Post by Khong DongFyi., https://qr.ae/pGmiyy
Well, I know a bit of number theory, but I have to admit I don't know much about this; hope some expert at Quora can give you a useful answer. Of course, number theory has a lot of problems which are very easy to state but extremely difficult to solve.
Note that the thread question doesn't ask for a proof, only some possible cause-explanation which could be wrong but shouldn't be a wild guess.
The explanation should be of the of the nature of probable-cause, sufficient logical ground to force the alleged finitude to stand trial (so to speak).
[Otoh, your comment "number theory has a lot of problems which are very easy to state but extremely difficult to solve" seems to indirectly yield some mild hint: logically speaking, the set of "extremely difficult to solve" problems is indeed a subset of that of "impossible to solve" ones.]
Again, we don't have to give any proof or disproof here; and I'm just giving some highlights of the probable causes for the apparent difficulty of the ABC conjecture. (In one post, anyway).
=======> First step
Let's assume the alleged ABC finitude problem can be translated into the ABC' finitude problem written in the language of arithmetic L(<,*,S,+,0) in the sense that:
(R |= ABC) => (PA |- ABC')
then if we can prove that
undecide(PA |- ABC')
we'd be home free: It'd be invalid for one to claim to have a proof of (R |= ABC) -- the ABC conjecture thesis -- IUTT or not.
=========> Second step -- Probable causes
What are the probable causes for the absolute undecide(PA |- ABC') or undecide(PA' |- ABC') where PA' is an _AXIOM_ set extension of PA?
---> First probable cause -- Trichotomy (multiple choice) axiom.
Axiom N8 (Shoenfield's PA axiom, pg. 22) harbors/embeds a provability incompleteness (undecidability) -- N8 being a multiple choice of 3 mutually exclusively negating-each-other formulas.
---> Second probable cause
We can't alter the axiom-hood of N8 -- due to the mathematics postmodernism position, w.r.t to the concept of "natural numbers".
---> Third probable cause
As a logical consequence of the 1st and 2nd probable causes, the entire set of PA-axioms constitutes a formal system _variable_ ranging over an infinite family of arithmetical sibling-axiom-sets --- *sibling*-formal systems:
(1) - any two of which would have common theorems
(2) - any two of which the union of theorems would be inconsistent
(3) - any two of which would have common undecidable formulas.
---> Fourth probable cause - Mathematical absolute uncertainty (undecidability)
"Yo soy yo y mi circunstancia"
("I am I and my circumstance")
https://en.wikipedia.org/wiki/Jos%C3%A9_Ortega_y_Gasset
The particular simultaneous combination of (1), (2), (3) w.r.t. to the well-ordering of the primes of the concept "natural numbers" would constitute the following absolute undecidability of PA-axioms as a formal system variable:
- If it's possible to know what the formal system variable PA denotes, it's impossible to prove alleged finitude induced by ABC' in that denoted system.
- If it's possible to prove the alleged finitude induced by ABC' in one system of the range the formal system variable PA, it's impossible to identify that formal system, hence the alleged proof is invalid -- there's still no proof.
Iow, the final uncertainty of proving the alleged ABC' finitude is a superimposition of both the local uncertainty ("I") of any one formal system in the family, and the global uncertainty of the whole family ("my circumstance") each local formal system is in.
=========> In summary ...
It's then only a matter of providing detailed information as to what such family of formal systems, and what local and global uncertainties might be, to show we have some probable causes.
And I think I have those details in:
- the formal systems 0PA, 2PA, 4PA, ....
- undecide(cGC)
- ....
in the "quantum_mathematics" (https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing).
Again, it's just one post ...