More than 10 years later, and I have not progressed in clarifying anything regarding the concepts of infinity, and consequently, of the universe/cosmology.

I just took a look at this post that I had written back in 2005:

# Infinity – Is this a Concept that Fits in Our Minds?

For me, the question remains, just as I had stated it then: “So how can the universe expand? This doesn’t seem logical to me.”

The idea of an infinite universe makes no sense to me, and neither does the idea of a finite one. In short, nothing makes sense. What’s even worse, nothing seems like it can make sense.

As I explained to Jack, if the universe is finite, where does it end? How can it have a boundary? And if it has a boundary, there must be something on the other side of the boundary, thus contradicting the very definition of “finite”. If it is infinite, how can infinity exist for physical things? It makes no sense to me. If the universe somehow loops onto itself, I’m lost as to how that could be.

“I was thinking along the lines of the problem that infinity is outside the scope of rational thought, or logical thought.”

That was another idea I had. But how could there not be a way to understand the universe rationally or logically? That doesn’t make sense either. In short, nothing makes sense. The more I think about it – not that this is a priority for me – the less anything makes sense, and I find no claim that lets me stand on firm ground.

Therefore, by exclusion of the two patently impossible explanations, that the universe is finite, and that the universe loops onto itself, I, for the moment, shall rest with the thought that the universe is infinite, even though my mind cannot comprehend or apprehend infinity applied to anything physical.

Who knows what the future shall bring!! 🙂

===============

update Feb. 22 2016

A reader left a comment mentioning the idea that the universe would be like the surface of a ball. Right there it already doesn’t make sense to me. A 2-D surface cannot be 3-D space, so it makes no sense to me how you could logically talk about a surface being something that, by definition, is not and cannot be space.

However, as I mentioned in my reply to this reader, his comment made me realize something more accurately: the concept of infinity, by definition, requires that there be no boundaries ever in any direction or sense. I had thought of this before, but now it simply stands out so bright and clear, in the most fundamental way. This has a very important consequence: any concept, image, or framework that contains any boundary automatically contradicts the concept of infinity as applied to anything physical when related to space. That’s why I can’t imagine a ball or any surface as a model for infinity. It’s boundaries and more boundaries. Why do other people claim this is a *logical* way of thinking about space regarding it’s infinite aspect, that is, the very aspect they claim has no boundaries? I don’t know – it makes no sense to me.

## 26 comments

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February 20, 2016 at 8:52 am

maffbloggerA most excellent topic! One way the universe could ‘loop back on itself’ is if it has the same topology as a 3-sphere. It would be like a ball ‘in 4d space’ whose surface is our 3d universe. When attempting to picture such a shape it is hard to imagine it without it sitting inside some sort ambient space, i.e. it would cause one to ask, if we live in the surface of a ball, what is ‘inside’ or ‘outside’ the ball? I think the answer to that is some like the following: we can imagine 3d space without insisting that it be sitting inside 4d space in the same way we imagine a 2d plane sitting inside 3d space. Similarly the concept of a 3-sphere doesn’t require that it be embedded in 4-space… Reading over that I’m not particularly satisfied with my explanation, but that’s probably as good as it gets at this point haha. When talking about the Big Bang, the image of all space emerging from a ‘single point’ actually requires something like this. If the universe is actually infinite then it is nonsensical to talk about it emerging from a single point, rather we would have an image of a universe that was always infinite but which started out with very high density and temperature. Hope something of what I said makes sense!

February 22, 2016 at 7:49 am

alessandrareflectionsHi Maff, sorry but to me it still doesn’t make sense. But your comment made me realize something more accurately: the concept of infinity, by definition, requires that there be no boundaries ever in any direction or sense. Any concept, image, or framework that contains any boundary automatically contradicts the concept of infinity as applied to anything physical, including space. That’s why I can’t imagine a ball as a model for infinity.

February 22, 2016 at 9:59 am

maffbloggerFair enough, but take solace in knowing the concept is well defined mathematically! If the universe were such a shape one could travel for as long as one liked in any direction and not ever hit a boundary, however it would be possible to end up in the place you started, so in that sense there are no boundaries involved. Personally I am a fan of an infinite universe, and the observable evidence does point that too.

