Is the universe finite or infinite?

The current, widely accepted model for cosmology is $lambda$-CDM. The universe appears (exactly) flat, and for simplicity the universe is infinite. Note that we distinguish between the observable universe (which is the local patch that light could have travelled between since the Big Bang) and the totality — we have constraints that even if the universe is not infinite, its size is many orders of magnitude larger than the observable one.

In the literature (especially the popular science one) the details are very muddled, because the consensus around $lambda$-CDM model is quite recent — relying heavily on detailed measurements of the cosmological microwave background radiation, largely done by WMAP in the last 8 years or so. In a sense, the lay reader should be exceedingly careful when she reads statements (even from heavy-weight scientists) regarding cosmology — it is (perhaps ironically) a fast moving field.

It's impossible to know whether the universe if finite or infinite because we'll never be able to see it all. Note that genneth says "and for simplicity the universe is infinite", and this is the key point really. It makes Physics simpler if the universe is infinite so we tend to assume it is.

But you need to consider what you mean by "infinite". It doesn't make sense to say the universe has an edge, because you then have to ask what happens if you go up to the edge then take one more step. That means the only alternative to the universe being infinite is that it loops back on itself like a sphere, so you can walk forever without reaching an edge, but eventually you'll be back where you started.

We don't think the universe is like a sphere because for that spacetime would have to have positive curvature, and experiments to date show space is flat (to within experimental error). However spacetime could be positively curved but with such small curvature that we can't detect it. Alternatively spacetime could be flat but have a complex global topology like a torus. The scale of anything like this would have to be larger than the observable univrse otherwise we'd have seen signs of it.

Incidentally, if the universe is infinite now it has always been infinite, even at the Big Bang. This is why you'll often hear it said that the Big Bang wasn't a point, it was something that happened everywhere.

Later:

I've just realised that you also asked the question about time beginning at the Big Bang. In the answer to that question I explained how you use the metric to calculate a geodesic, with the result that you can't calculate back in time earlier than the Big Bang. You can also use the metric to calculate a line in space at a fixed value of time (a space-like geodesic). Our universe appears to be well described by the FLRW metric with $Omega$ = 1 that I mentioned in the other question, and if you use this metric to calculate your line you find it goes on forever i.e. the universe is infinite.

But then no-one knows for sure if the FLRW metric with $Omega$ = 1 is the right one to describe our universe. It's certainly the simplest.

There is always the problem when answering this question that General Relativity, naively interpreted, allows you to speak about the part of the universe which is not observable from our vantage point, and this makes the question nontrivial.

But in a logical positivist perspective, the one suggest strongly by string-theoretic holography, the universe is exactly the stuff inside the cosmological horizon, and it is finite because the cosmological horizon is of finite area. There is no objective meaning to stuff outside the cosmological horizon, so there is no point in thinking about this--- it is meaningless in the sense of Carnap.

Nobody knows. But most probable theory says it is infinite.

UPDATE

The general theoretical description of Universe is given by Friedmann–Lemaître–Robertson–Walker metric. This metric allows Universe to be both finite (closed) and infinite (opened). This depends on k parameter. If k<=0 then the Universe is infinite. Current observations show that k is close to zero, which can mean either infinite or very big (much larger than 14 billions of ly which is observable space size).

I treat this as it is most probably infinite.

UPDATE 2

Lambda-CDM model cannot itself answer the question if the Universe is finite or not. This is just a summation of observable data about what Universe is consist of.

The universe cannot be infinite because everything has a limit and infinity is not apllied to stuff which is real

Although the general theoretical description of the Universe is given by Friedmann–Lemaître–Robertson–Walker metric, and although it allows the Universe to be both finite (closed) and infinite (opened), scientific observation has shown that the universe began a finite period ago (approximately 13.798 Billion years ago). There was a big-bang, inflation of the universe. All motion of stars and nebula are travelling away from each other from an apparent singularity, and this 'bang' left residual heat and still evident structure (to the universe).

Along the 'time axis' in the negative direction then, the universe is finite. Along the time axis in the positive direction it may be theoretically possible for the universe to age infinitely, but it has't yet, and at every point aging in that direction the universe will still be finite back to its origin. At no point in the future will the age of the universe be infinite, and it isn't currently.

So the age of the universe is currently and will only ever be finite. (Besides, David Hilbert proved that it is physically impossible to have an infinite change of events, so even as we approach infinity, the universe's age will be finite, perhaps very large, but still finite).

If the universe were spatially of finite and not too large volume then we could, in principle, discover this by observations in the future. For example, there might be evidence of light setting off in opposite directions eventually arriving at the same point after travelling around the universe, or something like that. This can happen even if the average curvature is zero or negative if the topology has the required form.

However if the universe is very large then we will not be able to make observations like that. We will not be able to tell whether it is infinite or just very large.

In fact I think it fair to say that no experiment, even in principle, could establish beyond reasonable doubt that the universe is in fact spatially infinite. Certainly it is not something anyone can claim to know for sure.

What happens is that people working in cosmology find that many of the ideas do not require one to know whether or not the universe is infinite, so if one is studying the average properties one might just say "oh well let's just treat it as if it is infinite". It has become so common to do this that people often forget that this is not a hypothesis that has been tested at all. It is just a working assumption, or a way of avoiding the need for more information. But if you query this assumption then it is not at all clear whether or not it is right. The idea of infinity is reasonably well-defined in mathematics, but it is not clear whether physical stuff can be infinite. Do we really know? No we do not.

So the short answer to your question is "no-one knows". And the longer answer is "very likely no one will ever know."