I'm not asking about how we worked backward from an expanding universe to the age of the big bang, but rather what is the meaning of time in a near infinitely dense point in the context of general relativity? Wouldn't time flow infinitely slowly for a theoretical (though physically impossible) observer?
Answer
No. The standard metric for cosmology is given by:
$$ds^{2} = - dt^{2} + a(t)^{2}\left(d^{3}{\vec x}^{2}\right)$$
where the term inside the parenthees represents the 3-metric of a homogenous three space. As you can see, there is no difficulty with evaluating the age of the universe:
$$ T = \int\sqrt{-g}\,\,x^{a}y^{a}z^{a}\epsilon_{abcd} = \int dt$$
where the integral is evaluated from the time when $a = 0$ to now.
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