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DuncanS comments on Query the LessWrong Hivemind - Less Wrong Discussion
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It's not completely so - it might become apparent that the universe was finite, and that would answer the question. But determining that it is infinite is different - after any finite amount of time, you can only see a finite part of any universe. So you will never know for sure.
How would this come about? The entire problem of attempting to make observations beyond our lightcone is that there are no interactions beyond that boundary.
If you started seeing earlier versions of the same part of the universe that you now are standing in at an apparently large distance out in space, you pretty much know the universe is finite. Light travelling all the way around the universe and arriving back where it started would be a large clue that the universe was finite.