This is a quote from Wikipedia:
"according to standard cosmological theories on the shape of the universe, it has no center."

As for a center that lies outside the universe, if we go for a moment to the point in time just after the Big Bang, or maybe at that very point, then at that point didn’t the universe have a center located exactly at the point of the explosion? And if so, how did the center move from a point inside the universe to a point outside it? And when did that happen?

Think of the surface of a spherical balloon. Such a surface has no center at all. You can define a center for it that lies outside it if it is embedded in three-dimensional space.
Now think of a balloon inflating (this is the standard analogy for the expansion of our space). At first it was a point, and that was its center. Once it began to expand, it no longer had a center (though one can define one outside it, as above).
Now think of Arik Einstein’s banana bender. A full cylinder begins to bend into the shape of a banana. At some point it stops having a center within it.
If the universe is finite but unbounded (like a circle or the surface of a sphere, but in three dimensions), then it has no center.

And what about the size of the universe? How can it be that it has no size at all?

I don’t know what it means to say it has no size. I’m not familiar with those claims.

If you think it has a size, how can it not have a center? After all, every three-dimensional shape with finite size has a center.

As I explained, a banana or the surface of a sphere has no center. Think of a two-dimensional creature crawling on the surface of a sphere. What is the center of its world?

The topic is fascinating!
A few difficulties with Michi’s answer.
First, in my opinion the analogy does not resemble the thing being analogized to (the balloon image does not serve the argument).
Second, even if the analogy did resemble the thing itself, Michi’s analysis of the analogy is hard to accept.

The analogy does not resemble the thing itself because when we talk about a balloon, we assume the existence of a material object given a priori in space and time. In other words: there is an external frame of reference for the discussion. By contrast, when we talk about the universe, I do not see any such frame of reference.

But even if there were similarity between analogy and thing analogized to, the claim about the surface — that supposedly it has no center — is unclear. The surface, by definition, encloses; that is, there is a center here (the region that is “enclosed”) and a periphery (the sheet of rubber surrounding that region). So in response to Michi’s example of the two-dimensional creature, the answer is simple: from the very formulation of the example it follows that his world does have a center, even if he can never know or grasp it, and of course would not be able to formulate it that way. From the point of view of the person formulating the example, the idea of center and periphery is already built in.

So what, then? If the universe is finite — even if it is in a process of development — then it probably has both size and center. That seems to me to be an a priori requirement.

I agree that two-dimensional shapes sometimes have no center, but three-dimensional shapes always have a center.

Oren,
I gave the example of a banana. Beyond that, there is no dependence whatsoever on the number of dimensions. You are simply used to three dimensions and unable to imagine the curvature of three-dimensional space (like the curvature of a spherical surface). But there is no problem here in principle. The example of the surface is in two dimensions curving within a third, and from there generalize to three curving within a fourth. And by the way, the sages of mathematics have already taught us that you do not need an embedding dimension in order to define curvature, so I added the third and fourth only to simplify.

A banana also has a center; admittedly it is outside the banana, but it has a center in the three-dimensional world. Regarding the universe, the claim is that it has no center at all in the three-dimensional world. For example, if our universe were shaped like a banana, then it would have a center, perhaps at a point where there is no matter at all, but it would have a center.

How do you know that things that are true in two dimensions are necessarily also true in three dimensions?

If the universe is shaped like an infinite banana, then it has no center within it (there is one outside it, but that is like the spherical surface, whose center is outside it). From this it follows that just as a two-dimensional surface has no center within it (only outside it), so too this can be the case with a three-dimensional universe. The dimension, of course, does not affect this in any way. If you want to claim that this can happen in one dimension and in two, but not in three, the burden of proof is on you. Beyond that, you raised the difficulty and declared that it is impossible for there not to be a center — you are the one who needs to prove your point.
But leave aside all the pilpul about burden of proof; let me show you, even though the burden of proof is not on me. It is very easy to construct such a three-dimensional surface mathematically. Define a three-dimensional spherical surface in four-dimensional space (X,Y,Z,T): X^2+Y^2+Z^2+T^2=R^2. This shape has no center in the three-dimensional universe (X,Y,Z), because the center is at the point (0,0,0,0), which does not satisfy that equation — meaning it is not on the surface I defined. That proves it. Note that for every value of T we have a two-dimensional spherical surface (with a different radius). So there is here an infinite three-dimensional volume, but its center is not inside it. It has no center.

Yes, I understood all that; that is why I wrote above that the claim about the universe is not that it has a center, only that it is outside it, but rather that it has no center at all. Everything that exists in three-dimensional space, including the examples you gave, has some center (even if it lies outside the shape itself). What I am asking is how can it be that the universe has no center at all? Especially if you hold that it is finite.

It is not only outside it, but not in three-dimensional space at all. Even if it is finite. The surface of a sphere is finite.

How is that possible? Every finite three-dimensional shape has some center in three-dimensional space.

I gave a counterexample: a three-dimensional sphere whose center is not in the three-dimensional space that it defines. Just as a two-dimensional shape does not necessarily have a center in its own two-dimensional space.

I understand that there can be shapes whose center is not inside the space they define. But the claim about the universe is not that its center is not inside the space it defines, but that it has no center at all.

In the case of the sphere, the center does not exist in three-dimensional space at all, not only in the sphere’s own volume/space. It is in four-dimensional space.

Regarding the example you gave,
you said it was a shape with infinite volume, whereas regarding the universe you said above that it is finite. Can there be a shape with finite volume in three-dimensional space and no center?

