How many real numbers exist? New proof moves closer to an answer

The results in the article are based on 1st-order logic,

which is inadequate for the foundations of mathematics.

Mathematical abstractions need to be characterized up a

unique isomorphism as in the theory Ordinals described in

the following article:

    "Theory Ordinals can Replace ZFC in Computer Science"

      https://papers.ssrn.com/abstract=3457802

Mathematical questions can be properly addressed only by

using adequate foundations.

Yes, what you are saying is indeed the case in the sense that if BB(30)=n for some particular numeral n is independent of whatever proof system we are focusing on, for the sake of argument let's say ZFC, (though 30 feels a a bit on the low side for ZFC), then we can consistently add BB(30)=n or BB(30)≠n as axioms to the system, the same way to can for any other independent statement. And there will be arbitrary large values of n that are independent, so we could add BB(30)=n or we could add BB(30)=m or so forth for any numerals i so long as BB(30)=i is independent of ZFC (or whatever axiom system we are considering).

But! that does not mean that all these systems are sound, for if you add the incorrect value as an axiom, specifically if you add anything but the smallest value 'm' such that BB(30)=m is independent of ZFC, then you are in a similar situation to adding ¬Con(ZFC) whereby you only have non-standard models.

They way this works is that, suppose BB(30)=m for some particular m is independent of ZFC, but BB(30) is not actually equal to m. What we will find is that there is some Turing machine M and some non-standard natural number q such that, while M doesn't actually halt in reality, according to some non-standard model it does halt in 'q' number of steps, and, in this model, when it does halt, it leaves 'm' 1's on the output tape.

That said, your comment did push me towards the limits of my comfort zone, so it might be helpful to double check what I'm saying. I don't see how it could be any other way though.

For this entire argument I will be operating under the assumption that ¬Con(ZFC) is independent of ZFC.

ZFC+¬Con(ZFC) proves "there EXISTS a natural number x such that x is an encoding of a proof in ZFC of 0=1".

Now suppose there is actually a numeral n, i.e. a specific written term written using symbols, e.g. (+,*,1), such that ZFC+¬Con(ZFC) proves that "n is an encoding of a proof in ZFC of 0=1". We can mechanically decode such a n and check if it is indeed a proof that that ZFC proves "0=1" or not.

There are two possibilities: (a) the check is successful and thus we have found proof of a contradiction in ZFC. But if ZFC is inconsistent, then it proves everything. In particular ZFC proves ¬Con(ZFC), which violates our assumption that ¬Con(ZFC) is independent of ZFC.

Alternatively (b) our check fails and n does not encode a proof in ZFC of 0=1. But that statement "n does not encode a proof in ZFC of 0=1" is a true Delta_0 statement, and we can prove by induction that ZFC proves every true Delta_0 statement. Thus we can prove that ZFC proves "n does not encode a proof 0=1", and hence ZFC+¬Con(ZFC) proves "n does not encode a proof 0=1". But now we have found a statement Q such that ZFC+¬Con(ZFC) proves both Q and also ¬Q. This means that ZFC+¬Con(ZFC) is inconsistent. But if ZFC+¬Con(ZFC) is inconsistent, by the deduction theorem, ZFC proves ¬¬Con(ZFC), or equivalently ZFC proves Con(ZFC). This contradicts our assumption that ¬Con(ZFC) is independent of ZFC (it also implies that ZFC is inconsistent).

Thus we are left with the conclusion that there is no such numeral n. However it is still the case that ZFC+¬Con(ZFC) proves "there EXISTS a natural number x such that x is an encoding of a proof in ZFC of 0=1", and the only way this can be the case is any such x is a "natural number" which has no numeral that denotes it.

Every integer can be written down with a finite number of symbols....right?

I find this very much fascinating, although I can't understand it. I hope I can, eventually, fully appreciate "higher math" questioning like this one, and would be really grateful if someone could point me what to study, as an "amateur mathematician" with only background on engineering calculus.