There could be an infinite amount of matter, right?

[Frylock]

Last I heard, if the universe is unbounded, then the "Big Bang" doesn't consist in an expansion of space in the sense of a growing region, but instead in the sense of lessening density. So, it would appear to me, there could be (and could always have been) an infinite amount of matter. This infinite amount of matter used to be much denser than it is now, but that has nothing to do with whether the amount is finite or infinite.

Am I right? Or has it somehow been ruled out that there might be an infinite amount of matter in an unbounded universe?