A black hole is a singularity. I guess you could define the event horizon as being the surface of a black hole and the size of an event horizon can be measured (and is non-zero), but the "black hole" itself is a mathematical singularity.

There is no evidence to suggest that the only Big Bang we know about came from a singularity and therefore there is no reason to think other singularities might generate other Big Bangs. Some theories do suggest this might happen, but they also suggest we would never know about it, because the new Big Bang would happen in an entirely other new universe (by definition).

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