February 22, 2016 at 10:10 am

alessandrareflectionsMy reply to that would be that math isn’t physical, math is like a fairy world, it’s abstract, it’s hocus-pocus. A theoretical concept has no materiality, but the universe does. Therefore, you could work out numerous mathematical concepts logically within the abstract world of math that would fall apart when applied to concrete and physical things, like the universe and it’s “container” or medium, space.

You cannot logically argue that 3-D is the same as 2-D, so either I haven’t understood what you are saying, or you believe in something that contradicts itself.

February 22, 2016 at 10:22 am

alessandrareflectionsBy the way, I just thought of something – like in a “Zeno paradox” kind of way. Shouldn’t the definition of a point be impossible? Because the minute you say a point, a dot, you can think, well, it could be smaller, and if it could be smaller, then it could be even smaller, and so it goes, infinitely so, to the smallest of the smallest right before it’s nothing. But that is so small that it’s impossible for us to ever “reach” because we, humans, are limited. So this means a point could never be a definite or stable thing. It just keeps going smaller and smaller and smaller – see what I mean?

I’ve just realized I’ve been lied to – ever since primary school when they told me a point is a dot. LOL

This might be a good question for quora 🙂

February 22, 2016 at 10:59 pm

maffbloggerMy apologies, I hadn’t meant it to come across as arguing that 2d is the same as 3d, though that suggests an interesting point: we haven’t mentioned what exactly we mean by the word ‘dimension’. There are a bunch of different definitions commonly used, but I think they all have it that 2d isn’t the same as 3d lol.

It’s useful to imagine a point as a dot, but it certainly isn’t one, as you’ve pointed out (pun intended) it possesses no properties such as length, area, etc., but its does have one thing: position! And that’s really all it is, a position. But you’ve made a good point, since we’ve come this far, what do you think about lines? They seem to have length, but no width. You could always imagine a line being narrower and narrower, until perhaps it disappears altogether.

What about this problem: Pick you favourite unit of measurement for distance, I’m going to imagine centimetres. Imagine a square width of 1 and height of 1. Now imagine a rectangle of width 2 and height 1/2. Next comes a rectangle of width 3 and height 1/3 and then one of width 4 and height 1/4, and so on. Each rectangle is shorter and wider than the last, getting closer and closer to resembling an infinite line. But each rectangle in the sequence has an area of 1 cm^2, no matter how far along the sequence you go! This is one example of many complications that need to be dealt with when dealing with infinity.

I tell you what, I love every minute of it, I appreciate the opportunity to discuss things like this.

February 23, 2016 at 5:16 am

alessandrareflections“It’s useful to imagine a point as a dot, but it certainly isn’t one, as you’ve pointed out (pun intended) it possesses no properties such as length, area, etc., but its does have one thing: position!”

And I went through years of school thinking it was a dot!!!! They lied to me! LOL My world is disintegrating 😉

“But you’ve made a good point, since we’ve come this far, what do you think about lines? They seem to have length, but no width. You could always imagine a line being narrower and narrower, until perhaps it disappears altogether.”

Yes, it’s the same problem as above. I had never thought of any of this through all my years in school. A new stage begins. You must know of the Zeno paradoxes, right? Actually, when I thought of the “dot problem” above, I also immediately associated it with Zeno’s racetrack paradox. http://www.iep.utm.edu/zeno-par/#H3

February 23, 2016 at 5:36 am

alessandrareflections“One way the universe could ‘loop back on itself’ is if it has the same topology as a 3-sphere. It would be like a ball ‘in 4d space’ whose surface is our 3d universe.”

Well, you don’t even have to go 3-D, do you? You could say, imagine that the universe is like a circle and it loops back on itself.

OK, I can see how a circle does that, but that’s not infinite, not in 2-D nor in 3-D. It’s just a closed loop. And if it’s closed, it’s finite, it’s contained, it doesn’t go on forever.

Which is why I can’t understand how someone could posit that a circle or a sphere could represent an infinite universe or space.

February 29, 2016 at 11:58 am

alessandrareflectionsI just thought of something really interesting – my world is saved! For now at least. Actually I wasn’t lied to in school – a dot is a dot. It deserves a separate post, which I will write later.

