The Baboon Nebula, Small Magellanic Cloud, orbiting the Milky Way.

*Hubble Space Telescope Photo.*

Since we've been kids we've been told we live in an infinitely immensi-huge-big Universe. And when told our Universe is still getting bigger, people ask:

1. How can a thing infinitely big get bigger?

2. If the Universe is bigger today than yesterday, what did it just bump into?

3. Why can't I get decent bus service from Augusta to Gardiner, Maine?

Astrophysicist Ethan Siegel provides the most up-to-date scientific understanding of the Universe here, here, here and here.

But even Ethan's explanations, as good as they are, do not really get at the two questions above, which are the most commonly asked and are the hardest to answer.

The best way I have found to ponder this is to think about the hierarchy of infinities. And for that we need to meet Georg Cantor, a 19th century German mathematician. Cantor upturned all of science and math by showing how bizarre infinite quantities are. For example, he showed how the infinity of odd and even numbers is no bigger than the infinity of just the even numbers:

When you line up all the odd and even numbers with just the even numbers, you can match up every odd and even number with an even number, and this one-to-one correspondence will continue infinitely. There will always be an even number available for each and every odd and even number you can list. This means there are as many even numbers as there are odd

**and**even numbers !!! This seems downright crazy, but that's because the concept of infinity is profoundly counterintuitive to the human brain (commercial fishermen excepted). You can also show the infinity of all prime numbers is equal to the infinity of all prime

**and**non-prime numbers:

Another way to think of this is to assume an infinity of blue blocks and an infinity of red blocks. Create two sets. One is the set of all the red blocks and all the blue blocks. The second set is just the blue blocks. In our normal way of thinking we would say that the set of all blue blocks is contained within the set of all blue and red blocks, therefore the set of all red and blue blocks has to be bigger than the set of just the blue blocks. But using a one-to-one correspondence, we can show that if you line up the set of just the blue blocks next to all the blue blocks and all the red blocks there will always be a blue block to line up next to each of them. How wicked bizarre !!!:

In the examples above, we see what appears to be a lock-solid proof which defies and defiles our intuitive sense of quantity. Apparently, all sets containing infinite quantities are equal, even when one set clearly contains items which are missing from the other set.

But then Georg Cantor pulls a fast one and shows that some infinities are truly bigger than others, although the term bigger is tricky, since it requires that you can count all the items in each set to make a final comparison. Since you cannot count to the end of an infinite quantity, the term "bigger" infinity is kind of an oxymoron. In his book "One, Two, Three ... Infinity," physicist George Gamow suggests the term "stronger."

What Mr. Cantor does is to extend the approach he used above: apply a one-to-one correspondence test to each set containing an infinite number of members and see what happens. When Cantor applies this test between rational numbers and real numbers (rationals + irrationals) the correspondence test fails. There is no way you can match up every whole number (or even every whole number + fractions) with every infinitely non-repeating decimal between zero and 1. Cantor does this with what is called a "diagonalization proof." His proof shows there is no way to assemble a complete, consecutive catalog of all infinitely non-repeating decimals between 0 and 1. No matter how complete and consecutive you try to make your catalog, there will always be at least one infinitely non-repeating decimal that is "between" any two numbers on your list:

Above are two simplified variants of Cantor's diagonalization proof. What he did was try to establish a one-to-one correspondence between all natural numbers with all infinitely non-repeating decimals between 0 and 1, just as he did with all integers and all even integers. But this time it doesn't work. The genius of Cantor's proof is that it shows that by whatever method you choose to compile your list of all irrational numbers between zero and 1, there will always be at least one that is not on your list. This is because the diagonal construction is guaranteed to generate a number that is different by at least one numeral from all the numbers it intersects and is built from. It's kind of a sneaky proof and for the past century there has developed a cottage industry of people who claim to have falsified it, but it still stands: some infinities are provably "bigger" than others.

The infinity of infinitely non-repeating decimals between 0 and 1 does not extend outward since it is bounded by 0 and 1. Instead, it extends inward between every nook and cranny between every decimal you can ever catalog. In this sense the infinite "space" between 0 and 1 is larger (stronger) than the infinite linear space of integers moving outward as they get larger.

