What do current theories have to say about any resolution to this question? The visible universe might be finite, but that says nothing about the totality. Should we include the Multiverse in this discussion (if there are many worlds)? What about the nature of quantum foam? Could that have a finite basis. And on and on... Is there any end to it all?
Judging from the WMAP data, we deduce that the universe is extremely close to being spatially flat. In the standard FRW cosmology, a spatially flat universe is infinite.
The question whether space is finite or infinite is not related to whether a spacetime is open, flat or closed. For instance a closed spacetime could be infinite and a flat spacetime could be finite. The fact that FRW models cannot handle finite spaces is obviously not a valid argument here.
Welcome to Physics forum. That question has not been answered yet. My view of the universe is that it will expand forever. Therefore being infinite.
If the time component of space-time is infinite, then indeed the universe must be infinite even if the space component is finite every step of the way. Although, then the space component could be potentially infinite.
I don't see how a flat smooth manifold would not be infinite (although perhaps that is a failure of my imagination). Maybe you refer to a non-smooth model? Could you give an example of such a model embedded in Rⁿ?
It seems that there is no relationship between expand forever and infinite. For example, the tree is growing forever but it isn't infinite high for the reason that the time isn't infinite long, is it?
A tantalizing question. My gut instinct is we will never entirely resolve the question of flatness. CMB anisotropy gives us conficting hints. Too close to call, IMO.
The Big Bang has taken care of that issue, you can not resolve the issue of finiteness/infiniteness of space and/or time from that paradox. I think the question is even theoretically unanswerable, and we just use the gut feeling that spacetime is unbounded (there is no *edge* to spacetime) but that also leaves the issue of finiteness/infiniteness open. My gut feeling is that spacetime is infinite.
How could it be? maybe spatially, but, doesn't the Big Bang make it time-finite, at least in one direction?
Nope. There is nothing in the Big Bang Theory that implies that. The Big Bang Theory is not an explenation of the origins of the universe, but of it's development.
On the subject of infinites in the realm of physics, you have to be careful about some subtle aspects of infinites. First of all, real infinites (like the mathematical infinite) do not belong to physics. Any measure in physics always is of finite proportions. However note that this does not preclude that space and/or time cannot be infinite. Look at it like this: consider the natural numbers and select a number. Now, whichever number you came up with, it definitately is a finite number, and also, you can always select a number bigger then the number you just selected. So the seemingly contradictionary conclusion is that you can select an infinite amount of numbers, yet no number you ever get is itself infinite. Infinite itself is not considered a number. The infinite exists only in the forms of finite elements. So, if we consider space-time like the set of all possible spatial and temporal measures that can be made, the set itself is clearly infinite, although any element in the set is of finite measure. There does not exist a point in time or space infinitely far away.
This is not entirely true but i can see where you are coming from. Yes the infinate can be measured using finate elements, i.e. your number example. But lets say there are no numbers left to pick. Lets say that there is no finate elements which we can use to measure the infinate, the vastness of space. What do we get then? Well, absolute zero because there can be no set of elements that can define the measurement. Furthermore, you can't even call it Zero for Zero is a measurement of some sort. In fact it is utterly impossible to measure something without giving it a set of measurment, i.e. distance, time, dimention, space etc. Thus don't think of Universe being a set of simple measurements that one can simply use for one's own convenience because the moment you do that, you've restricted yourself from the bigger picture.
(heusdens=robheus) Perhaps anyone familiar with the Kalam Cosmological Argument. It's a famous but false argument against the infinity of time. The argument goes something like this: if we suppose that time did not have a begin, we could have never arrived at the moment of "now" because it is impossible to have traversed an infinite amount of time. For people that do not immediately grasp the incorrectness of the argument, just ask yourself, at what point on the time axis is it supposed that we have started the traversal of time. The point is of course, that wherever you have chosen to start traversing the time axis, you already smuggled in as a premise that time had a beginning, since else, you could not have started traversing the time axis at all. The only validity of the argument is that there is no point on the time axis in the distant past that is infinitely far before the present point in time, since we cannot traverse an infinity of time. [ and pls. note, that is just what infinity is by definition, that it can never be exhausted or completed, no matter how hard or how long we try. A "completed" or "exhausted" infinity is nothing more as a contradiction in definition. ] Yet, at the same 'time' this is not to be held against the infinity of time itself, since we can always design a point farther back in time on the time-axis, and thus show that there is no upper limit to a past point in time.
