If the universe is infinite in size how does that fit with the Big Bang?

You would need an infinite rate of expansion but currently the rate of expansion seems quite finite. Did the inflation period include a moment where there was an infinite expansion? And even then any finite amount of matter would be stretched, on average, infinitely thin. An infinite universe seems quite... show more You would need an infinite rate of expansion but currently the rate of expansion seems quite finite. Did the inflation period include a moment where there was an infinite expansion? And even then any finite amount of matter would be stretched, on average, infinitely thin. An infinite universe seems quite problematic to me. What do you with your superior knowledge of physics say?
Update: DLM - Missing the point, I'm you asking to reconcile two things that don't seem to fit together in my mind. Not making an argument against the Big Bang.
Update 2: Petrusclavus - The observable universe, yes. But some people, scientists even, think that the universe continues on beyond what we can observe due to light's finite speed and is probably infinite. This to me seems to create problems.
Update 3: Erica s - Reading comprehension may be required. I used the word if deliberately. If the universe is infinite, that seems problematic to me. Hence the question is to either resolve the problems or say they're unresolvable in which case we get reasons for thinking it's not infinite.
Update 4: Krae - There is every reason to think the Big Bang happened, Cosmic microwave background radiation and redshift for example. The Big Bang is not in question. What's in question here is how could the universe be infinite when all space and time was contracted into a single point in the distant past?
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