*Long note on math*
ZF(C) set theory is the basis of modern math and includes the axiom of infinity; something we grant as obvious. "There's no number so big that you can't add 1 to it."
Yet nothing we can observe or measure in reality is infinite. Not the size of the observable universe nor all observable particles in it nor even all possibles states of all observable particles (hot take; see below for explanation).
Should we really use that axiom?
Even the digits of pi can’t be fully observed. Spend all available resources building as many computers as you can and use all matter for computation from now until the heat death of the universe and nothing and nobody will ever experience or observe more than a finite number of digits of pi.
You could argue that we need irrational numbers for algebraic closure. Fine. But even then, in practice on computers or in hand written solutions only finite-precision representations exist. Not a single engineering project in existence relies on full precision of pi or the square root of 2. Yes, even the LHC and LIGO.
"But the ratio of the circumference of a circle and its diameter is pi. That ratio has infinite precision." Well, almost! That assumes a Platonic ideal circle; you're introducing new assumptions even in such a basic concept. Good luck creating such a perfect circle in reality (nothing against the band with MJK). The total curvature theorem that gives us some multiple of pi for any closed loop rewrites integrating over a continuous set so implicitly assumes infinity (ie. infinitely divisible).
What about quantum mechanics? Hilbert spaces as far as the eyes can see! We need a state space rich enough to represent continuous observables like momentum. But that's a circular argument! We claim the observables are continuous but as far as I can tell that's just an assumption. Our theories break down below the planck length which means we can't just assume spacetime is continuous; at best it's a useful approximation.
Continuous spacetime also creates problems like black hole singularities. I think most physicists agree singularities are a flaw in the models and don't represent reality. Quoting Oppenheimer:
"Physically such a singularity would mean that the expression used for the energy-momentum tensor does not take account of some essential physical fact which would really smooth the singularity out. Further ... it is impossible for a singularity to develop in a finite time" - 'On Continued Gravitational Contraction'
And according to Hawking eventually black holes dissipate thus removing the singularity. You can't go from infinite to finite density in finite time yet that's the best our theories have come up with.
We have tools like renormalization to cut out the infinities in some sense and to be honest I don't fully understand the logic there, but it ultimately boils down to using infinities to cancel out other infinities that pop up in our models that, yeah you guessed it, are built on continuous space and therefore more infinities!
In conclusion, as far as I can tell all uses of infinities are useful approximations to make math easier but have quite possibly led us astray.