February 23, 2016 at 6:02 am

alessandrareflectionsAnd you know what I just thought? There are infinite ways of tracing a path around a sphere (I mean, on the surface of the sphere).

I was picturing your sphere and a path going around the sphere. Actually I pictured a red party balloon (it’s just prettier). So then I went around the balloon, like in a perfect circle, looping back to the beginning. And then I thought, well, we could wiggle around the perfect circular path, like zig-zag and stuff, so you are covering a much longer path to loop back to the point where you started. Now you could do this, i.e., trace paths, in infinite ways. And then I thought: maybe you can’t. But you should be able to.

I can’t decide. Because we’re back to the dot/position problem. If you think that a point is just a position, you can’t trace it, because it has no width or length. But we can trace a path. So I guess you could trace infinite paths. But then you think, but what is a path? It’s a line. And what is a line, it’s a bunch of points. And what is a point? It’s nothing – or is it one of those weird things in math that “tends to nothing”? And what is a position? It’s math, it’s abstract, it’s not real. Everything just disintegrates.

Anything that has to do with infinity is mind-boggling to me. I’ll have to think about this path/line problem a bit more.

February 24, 2016 at 12:09 am

maffbloggerIf you can be bothered committing to watching a 24 minute video, here’s a nice one about something called the “Banach-Tarski” paradox https://www.youtube.com/watch?v=s86-Z-CbaHA

Given what has been discussed here, I think you’ll find it really interesting! It’s all about points on the surface of a sphere. It’s a bit confusing but the guy in the video describes a lot of cool ideas, as well as a thoughtful conclusion at the end.

February 24, 2016 at 7:21 am

alessandrareflectionsHi, I’ve just watched it. I was glad he asked the question about math diverging from the physical world. That’s what I think happens. But then I don’t know any math. 🙂

And also, as I mentioned above, I’m not at all convinced that math can properly deal with the concept of infinity, because the postulations just fall off the logical plane, and it seems no mathematician notices it!!!

Someone had mentioned to me before this concept of different types of infinite sets (what he calls countable and uncountable in the video, if I remember correctly).

But what I am not convinced is when mathematicians say: look here is the set of integers, going to infinity. Now if we subdivide infinitely between 1 and 2, and so forth for all other integers, we will get a much bigger set of numbers, so we end up with the second set of infinite numbers being bigger than the first set of infinite integers. But this contradicts the definition of infinity of the first set – it cannot, by definition, be smaller than anything.

If I understood what mathematicians are saying, they are saying this contradiction does not pose a problem for them. We can have one set of infinite numbers that is smaller than another set of infinite numbers. Is that what they are saying?

I would say you cannot have that, because, by definition, your first set cannot be smaller than anything. So their answer is to say, but we just proved it! And I would say, I’m not convinced that math can deal with infinity properly, because what this mathematician just stated is a patent logical contradiction: an infinite set is smaller than another infinite set. It’s like 2 does not equal 2.

So contrary to what mathematicians argue, what this shows to me is that math is the one who can’t deal with infinity. Therefore, mathematicians would never know if the universe is infinite nor how it could be infinite. They could take the distance between one end of my desk and the other end and divide it infinitely. But it doesn’t answer anything about the universe.

To me, it’s almost like mathematicians are saying: we just showed you we can divide the distance from one end of your desk to the other infinitely, THEREFORE, we just proved your desk has an infinite length. And I would say, no, you have proven that playing with numbers is not the same as understanding physical reality. And you have just proven that math may not be able to deal with the very concept of infinity, because maybe we are limited as humans in that respect, or, mathematicians, of all people, have a problem with logic!!

The thought that we humans have a limitation to grasp infinity is also a thought I don’t like, because I don’t think that the limitations of the human mind should preclude it from understanding infinity. I mean, we aren’t bacteria. Aside, as I mentioned, from the issue that I don’t know any math – so I look at these issues from the outside.

So, what do you, who, guessing from your blog, is someone who knows math, think about what I stated above?

