I ask this question in the context of thinking about infinite ethics. It seems to me that I would need to give up at least one of three beliefs about the universe in order to coherently think that it is infinitely large. They are:

(i) at some past point in time t the universe was finitely large

(ii) since t a finite amount of time has passed

(iii) since t the universe has always expanded with finite speed

To the extent that I continue to believe all three, I would have to be convinced that the universe is finite. Right?

My very layman's understanding of the big bang theory is that it implies (i) and (ii), i.e. that the universe was initially a finite matter-blob and that the universe started to expand a finite amount of time ago (~13 billion years give or take).  

So, which of these assumptions are given up by e.g. cosmologists who take seriously that the universe is infinitely large? Or, put another way, which of these assumptions is least likely given our current understanding of physics?

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I believe the key issue here is with (i). Standard theories where the universe is infinitely large also suppose it was infinitely large at the moment of the big bang. 

The discussion here may be helpful.

Thanks! The StackExchange discussion is actually very good!

It's not clear that the universe was ever finitely large. At the time of the big bang, everything was packed together very densely, but there still could have been matter going out forever in every direction.

Do you know how common a position this is among cosmologists?

As far as I know, this is the standard position. See also this FAQ entry. A lot of people sloppily say "the universe" when they mean the observable part of the universe, and that's what's causing the confusion.
I think pretty much everyone agrees that the universe could have been infinitely large at the time of the big bang, but I don't know any more than that. I'm just going off an intro to astronomy class I took in college.

We don't know that the universe is infinitely large, only that it seems extremely likely to be very much larger than we can observe. We can model it as being infinitely large, which greatly simplifies the models in many ways. That isn't the same as saying that it definitely is infinitely large.

On to your question, we actually don't know any of those three properties, but (i) is the one that has the least evidence for it. We have no evidence at all that the universe was ever finite. Our best models don't extend backward that far. For as far back as we can gather evidence, what we observe is consistent with space having been denser in a technical way that if extrapolated would have an instant (t=0) of zero size and infinite density, but we know for certain that our models break down before that point. Not just in the sense of no longer matching observations, but in the sense of not having a model that is mathematically consistent.

We do have evidence for (ii) and (iii) in the sense that we are unable to observe anything older than 13.8 billion years, and that the universe at those earliest observable times appears to have been incredibly denser and hotter than it is now, fitting very well a model that when projected backward has finite time since a singularity of density. So we can have pretty good confidence that something spectacular happened, and our most sensitive measurements can't find any hints that it happened even slightly differently or with even microscopically different timing anywhere in the universe we can see.

If the universe was finite and not staggeringly larger than we can see (including "wrapping around"), then we would expect to see signs of it from when everything was closer together. An infinite universe is a simpler model than one of some special finite size that happens to be larger than we can detect, so we go with a model in which it is infinite until we're shown otherwise.

If the universe were once finitely large, what kind of experience would it anticipate?