calculus, etc) is not the same as the aleph numbers which represent cardinality, i.e. the “size” of infinite sets. Your argument blurs the two.
You have a beef with cardinality? Good luck – Godel and Cohen have already shown that the continuum hypothesis can neither be proven nor disproven, respectively, within the current framework of set theory. You can choose not to believe – but you CANNOT “have the math” to disprove it.
Great post WVS. Looking forward to more. I think the abstraction of infinity is exceedingly useful.
Here’s a question: abstractions seem to be used primarily as a way for us to deal with things we either can’t realize ourselves or with our math and technology. But many abstractions go away once we have the ability to deal with the reality. For example, an inertial frame is an abstraction allowing us to calculate equations of motion of things. But now that we understand many of the subatomic forces, it seems we may be able to do away with the appeal to an inertial frame and instead rely on particle interactions to describe motion.
You’ve outlined several useful abstractions for infinity, it’s use of representing a really really big number as only one of those abstractions. Nevertheless, do you think it is impossible that math/technology could advance to the point where we no longer need to appeal to infinity?
jmb275, the question of computational limits and their relation to the infinite is interesting. Presumably there are limits to computational efficiency. In modeling physical processes we run up against the question of whether our modeling constructs (lines of code, say) tell us whether solutions “exist” or whether they are subject to subtly chaotic processes — and of course the same question exists for “actual” physical processes. We hope our approximations, if fine enough, really are approximations to both theory and physical reality (whatever that might mean). So, my guess is, we will never discard the infinite as a source of assurance about our physical theories and at least in some cases our models of the physical. Mathematically, many theoretical superstructures are embedded in the infinite and derive their meaning from it. And then there is theology.
Can we use irrational numbers to report home teaching results?
tubes,
Saying “cardinality” or “size” is the same thing. The cardinality of an infinite set is “all” the numbers in that set. So excuse me if I say it one way- it’s the same thing!
My beef is with “infinity”. You keep getting caught up in the semantics of it. Try addressing what I have put forth for a change.
I did think of one more test to prove Cantor’s theory wrong in practice.
Suppose you had a glass that was measured on the side of up to 1 ounce. All of the numbers in Cantor’s diagonal argument are going to represnt different levels up to that 1 ounce as a percentage. Such as – .9736352412… and then we are going to count with numbers how many incremental steps or percentages in infinitessibly small amounts up to the 1 oz. mark. Each one of those steps we are going to count and see if the two match up- a one to one correspondence. Now, if Cantor’s theory is right he should be able to come up with a new percentage amount out of his existing infinite set, right? He should also find that there isn’t enough numbers to count all the steps, right?
This is where applying his theory to real applications it falls entirely apart. Of course we should know that every percentage possible in the 1 oz. test would already be included in his infinite list. We also know that each one of those steps- those infinite steps could be counted- if it were possible to do such. After all- you can’t never run out of numbers- they are endless. The numbering system can count all things forver and ever withou having any capacity to ever be full or complete.
It’s all ridiculous if you ask me.
Wow, Rob, next you’ll be telling us that you think evolution is false!
Rob, wouldn’t you say that your conclusions against Cantor depend upon what counts as an acceptable style of argument within mathematics? This is hardly a new debate but goes back to the debate about foundationalism back in the days of Russell. I notice such foundationalist critiques haven’t been popular in decades. If anything mathematics has actually “liberalized” what they allow as a proof. The semi-empirical methods that Putnam suggested be allowed in mathematics back in the early 90’s now seem to be status quo. While there is some debate about the place of infinitity in mathematics I think the main reason so many accept it is simply because it is so useful for certain ways of thinking.
Perhaps that’s insufficient for people who want to demand a certain type of rigor within mathematical proof. However it does seem like that debate was had and lost decades prior.
To add, I’m not saying people still don’t write papers on logicism, constructivism, intuitionism or mathematical platonism. Rather my point is that the debate doesn’t seem to have significantly changed what’s treated as acceptable mathematics. A constructivist may dislike certain proofs and be skeptical of certain areas of mathematics and yet the journals are still filled with avenues they may dislike.
“My beef is with “infinity”. You keep getting caught up in the semantics of it.”
That is the entire point of this series, you dummy.
Rob has demonstrated that a fundamental principle of set theory is unsound because the fundamental principle in question strikes Rob as “ilogical.”
Check and mate.
Jacob (#32),
Rob has seen dogs and Rob has seen cats. Rob has never seen a dat, and a dat doesn’t even make sense. QED.
