Modern cosmologists don't generally believe the Big Bang involved infinite density, nor do they believe that physical singularities of infinite density exist.

Confusion arises because theories predict these things, but one has to be careful to distinguish cases where the theory is being used in a regime beyond its effective domain, from other cases where infinities are less problematic. For example:

> in actual observation nothing infinite was ever seen.

"Real" infinities do show up in various places in physics and cosmology, not in the sense of objects with infinite density, but in more abstract ways. For example, it takes infinite energy to accelerate an object with mass to the speed of light - which we interpret to mean that it is not possible to do that.

A more relevant example is that beyond our cosmic event horizon, light that is emitted "now" will never reach us, even if we wait an infinite time. That is another kind of abstract infinity that doesn't pose any serious conceptual problem.

One effective way to model this case is that the light is infinitely redshifted, very much like light from the event horizon of a black hole. In fact, some models equate the two cases.

You can argue that infinitely redshifted light doesn't actually exist, and in a sense you'd be right - but the point is that very good models predict that the light tends towards being infinitely redshifted, and the effect is that we never see it, even after infinite time.

As such, talking about infinite redshift effectively means that the light in question doesn't exist in the region in question, and "infinite redshift" is a consistent way to model that. Infinite redshift also implies zero amplitude, which again implies that the light in question doesn't exist in that region.

(As an aside, Heisenberg tells us there's no such thing as a consistently zero amplitude quantum field, which gives us a direct link to Hawking radiation from event horizons!)

> I think what he meant was that we have no proof that "singularity" even exists.

As I mentioned, that's not controversial. But what the original commenter wrote was "you can escape [black holes] just fine !" That contradicts everything we know about physics, including our observational evidence of actual black holes.

Importantly, the infinities that arise at the event horizon are also observer-dependent. The infalling observer doesn't see anything special at the event horizon, it's only an external observer that observes time and redshift tending towards infinity.

This is again analogous, if not equivalent to, the cosmic event horizon I mentioned - the cosmic event horizon isn't common to all observers. If you could travel to the location of the cosmic event horizon we observe from Earth, you would find nothing special there, and you would observe another cosmic event horizon 46.5 billion light years away. It's similar to the ordinary visible horizon on Earth in this sense.

So there's no fundamental physical problem in this case with taking the infinities literally, as long as you correctly understand what that means. Besides, for the question of escaping a black hole, it doesn't really matter whether these values actually "reach" infinity - you still aren't going to be able to escape a black hole.