February 24, 2016 at 10:49 am

alessandrareflectionsI found this wikipedia page “List of paradoxes” related to infinity – I guess it covers exactly the question I posted above – about how little sense infinity makes in math or physics in so many different ways: https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals

February 25, 2016 at 11:38 am

maffbloggerI quite recently wrote a post which related to the different types of infinity actually (Cantor’s Theorem: That’s Pretty Neat!), so it’s been on my mind a bit lately.

Well, for someone on the inside everything seems pretty good. I think part of understanding the idea of ‘bigger’ infinities is realising that the terms ‘bigger’ and ‘smaller’ are actually a bit misleading. They do make good metaphors, but I think that’s a problem, because they’re so good it’s easy to forget that in this context they actually have very specific definitions. What we usually mean by one set being bigger than another is that one has more things in it than the other. One has 3 elements and the other has 5, and 5 is bigger than 3 so there ya go. As you’ve pointed out this doesn’t make sense when both the sets are infinite, because neither of them really have any number of elements. One thing mathematicians like to do sometimes is to take an idea which makes sense in a known context and try to find some way of charecterising it which also makes sense in more general settings. So with finite sets it’s easy to say some are bigger than others, but is there some other way of comparing the number of elements without having to refer to things like 5 or 3? Once we’ve found something like this, it might be that we can use it to compare infinite sets in a way that does make sense and gain some novel insights.

Such a charecterisation of set size does exist, and it comes down to ways of pairing the elements of one set with those of a another. We can say two sets are the same size if we can exactly match up the elements of each set, each element of one can be paired with exactly one element of the other with none being left out. On the other hand, if it turns out that no matter how we try to form pairs there’s always a few unpaired elements left over, then we can say that one set is ‘bigger’ than the other. For finite sets, this way of thinking matches up exactly with our usual notions of size. The thing is, we can now take infinite sets and say very rigorous things about the ways we can pair up elements. Leaning on our familiarity with finite sets, we still use words like ‘bigger’ and ‘smaller’, but we’ve really taken the words to a place where the concepts they usually represent break down.

Regarding the sort of logic a mathematician might be using, you said “if we subdivide infinitely between 1 and 2, and so forth for all other integers, we will get a much bigger set of numbers”. There’s more to it than that! Imagine all of the rational numbers (fractions), think about how many their are. All of the integers are rational, but then between every integer there’s infinitely many more rationals. Just like Zeno’s paradox, between 0 and 1 there’s 1/2, 1/4, 1/8, 1/16, 1/32, 1/64… and on and on, and that’s not even close to naming all of the rationals between 0 and 1. We can name infinitely many of them and still leave infinitely many out. Now you might be tempted to think that this is the sort of thing mathematicians call uncountable, but it isn’t so! The rational numbers are in fact countably infinite, there are just as many rationals as there are integers! The real key to understanding is in seeing the difference between the rational numbers and the real numbers, and that is not so easily perceived.

Have you ever heard of the work of Kurt Gödel? He published a paper in 1931 which demonstrates that we can never hope to have a finite system of axioms from which we can derive all truths, without also being able to derive falsehoods. Consistency (no contradictions) implies incompleteness (there are truths that can’t be proved) and completeness implies inconsistency. I think this insight says a lot about the nature of mathematics (=

February 27, 2016 at 11:52 am

alessandrareflections“The thing is, we can now take infinite sets and say very rigorous things about the ways we can pair up elements. Leaning on our familiarity with finite sets, we still use words like ‘bigger’ and ‘smaller’, but we’ve really taken the words to a place where the concepts they usually represent break down. ”

Well, thank you for explaining this. I think I can see how this might work – from a logic perspective.

I am now more relieved for the sanity of mathematicians the world over 😉

I had this thought: You have set A that goes from 1 to 3 (3 integers). And set B that goes from 1 to 3 (3 integers), so each has 3 integers: 1,2, 3. Except that now you say: in set A, we’re going to consider all the real numbers between 1 and 2 as well. See, both sets still just go up to 3, but now set A has a lot more numbers, an infinite quantity of numbers. I could extrapolate this to represent what I described as the seemingly crazy logic of saying one infinite set was bigger than the other infinite set. What would you say of my reasoning here?

About Gödel, I had only heard the name, but I’ll check him out and what you just mentioned.

And I also found this page (https://www.mathsisfun.com/numbers/infinity.html), which was amazing, because of one thing that they said:

================

Infinity is the idea of something that has no end.

