I give back what I recieve just as the law requires

Starter mathematicians might think everything is math.

Senior mathematicians lose their sleep whenever they think about what our math is missing and they don't know what it is yet

Do we have ways of computing orders of infinity higher than ℵ₀?

Crap. Markdown isn't markdowning. Fixed with unicode.

Symbolically, sure, but then you're not dealing with infinities you're just representing them.
It's a meme it's playing fast and loose with things but the general gist is that mathematics, to this day, doesn't really care about Gödel/Church/Turing, incompleteness, the halting problem, whatever angle you want to look at it from. Formalists lost the war and they simply went on doing maths as if nothing had happened, as if a system could be simultaneously complete and consistent. There's people out there preaching to the unenlightened masses but it's an uphill battle.
- Math went on because it doesn't matter. Nobody cares about incompleteness. If you can prove ZFC is inconsistent, do it and we'll all move to a new system and most of us wouldn't even notice (since nobody references the axioms outside of set theorists and logicians anyway). If you can prove it's incomplete, do it and nobody will care since the culprit will be an arcane theorem far outside the realm of non-logic fields of math.
- We have sorta the same problem with imaginary numbers, and I remember some programmable calculators can process complex numbers using symbolic representation (which happens to work similarly to Cartesian coordinates, so that's convenient)
  But from what I remember any infinity bigger than counting numbers (say the set of real numbers) cannot be differentiated from each other, so we don't have established rules.
  To be fair, I last tinkered with infinities in the aughts and then as a hobbyist. The Grand Hilbert Hotel can accomodate more compound infinities and still retain perfect utilization since the last time I visited.
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Your meme doesn't work in the age of floating point numbers.

Suggestion: "1.0/x for x>0→0.0 goes against infinity, so 1.0/0.0 is infinity."

[TREE(TREE)] ↑↑↑ (Busy Beaver)
- Any sufficiently large number is indistinguishable from infinity.
  Wait. Something's wrong...
Floats are actually fairly limited in what they can do and you lose a lot of accuracy at some point

So that they can be taxed, of course.
- Well... tax on infinity dollars is (crunches numbers) infinity dollars! Government deficit averted!
Computable. There are countably many computable numbers since there are only countably many possible programs. Non-computable numbers can't be exactly referred to / described / constructed by a program, so if your point of view is that everything is a program, you would say they don't exist.
- homotopy type theory has a way to describe the reals in full generality via cauchy sequences, so if type checking ever gets proven to be decidable, this won't be true any longer. it's chapter 11 in the hott book.
my point exactly
- The meme is about computability of numbers though, not countability of sets of them

The universe is actually a computer program

The mind is actually a computer program

When someone has a hammer everything looks like a nail.

When someone is paid a lot of money because they have that hammer, the belief that everything is a nail seems backed by self-assigned genius and unique insight.