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This article is inaccurate I believe. It's inconsistent to state that we believe that the Big Bang theory is a theory about the origin of the Universe and in the same time state that before some point in time we don't know what happened. If we don't know what happened before some time t, then we don't know the origin, by definition.

I vaguely recall a physicist stating that a more accurate description of the big bang theory is that a long time ago, about 13.8 billion years ago, the whole universe was much, much more dense and hot, actually denser and hotter than the core of any star, and that it very quickly expanded (or rather diluted) and cooled.

Back then it very much is possible the Universe was still infinite in space anyway, it was just much denser and hotter.

And before that, we simply don't know.

> It's inconsistent to state that we believe that the Big Bang theory is a theory about the origin of the Universe and in the same time state that before some point in time we don't know what happened.

If that rule were in force, mathematics would disintegrate because it's built on a foundation of axioms. But scientific theories are often predicated on axioms whose consequences are examined, but whose existence are accepted without explanation until a more encompassing, more global, theory is created. Like relativity replacing Newtonian gravitation -- the new theory still has axioms, but the theory explains and encompasses more.

> If we don't know what happened before some time t, then we don't know the origin, by definition.

A scientific theory is by definition a testable, falsifiable explanation of what can be observed. For each such theory, there are limits to what can be observed.

> Back then it very much is possible the Universe was still infinite in space anyway ...

Yes, unless (imagine time running in reverse) spacetime collapsed to a point, as is often claimed, and as relativity theory requires. In that case, an infinite universe would occupy no space or time at all. The notion of infinity doesn't require infinite size.

As I understand it, space and time didn't exist until the "instant" of the big bang - and that space itself began to expand.

So it's not like a firecracker gases expanding to fill an empty box.

It's true that space-time itself did expand, but it's only speculation that space itself was created at the big bang. There may be theories that go far enough to explain the creation of space-time, but that's not the big bang theory.

These kinds of infinity concepts are throughout math and physics. For example, the Universe is "infinite" yet it is also expanding. There is very little that is logical about infinity itself and it has been the source of much philosophy throughout time. It took a couple thousand years just for mathematicians to theoretically prove that some infinities are larger than others, for instance.

> There is very little that is logical about infinity itself ...

On the contrary, infinity is a very logical notion -- much more logical than many other ideas we accept. For example, what value must "m" have for this statement to be true:

sum(n:1 to m inclusive, 2^-n) = 1 (i.e. 1/2 + 1/4 + 1/8 ...)

What value must "m" have, for this statement to be true:

lim(n -> m), (1+1/n)^n = e (base of natural logarithms)

For both these examples, for the described outcomes, m must equal infinity.

Also, you say, "For example, the Universe is 'infinite' yet it is also expanding", but you also say, "It took a couple thousand years just for mathematicians to theoretically prove that some infinities are larger than others ..." If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding.

Imagine a cube of space, in isolation, initially measuring one meter in any of its dimensions. If measurements reveal that this value is increasing over time, one may conclude that the cube is expanding. Now imagine that the cube is embedded in a space consisting of an infinity of such cubes in three spatial dimensions, all increasing in size in the same way. Does the fact that the cube is part of an infinity of such cubes prevent it from expanding? If this were true, it would be an easy test of the existence of an infinite spatial manifold -- but the idea of an expanding infinity is logically consistent.

I think that saying that m "equals infinity" is probably a bit of a simplification.

One can say that some values are infinite, such as omega in the ordinal numbers and the surreal numbers, but neither of those values are what is meant in the examples you gave, which are both limits.

If you plugged in omega (from the surreals) for those expressions instead of taking the limit, I think you would get a value which is off by around epsilon (i.e. 1/omega). I'm not sure of that but I think that's how it tends to work out.

In the surreals one can indeed have a continuously increasing infinite value. But I'm not sure that that is a proper way to describe the "total volume of the universe"? Maybe it is, but idk.

> I think that saying that m "equals infinity" is probably a bit of a simplification.

Yes, it is, I agree. For the limit example, one might instead say, "As m approaches infinity, the result approaches 1." By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things.

The other reason I agree is because infinity isn't a number. I'm guilty of compressing the lexical description of a limit to a ... ahem ... finite size.

> ... what is meant in the examples you gave, which are both limits.

One of my examples was an infinite sum, not a limit per se, but the same argument can be made, i.e. that infinity is addressed without being literally achieved.

I believe that infinite sums are usually defined in terms of limits, which is why I said that both examples were limits.

> By definition, limits approach their target values, they never equal them -- otherwise we would accept division by zero and similar things.

I think I would say that a convergent sequence (most of the time that one cares) does not ever reach the target value (though, eventually constant sequences do, but those aren't interesting), but that the limit of the convergent sequence is equal to / is the target value. As opposed to saying that "the limit approaches". The limit isn't the thing that changes with another thing, but the limit of some changing?

>If one infinity can be said to be larger than another, then this admits the possibility of any single infinity expanding.

True, and that is the point I was trying to make.

My use of the word "logical" was perhaps a bit rushed, as another pointed out, more likely "intuitive" was a better word.

> There is very little that is logical about infinity itself [...]

I assume that by "logical", you mean "intuitive"?

The Universe is not infinite

> it doesn't make sense to me how something could be described as infinite (density) and finite (time) at the same time.

The sequence A_n = 2-n = 1, 1/2, 1/4, ..., has infinitely many terms, yet a finite sum 2.

Different properties of a thing are different after all.

After the Planck's time (10^-43s) Physical laws already existed

Planck time is a red herring anyway, it has no know physical significance and is really heavily overused. The same holds for all the other Planck units.