In our world we don’t have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.

So don’t think like that (it just hurts your brain!). Just think “endless”, or “boundless”.

Infinity does not grow

Infinity is not “getting larger”, it is already fully formed.

Sometimes people (including me) say it “goes on and on” which sounds like it is growing somehow. But infinity does not do anything, it just is.

=======================

!!!! :-OOOOO !!!!

“But infinity does not do anything, it just is.”

I had never thought of infinity that way!

And I don’t like it. I’m not saying it’s wrong – but I don’t think I have ever thought about this, although some vague memory on some article about Einstein comes to mind that would point to this…

But I just don’t like it. You see the image I think of for infinity is exactly this huge black space maybe with little light flickers here and there and you are just moving, zooming farther and farther away in that space, but as the person says, you are “trying to get there”. And you can’t because… it’s infinite. But clearly this idea of motion was fundamental to my imagery for infinity, the “getting there” part. And now he has thrown a wrench in my imagery. And the idea that it’s just static, sitting there, is highly disturbing. The thought that things are just sitting there and *that’s* infinity – my goodness, that is weird (but again, a vague recollection of something related to Einstein pops up, how the position of things in space are relative or something like that…).

This means that every little rock is infinity. Which leads us to the fact that even if everything was still, was completely static, the universe could still be infinite.

However, nothing concerning objects moving or not moving in space matters, because the larger question is if space itself is infinite. If the universe had no objects, it could still be infinite, obviously, in terms of space (the container).

I favor the idea that time is independent of space. That is, if you could make space disappear for five seconds, time would still continue, on its own. And what, Mr. Mathematician, do you think of that? 🙂

February 29, 2016 at 8:30 am

maffbloggerWell, if set A has 3 elements, but then we add uncountably many elements to A, we get a set with uncountably many elements. B still only had 3 elements. However if we scale this up and say that instead of 3 elements both A and B start with countably many elements (like the positive integers), it indeed becomes less obvious what will happen when you add more elements to one of the sets. If we add all the real numbers between 1 and 2 to A, we get an uncountable set, but if we only add all the fractions between 1 and 2 we will still have a countable set. So A and B will still be the same type of infinity even if we add infinitely many rational numbers. Adding all the real numbers, however, makes a difference. So I guess I’m trying to say that the logic used by mathematicians isn’t clearly displayed in that example, or something.

I think I’d like to have a good think about this idea of static/dynamic infinity before I let you know what I reckon, so stay tuned!

February 29, 2016 at 9:46 am

alessandrareflectionsVery good.

For some unknown reason, I couldn’t register in the math forum I mentioned above, but I found another one. Just posted a question, but I don’t see it – maybe they moderate the posting. I was looking over some of these statements with infinity on the “math is fun” site, including the very famous one “infinity + 1 = infinity” and I realized this is not a logical statement, it’s false. Below is my question that I left on the forum:

Hello good people of the math world!

I was recently having an exchange with a mathematician about infinity-related ideas. I don’t know math. But it seems to me that the statement “infinity + 1” is not possible in math. Which means that neither is “infinity + 1 = infinity”.

It’s a question of examining the definition of infinity. If infinity is, by definition, something that could not be larger, nothing can be added to it. Otherwise you are no longer making a logical statement. Which leads me to conclude that math cannot deal with infinity, it’s beyond the framework of mathematics.

My question to you guys is: has any mathematician said the same thing already? Who was that?

February 29, 2016 at 10:22 am

alessandrareflectionsI found a good answer! on Reddit https://www.reddit.com/r/askscience/comments/2zpfej/what_is_infinity_plus_1_minus_infinity_plus_1/

I don’t understand all their discussion (the different types of numbers, associativity, etc.) – but they do say “There’s generally no answer” – which seems like the right answer to me. And which means that this “math is fun” site is only misleading and confusing people as to the right answer. A pity.