“Can we use irrational numbers to report home teaching results?”
I don’t know about home teaching, but I experimented making a fast offering with a check made out for something like “ten and one seventh dollars.” That worked, so next I tried “ten and 1/π dollars.” My account statements showed these transactions rounded to dollars and cents. The checks were drawn on an account at Los Alamos National Bank, so it probably wasn’t the first time they’d handled dumb stuff like that.
> It’s all ridiculous if you ask me.
We didn’t.
John, some guy at the bank actually calculated the pi one?! That is too awesome.
Of course, the problem with reporting your home teaching results in irrational numbers is that you will consign that poor clerk to die in that little office, forever typing in your report. No rounding in the church!
Hey Rob,
Given your water example as I understand it, we face the problem that it is physically impossible to create an infinite number of subdivisions of the glass. If anything, in the realm of empirically demonstrable, infinity does not exist.
However, as a theoretical construct, it most definitely can be given rigorous constructive definitions, and from there (depending on the axioms you are using) real numbers can be constructed which exhibit the uncouth properties Cantor demonstrated. I believe Kronecker is the one that advanced the kind of skepticism you raise.
Cantor’s diagonalization is not the first proof he made for the largeness of transcendental numbers. It is just the easiest to wrap your mind around. In it, you need to assume that never-ending decimals exist, and that we are able to gather them into a single collection. Further, you must assume that the infinite collection of natural numbers exists. Finally, assuming that each natural number is capable of being paired one-to-one with the collection of never-ending decimals between any two real numbers (which we usually put as 0 and 1 for simplicity), Cantor simply showed that you can construct a number using any pairing which should be in the pairing but is not. Now this means that one of our assumptions was wrong. For Cantor and most mathematicians, the assumption taken to be wrong is that it is possible to pair the set of natural numbers with the set of never-ending decimals.
It is perfectly possible to say that we were wrong in assuming a set such as the irrationals even exists, or that, given such a set, that it is possible to pull any element out of it for the purpose of matching with another set. In fact, that last assumption, known as the axiom of choice, if taken to be false destroys Cantor’s argument, but it destroys a lot of standard number theory as well. There is a similar assumption we could make about ordering, which is that any two real numbers have exactly one order relation that is true between them: less than, greater than, or equal to. It turns out that making that assumption will force us to accept the axiom of choice, at least among real numbers (the definition is a bit different among sets in general, but implies just as much).
Limits are, I believe, the only real way to deal with the infinite (in a mathematical sense). In analysis we have to be extremely careful about how it is dealt with. If you are interested, Rob, in having a further discussion of the infinite, I’d be happy to get your email or something.
There are enormous problems in trying to place the infinite in a real setting. Just think Banach-Tarski paradox. However, even if infinite is only an imagined thing, it still has properties which are consistent, one of which is cardinality. You might benefit from looking into the construction of the real numbers out of the rationals.
The irrational numbers complete the number line. The real numbers are the closure of the rational numbers, which means any sequence of rational numbers I make which eventually settles to a value, is guaranteed to settle to a real number, but not guaranteed to settle to a rational (such as the sequence {3, 3.1, 3.14, 3.141, 3.1415…}, which I define as the sequence of decimal approximations to pi. It does not settle at a rational number). I personally prefer to think of a real number as the collection of all sequences that settle to that number (circular, but easy to understand).
Then I am satisfied that although I pretend to work with irrational numbers, I really am working with sequences of rational numbers, which I understand. And every irrational number can be approximated with a (really, any number of) sequence of rational numbers, to any degree of accuracy that I choose (we’d have to discuss delta-epsilon proofs here).
I hope you can wrap your mind around the infinite, and accept the grounds for differing cardinalities. And then assuage your sensibilities by knowing it is strictly a theoretical acceptance that is profoundly useful to the real world, but perhaps not really present. And then realize that the reason it is useful is precisely because of approximability — the real world, at the quantum level, may in fact be a huge lattice of finite states and positions, but it finely approximates a continuum at most levels. And so we can use continuous mathematics to characterize it, which in turn relies on using sequences of finite numbers to approximate the continuous to any degree of accuracy we choose. This result, and results like the Weierstrass Approximation Theorem allow us to accept the theoretical existence of infinity / multiple infinities and yet only have to deal with the finite in characterizing it.
Rob, take it from me (a PhD mathematician) that you don’t know what you are talking about and it sounds like you really have no business discuss this