— How exactly is a singularity (and electrons) a point with no volume? Space is in 3 dimensions, so everything has to has some thickness, right?
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In a quantum mechanical sense, electrons necessarily have volume.
They exist as a sort of cloud of probability in an atom, and the volume of that “cloud” can be worked out.
You can find the electron as a pointlike particle at a random location in that “cloud”, but the key word here is pointlike. It displays some some behavior as if it was a point, but if you measure its properties you will always find a length of some sort.
You will run into the wavelength of the electron, something called the de Broglie wavelength. This wavelength is equal to planck’s constant divided by the momentum of the particle.
If you think about it, and if you know a bit about quantum uncertainty, you might realize that trying to measure the electron smaller than that quickly becomes fruitless.
The more accurately you try to pinpoint the position of the electron, (in order to confirm it has no volume and is indeed a point) the less you know about its momentum. And the less you know about its momentum, the less you know about its de Broglie wavelength.
The fundamental limit as far as quantum mechanics goes is what’s called the planck length, or planck volume for volumes. Smaller than that and we end up with 100% quantum uncertainties. So it’s impossible to measure the wavelength of an electron to be any smaller.
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