With infinitely non-repeating decimals between 0 and 1, there is no precise "next" in the series because the number of digits in each decimal is infinite, and you need to know what the "last" digit is in each number to correctly assign the number to its place in the series, which you cannot do. It's this "infinity of between-ness" which caused Georg Cantor to declare the infinity of points on a line to be, as Gamow says, "stronger" than the infinity of integers. We can say that a point on a line from 0 to 1 denoted as:

.11111111111111123456789 ...

must be "larger than" or "farther to the right" than:

.11111111111111123456788 ...

In the above case, no matter what comes after the last digit shown, we know the first number is farther to the right on the line segment than the second number because they are identical up to the last digit and 9 is bigger than 8. But that's all we can ever really know. Unlike integers, there is no exact "next" in line with infinitely non-repeating decimals. No matter which number we try to catalog as the "next" in line, there are an infinite number of others in between.

**So what the hell does this have to do with the Universe?**

Cantor's hierarchy of infinities gives us a possible analog (and possibly totally wrong analog) to conceptualize the Universe.

We can say that if the "space" component of the Universe is infinite and the amount of "stuff" in the Universe is also infinite, then the Universe should be so tightly packed with "stuff" that we can't move. Without a strategically placed lack of stuff we would have no room to move our stuff. I used to live in an apartment like this.

But if the infinity of space is of a different hierarchy than the infinity of stuff, then we could have more space than stuff, even though we have infinite amounts of both. Cantor's proof shows the infinity of real numbers is "bigger" than the infinity of rational numbers in the sense that it is impossible to establish a one-to-one correspondence between rational and real numbers, particularly infinitely non-repeating decimals. By analogy we can suggest that the infinity of space is bigger than the infinity of stuff in the sense that you can never establish a one-to-one correspondence between space and stuff.

Our efforts to catalog all the stuff in the Universe with telescopes is analogous to writing down all integers starting from zero. While you can never count every integer, you can at least compile a sequential catalog starting from any one spot (say, zero) in which you are certain you haven't left any out. Although your list will always be incomplete (there's still more to count), you can be certain it is complete in an inward direction (of what you've counted so far, you haven't left any out).

We can do the same with all the stuff in the Universe, and actually we have. [1] Starting from Earth we can conceivably catalog every piece of stuff moving outward from Earth and assign it a unique label, such as an integer: particle 1, particle 2, particle 3 etc. If we did this carefully enough, we could be certain we weren't missing any stuff. We would only need to continue moving outward and count more stuff. And because integers are infinite we will never run out of unique labels for every bit of stuff we count.

Assume we have built a Commodore VIC-20, sorry, a supercomputer, that can count, compile and store a catalog of every subatomic particle from Earth outward and assign each particle with a catalog number corresponding to an integer and store this catalog in its hard drive. But eventually our computer will run into a memory space problem. Even if you turn all the atoms in the entire accessible Universe into a hard drive to store our catalog of stuff (each labelled by a unique integer), eventually we will use up all the nearby atoms and our computer will run out of hard drive space. Our ability to store our catalog will be limited by how fast we can go farther out into the Universe and gather more atoms and convert them into new hard drives so we can keep counting and cataloging all of the "stuff" we find. At this point our counting exercise and our hard drive building exercise will become one since we need to collect every subatomic particle to add to our hard drive to extend our catalog of all the subatomic particles we find. At this point, the map becomes the territory and vice versa.

But how do you count empty space? The only way we know how to "count" space is by estimating the amount of stuff that would be in that space if it was filled with stuff, which it isn't. This brings us back to Georg Cantor and the problem of counting rational numbers vs. counting irrationals. Like "stuff" you can count rationals, like 1, 2, 3 etc. because there are gaps between them. Cantor proved you cannot count irrationals like you can integers because no matter how tiny a region of number space you select, there will always be an uncountable number of irrationals within it. Space is analogous to irrationals in that there are no gaps. What do you call a gap in between space?