February 29, 2016 at 9:57 pm

maffbloggerOne point that perhaps I haven’t made explicitly enough is that in mathematics infinity is made up, it plays the role it is defined to play. The statement “infinity+1” usually has no meaning, but when it is written that “infinity+1=infinity” this is a definition rather than a consequence of some other maths. Once more I find myself not having enough time to write as much as I’d like, but there’s something to think about! Maybe I should write a post too, about other examples of making stuff up in maths. I sort of did one about adding ‘infinity’ to the real line in order to make it ‘compact’, but it’s not easily readable, so maybe I’ll have a think about writing something similar with a different focus (=

February 27, 2016 at 11:57 am

alessandrareflectionsBTW, were you familiar with this not very well known episode from Einstein’s life? https://alessandrareflections.wordpress.com/2006/03/20/a-very-clever-chauffeur/

February 27, 2016 at 1:10 pm

alessandrareflectionsHere’s another thought. Using my example of the length of my desk, as infinite, because one could subdivide it infinitely, like in Zeno’s paradox, this means that the length of every single thing is the same, that is, infinite. 🙂

February 28, 2016 at 9:34 am

alessandrareflectionsAnother question: https://www.mathsisfun.com/9recurring.html

Does 0.999… = 1 ?

What do you think? I don’t agree!! I shall ask the great mathematicians of the forum…

And look at what I found on a link from this site:

“I am nobody.

Nobody is perfect.

Therefore I am perfect.”

Ha! I knew the proof of my perfectness was out there somewhere.

February 29, 2016 at 10:01 pm

maffbloggerI’d say true!

What about this one:

“Nothing is better than coffee.

Tea is better than nothing.

Thus tea is better than coffee.”

February 28, 2016 at 9:58 am

alessandrareflectionsI also favor the idea that nothing does not exist and has never existed. Why do things exist however? I don’t know. But time is the primal element to me. How could things have always existed? I don’t know. But that seems less far fetched than any “big bang” theory that out of nothing came everything – with a bang, no less! Why not add party hats and noisemakers to the imagery? It would sound less crazy.

February 29, 2016 at 3:30 pm

My entire world almost desintegrated during an exchange about infinity and math | Reflections, Reflections by Alessandra[…] this very interesting exchange in the comments section of this post about infinity (The question remains: how could the universe be infinite), for a second, my world entire […]

March 8, 2016 at 7:55 pm

alessandrareflectionsMore thoughts:

I really don’t agree with what’s explained in this site: http://www.cwladis.com/math100/Lecture5Sets.htm

Or in the wikipedia page for Cardinality: “there are more real numbers R than whole numbers N.”

The flaw in this statement is that people are still using counting to see which set is bigger and infinity cannot be counted. So any time someone comes along and starts counting to figure out things about infinity, in the same way as they do with finite quantities, they mess up.

By definition, since whole numbers are infinite, there exists no other set that is bigger (i.e., has more elements) than whole numbers. In fact, every infinite set has the same number of elements. To say otherwise is to pretend that one of the sets you are stating to be infinite is actually finite.

stapel wrote: “It seems as though you are thinking that “infinity” is itself a fixed (and finite, but “the largest”) value. This is not true. “Infinity” is a descriptor, a cardinality.”

Well, it certainly seems to me that cardinality is a concept that is based on finite numbers, and on counting. So it doesn’t work for infinity, just like most math doesn’t work when it comes to infinity. The minute anyone says an infinite quantity of numbers is greater than another infinite quantity they have contradicted the most fundamental aspect of the concept of infinity – which is: infinity CANNOT ever, in any way shape or form, be less great than anything else. A set that has a certain quantity of numbers that is smaller than another set is simply not infinite. I just see a major contradiction there. It makes no sense.

As a result, this is what I was thinking today: one way some math operations can deal with infinity is if the basis for the quantities expressed is actually infinity and not finite numbers.

So:

∞ – ∞ = 0

∞ * any number greater than 1 = impossible operation, since infinity can never be greater

∞ + 0 = ∞

∞ + any positive number = impossible operation, since infinity can never be greater

∞ + ∞ = impossible operation, since infinity can never be greater

∞ * ∞ = impossible operation, since infinity can never be greater

∞ / 2 = half of infinity 🙂 we can never express the result with finite numbers, because it would blow up the concept of infinity, making it finite as well. The same goes for subtracting numbers other than zero from infinity, it can’t be done.

∞ / 1 = ∞

That’s what I think so far.