But the electron field has different values at different points in spacetime, and we have no evidence that either the number of points (locations) or the number of different possible values at those points is finite. Unless we posit that they are finite in number, infinity is quite present in the universe.

Would not an infinite electron field in a finite (observable) universe result in an infinite energy density and therefore the entire universe would collapse into a black hole?

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

Well, I was referring to the electron field, not to electrons. According to QFT, particles are excitations of an underlying quantum field. It’s the field that is fundamental, not the particle. See e.g. [0]. And those fields are continuous, not discrete, i.e. can only be described by an infinite number of points and values.

[0] https://www.quantamagazine.org/what-is-a-particle-20201112/

Great read, thanks. I think I see the point you're making now - any continuity necessarily requires an infinity by virtue of being continuous.

I'd argue you can't cram any electrons into any finite space without infinite energy. But we're not really talking quantum physics, here.

I'm not sure what you mean - any battery you have would seem to contradict you - would you elaborate?

The electrons are not wholly within the battery; they are mostly within the battery. (Batteries discharge over time for an almost completely unrelated reason.)

I think you're misinterpreting the meaning of the wave equation. Batteries self-discharge due to chemical reactions not quantum tunneling.

That sounds like a weird interpretation of "to be present in the universe" to me. Also I was under the impression that it's unknown whether the universe contains an infinite number of electrons or not.

It's certainly known that the observable universe does not contain an infinite number of electrons, as it has a finite size and finite mass. And it's rather moot to talk about the space beyond the observable universe that can never affect us or anything we can observe in any way whatsoever, so any other statements about it are inherently unfalsifiable, so all the science of physics is relevant only w.r.t. the (finite) observable universe.

Weirder than 'numbers are physical objects'? Can you suggest a more suitable interpretation of the OPs intent?

it's OK. electrons don't "exist" discretely, either.

At best, when you "hand me 7 electrons", you're directing me towards the fat part of 7 probability distributions, so we're back to math again...

Well that's a bold assertion about a theory with whose implications we are still grappling; electrons may not be point-like entities but they are nonetheless quantifiable 'packets' of energy, are they not?

But is that not the locations that are inexact, not the quantity of electrons?