Absolutely yes. The sphere I described. The volume is finite (I was mistaken there. When T exceeds R there are no solutions. We are talking about a collection of spherical surfaces, one surface for each T up to R). Exactly like the volume of a two-dimensional spherical surface, which has finite area and no center in its own two-dimensional space.

A basic cosmological principle is that there is no region in the universe that is more special than any other region.
One can imagine the surface of the Earth as a two-dimensional universe. That surface has no center within that two-dimensional universe.

But if you relate not to the two-dimensional universe, but to the three-dimensional universe, then the example you gave does have a center.

The two-dimensional surface has a center in three-dimensional space, and the three-dimensional sphere in four-dimensional space. And even that is only if there really is a four-dimensional space in which the sphere is embedded. If there is not, then it has no center at all.
We are repeating ourselves again and again.

The three-dimensional sphere has a center not only in four-dimensional space; it also has a center in three-dimensional space (unlike the two-dimensional surface, which has no center in two-dimensional space).

That is incorrect. Its center is at the point (0,0,0,0), and that is of course not in the three-dimensional sphere (proof: those values do not satisfy the sphere’s equation — that the sum of the squares equals R squared).

I did not claim that the center of the sphere is inside the shape it defines, but that it has some center in three-dimensional space. As for the two-dimensional surface, it has no center at all in two-dimensional space, not merely that the center is not on the surface. The same applies to the universe: the claim is that it has no center at all, not that its center is not inside it.

Wow. It has no center in three-dimensional space. Not even outside the shape.

It does. That center is at the point (0,0,0).

Incorrect. Just as for a two-dimensional surface the center is not (0,0) but (0,0,0).

I agree regarding a two-dimensional surface that its center is not (0,0) (and in fact it has no center at any point in two-dimensional space). But regarding the spherical surface you gave in three-dimensional space, the point (0,0,0) is the “center of mass” of the shape, or the point that is equally distant from all points of the shape (that is, the center of the shape).

I understand that the two-dimensional surface example is brought as an analogy, but I claim that not everything that is true in two dimensions is true in three dimensions. There may be phenomena that are true there and not here, and vice versa.

Returning to the heart of the discussion: from what I read online, the reason scientists say the universe has no center is because it has no edge or boundary, and in their view it is infinite (which also does not sit well with me). But if the universe were finite, as you argued above, then clearly it would have some center of mass. They also claim that the Big Bang did not happen at some point in space, but happened everywhere in space (which also does not sit well with me).

What I may perhaps have managed to understand from what they are saying is that space is like Hilbert’s hotel. Already at the moment of the Big Bang, space was infinite, and after the Big Bang every point in space moved away from the neighboring point and created new space, and that process continues to this day. In such a picture it is clear there is no center. The question is whether I understood correctly.

Everything that happens in two or three dimensions happens in every other dimension as well. There is no difference at all in these contexts. You are simply finding it hard to imagine.
The point (0,0,0) is definitely not the center of mass. Think, for example, about the distance between it and the point (X,Y,Z,T) on the surface. Is the distance R? Obviously not. There simply is no such distance. It is not defined at all. So that is not the center of mass.
It seems to me that what is confusing you is the idea that this is a full three-dimensional sphere. But that is not true. What leads you to that mistaken conclusion is the assumption that for each value of T (up to R) we have a spherical surface, and therefore the sum of all of these is a full sphere of radius R. But that is a mistake. Think again about the two-dimensional analogy:
X^2+Y^2+Z^2=R^2. Fix a value of Z (less than R), and you get a one-dimensional circle on the X-Y plane. But note that this circle has a different radius (the square root of R^2-Z^2), and it is located at a different height. Therefore all these circles together do not add up to a full two-dimensional disk centered at (0,0). It is a collection of circles at different heights. Exactly the same thing happens in three dimensions. Every value of T determines a two-dimensional spherical surface, but at a different T value and with a different radius. So they do not add up to a full sphere. Therefore their center is not (0,0,0) either.
That’s it. I think we’ve exhausted this. Forgive me, but I’m completely worn out. If you want, we can talk verbally.

You are trying to imagine these mathematical pictures, and that is very difficult. You have to look at their mathematical description. We got very tangled up here even with a simple circle.

Michi,
I understand that you are claiming there is no mathematical problem with assuming the existence of a finite three-dimensional body that has no center (inside or outside itself).
What I do not understand is whether you are claiming the possibility of such a body also outside mathematics. If such a body does indeed exist there as well, what is its status? Physical? Metaphysical? Logical? Or something else?

There is no difference at all between the questions. Anything that is mathematically possible can also be realized in practice. The question whether it is realized or not is an empirical question (you have to check and see whether such a thing exists).
Of course, as part of the empirical check I examine whether it contradicts some law of nature, in which case it probably cannot be realized. That is not the case here. There is no law of nature that forbids the existence of such a body. Certainly if scientists claim that this is in fact the state of the universe, then there you have it — it has also been realized in practice.

I figured that’s what you’d say. In my opinion, you’ve opened the door here to another difficulty, more fundamental, but since that leads us to a somewhat different topic I’ll present it in another thread. Many thanks.

Oren

What Rabbi Michi is calling here a three-dimensional sphere that has no center within itself is not precise. He is talking about the boundary of a four-dimensional sphere, which is itself a three-dimensional entity — a spherical surface — but three-dimensional (just as the boundary of a three-dimensional sphere is a spherical surface and is two-dimensional. But it is not a circle. So too the boundary of a four-dimensional sphere is a three-dimensional entity but not a sphere, and it has no center within itself. At the center of the four-dimensional sphere, yes, but within itself, no). So that is the example of how our universe can have no center even though it is three-dimensional — if it is, for example, the boundary of a four-dimensional sphere.