Even though integers are infinite, there are gaps between them, like between 1 and 2. And there are certainly gaps between two prime numbers, even though primes are also infinite. But Cantor's diagonalization proof shows there are no gaps between any two infinitely non-repeating decimals. Instead there is an infinity of decimals in between any two you select, no matter how "close" they are together, which is why they cannot be cataloged.

So if we have a model of the Universe as containing infinite space containing infinite stuff we can still have lots of gaps in between the stuff if what we call space corresponds to the infinity of real numbers and what we call stuff corresponds to the infinity of integers, or primes or even numbers, etc. Both are infinite, but one is bigger than the other. One has gaps, the other has none.

So if space is like an "ether" of all infinitely non-repeating decimals and "stuff" is like the integers, we now have a way to explain how you can have an infinite amount of stuff but still have gaps so the stuff can move around.

**Just keep telling yourself, "This is only an analogy."**

What's more bizarre is the Universe may actually be finite. Vastly immensely huge, but still finite.

The state-of-art "inflationary" model of the Universe posits that around 13.75 billion years ago, a tiny section of the Universe, perhaps only the size of a proton, expanded within a few trillionths of second into the Universe we exist in today. As crackpot as that sounds, hard data from the Wilkinson Microwave Anisotropy Probe (WMAP) shows this idea is by far the best fit for what the evidence shows. And this data effectively rules out a whole lot of competing ideas.

What's interesting about the inflationary model, as buttressed by recent WMAP data, is that the Universe may not be infinite

*per se*, but "our portion" is so huge that it may as well be infinite for anyone within it, in the sense that only a tiny portion the Universe will ever be observable. This is partly because of a weird thing where the most distant galaxies now visible to us (the light we now see from them is from 10 billion years ago), are actually moving away from us at faster than the speed of light. Yes, that's not allowed by E=mc2, However, only "stuff" must obey E=mc2 and space is not stuff. And technically, it can be said that we are moving faster than light speed from these distant galaxies. This is not a big deal, actually. If this were not the case we would enter Olber's Paradox, which says that if there is infinite stuff in the Universe then the entire sky should be blindingly bright with an infinite number of stars and galaxies. And we know that is not the case so we must consider the alternatives:

a) The Universe is pretty big.

b) The Universe is so immensely huge in its big immensity of large bigness and is biggening so fast that some parts of it are moving faster than the speed of light from us and can never be seen by us again.

c) The quick brown fox jumped over the lazy dog.

The WMAP probe tried to see if any part of very early microwave emissions of the Universe appear to be moving faster away from us than others, which would imply some directional movement, ie. that we are not at the very center of the expansion. Since it is obvious that we are not at the center of the expansion of the entire Universe, the lack of any directionality shows that if there is any "shape" or "edge" to the Universe it is so big and far away that it is completely and utterly unobservable or confirmable. This does not prove there is no "edge," but if there is, it is so distant that it will forever remain outside our detection ability. The question of "edge or no edge" will be an unknown that remains an unknown forever.

The inflationary model, by postulating a sudden expansion from a tiny point of space, does "sort of" postulate the existence of a center and edge, unless the point is truly a dimensionless point. It's at least possible that there are parts of the Universe where a WMAP actually shows a slight preferential direction, ie. a pointer toward some "edge." WMAP shows the Earth and Sun and Milky Way Galaxy are definitely not in a place where such a differentiation can be seen.

From our feeble brains' perspective, the idea of a vastly immense and unknowably large and expanding but still finite Universe is much easier to grasp than a truly infinite Universe. But Georg Cantor's hierarchy of infinities provides a way to at least begin to conceptualize the latter.

[1] Thanks to Newton and Einstein's laws of gravity, astronomers have already calculated the entire amount of stuff in the visible Universe and found there to be far less stuff than there should be. In fact about 75 percent of all the stuff that should be in the Universe is unaccounted for and has defied all efforts to observe it. It is provisionally called "dark energy" and "dark matter." Nobody knows what the hell it is.

**UPDATE**: I wrote and posted this with lots of trepidation because earlier drafts kept veering far closer to 'crackpot land' than I would prefer. Devising analogies is a way that helps me think about things that are not easy to grasp in and of themselves. That said, the analogy is not the thing, just like the map is not the territory.

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