If the universe was infinite, wouldn't it, according to the 2nd Law of Thermodynamics, have no heat? And if it isn't infinite, that means it's finite, and thus had a beginning. Which also means time had a "beginning." So to speak
That is a famous arguments from the creationists, they say that (acc. to physics) the world needed to have a beginning, based on the 2nd law of Thermodynamics. However, their idea clearly contradicts the 1st law of Thermodynamics, which claims that the total quantity of energy in any system, is constant, and thus no creation or destruction of energy is possible. The issue onhand is however much more complicated. Based on E=mc^2 the 1st law of Thermodynamics we need first to consider physical matter too, so the total quantity of both physical matter (mass) and energy is a constant. Second, both laws of Thermodynamics were originally constrained to laboratory scale thermodynamic systems which were closed and had a thermodynamic boundary. The 1st law (in it's contemporary form, based on GR and QM) still holds for all systems, including the universe. Physical matter creation, as what happened on a mass-scale in the early universe, does not contradict that, because it was a conversion of energy in another form. The problem with the 2nd law of Thermodynamics however is that it is still constrained to thermodynamic systems wich have a thermodynamic boundary. This does not apply directly to the universe, because apart from the cosmological issue of open or closed universe, there is in the strict sense no thermodynamic boundary to the universe. There's no border or boundary to the universe (cosmological principle), and that is true even when the universe turns out to be a multiversum. In the Thermodynamic sense the observable universe is an open system, since it does not have a boundary and is in thermal contact with the rest of the universe, beyond our horizon. For the universe in total the terms open or close with respect to it's Thermodynamic behaviour makes no sense since there is no boundary to the "rest of the universe" (since the total universe already encompasses that) so that it is neither open nor closed. But there is also something else peculiar about the universe. You might have seen these pictures of a sequence of moments of time in which a gas in a container spread outs through the container (due to entropy or the 2nd law of TD) and becomes uniformly spread through the container. At the microscopic level however, all physical laws work both ways, so how do we know the progress of time? When we have to order the pictures (let's say there are 3 pictures, one with local concentration(s) of molecules, the next with a medium spread of those molecules in the container but not yet uniform, and the third a uniform distribution) we would clearly say that the progress of time is from picture 1 to 2 to 3. Yet, when this picture was not of a small gass container, but was of cosmological size, we would need to arrange the pictures in the opposite order! Due to gravity, local matter clumbs together forming stars, galaxies and clusters of galaxies. So if we were not told what scale the picture represent or that the scale is a varying quantity, we could not say what the right direction of time was! In the example of the cosmological progress of distribution of matter, there are two important differences with the example of the small gas container. First, the progress of time is from a uniform distribution towards local clutterling of matter, forming galaxies, stars, etc., and the scale of the 'container' grows, due to cosmological expansion. In terms of entropy, this in fact means that the growing metrics of space allow for more possible states, so this in fact means a lowering of entropy at a cosmological scale! If you search online, you might find a lecture of Roger Penrose on this issue of cosmological expansion and entropy, which is very interesting. This is just some basic information, the issue itself is far more complicated as I can explain, but at least I think you get the basic idea that you can not simply scale up our laboratory scale experiment and conclusions based on the 2nd law of Thermodynamics to cosmological scales. I don't give a proof of it, but one could suspect that on truly cosmological scales (the universe as a whole) the issue of entropy is different as expected, and might lead to the conclusion that entropy is a conserved quantity throughout the cosmos, even if at local scales the 2nd law still applies. But perhaps someone else more educated on this subject can explain more details.
Your questions of "finiteness" and "end to it all" restrict the possibilities I believe. 2000 years ago, a farmer would have thought walking far enough straight-ahead would result in falling off the "end" of the world. But in fact, something qualitatively different than "flat" emerges at a large enough scale. Can that also be true of the Universe as well? At a large enough scale, our local concepts such as distance, size, and volume may loose meaning as something qualitatively different emerges. I know that's hard to imagine but it would also have been hard to imagine to the farmers long ago how the earth could be a sphere and not flat. To me, I find that possibility comforting: I no longer wonder how large the Universe is, how it started, and where it's going. I simply believe it is likely, based on historical trends such as the flat earth, wandering planets, moving sun and moon and milky-patch in the sky, as well as all the other critical-point phenomena I see around me, that perhaps something qualitatively different then our current understanding of the world is needed to answer these questions. So my answer is that it may be neither finite nor infinite but rather something qualitatively different.
For your last remark, we could refer to the analogy of the theoretical issue wether or not matter was infinitely divisible or not. Some philosophers argued pro, the other against. What we found - indeed - was neither, as per quantum mechanics and the uncertainty principle. You can split molecules, and then atoms, but when we try to split protons or neutrons into their individual parts. we need such energies to overcome the quark gluon force, that this energy will create new particles.