How can something infinitely big have expanded from an infinitely small? Please help me reconcile what I see as contradicting theories:

The universe began with the Big Bang and expanded from an infinitesimally small point.
The universe is infinite.

How can something infinitely big &quot;expand&quot; from an infinitely small point? There has to be some sort of a 'blast wave', right?
Also, if we think we know the total amount of matter the Big Bang must have produced - how can a limited amount of matter fill an infinitely big universe?
P.S. One idea I have is that the universe may be infinite, but the Big Bang didn't create it - it created matter within the infinitely big universe. This would explain the expansion we're observing now: the matter will continue being distributed across the infinitely large universe. So I guess my question is more of a &quot;why this assumption is incorrect&quot;... :)
The universe started with &quot;big bang&quot; is a slogan for &quot;There was a time when the universe was hot and dense, and this time may be considered to be the start of the universe, and the start of time.&quot;
The universe has (as far as we know) always been infinitely large.  That's not certain, but we have no evidence contradicting it.  The universe was never an infinitesimally small point. It was always infinite at every time t&gt;0.
We know how much matter the big bang has produced in the observable universe, and we can model this. (We don't really know why it produced this much matter, but we have enough parameters in our models to get a universe like ours).
We also think that the universe is kind of the same everywhere.  Again, there is no proof of this, but nothing contradicts it, and so there would be an infinite amount of matter in the universe.
There was no &quot;blast wave&quot;.  Because the whole universe was hot and dense at the same time.  The whole infinite universe, filled with an infinite amount of matter, which then expanded. (and yes, an infinite thing can expand, it just means that things within it get further apart.)
As for t=0 or for t&lt;0  We have no idea.  I don't know if those are even things that we can talk about. It seems that we can talk about the state of the universe at time t=1 second.  But we can't talk about the state of the universe at t=0 seconds, at least, not with the mathematical and physical knowledge that we currently have. Time t=0 is a singularity.