Completeness Theorems of Model Theory, a branch of Mathematical Logic. Together, these two Theorems show that: under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable ,Now I want whethere exist such new theorem in philosophy which prove or disprove existence of complex quantities and irrational numbers in nature ?
Irrational numbers were discovered by Classical Greeks in simple things like the ratio of sides in a 45 deg right angled triangle, which is the square root of 2. In some cultures it was declared heresy to admit such numbers were irrational and the advocate was put to death.
In an electrical resonant "Tank" circuit, the voltage and current are 90 deg out of phase. Mathematically we model this using complex numbers, with real voltage and imaginary current. I once set up such a circuit with a period of around half a second and moving-pointer meters wired in. I happily sat and watched as the needles oscillated to and fro, 90 deg out of phase with each other. The ammeter thus demonstrated that the mathematically imaginary current was in fact as physically real as anything else.
So these kinds of number clearly reflect the realities of nature every bit as much as the rationals do. Your theorem results are confined to what may be called discrete mathematics. To deal with continuities, you must add one or more "axioms of continuity" to your formulation. A similar issue occurs with projective geometry, to which axioms of continuity - among others - must be expressly added before one can apply homogeneous coordinates and hence analytical methods.
But all this is really a mathematical issue. Philosophers can do little more than ask why these various axiom sets are necessary.
I am hardly as 'deep' WRT the philosophy of mathematical logic as many here. That said ..
I am not sure what 'numbers existing in nature is', exactly. But it is interesting that the best 'representation of how things are' that we have so far is quantum mechanics. Until you get to that tricky 'observation part' it is exact (as far as we know) and I would think (but do not know how to prove) that irrational numbers would have to be part of the solution set.
My initial thought was that the use of complex numbers had to be part of that representation of 'how things are', but the Schrodinger formulation is not the only one that works.
I'll try to reconstruct your argument to show where it "goes wrong":
under the Field Axioms (the rules of the game for scalars) existence of rationals is provable, but existence of irrationals and complex numbers is neither provable nor disprovable ,Now I want whethere exist such new theorem in philosophy which prove or disprove existence of complex quantities and irrational numbers in nature ?
So, if I'm not mistaken, the reasoning behind the question is:
- Rational numbers can be deduced from the axioms for scalar fields, but irrational and complex numbers cannot be.
- If a number cannot be deduced from the axioms for scalar fields, it does not exist in nature.
C. Therefore, rational numbers exist in nature but irrational/complex ones do not.
And then you ask whether there is a philosophical theorem that can supplement the theorems used in the background of the relevant field axioms, such that this theorem would go on to allow for a relevant deduction of irrational/complex natural numbers. So there is a meta-claim that seems like:
If a theorem T satisfies the completeness theorems of model theory, then T is relevant to what exists in nature.
I suppose the intuition behind this might be that model theory, as a branch of mathematical logic, might be ontologically fundamental in some relevant way, but I feel like there's a lot missing, here, with which to make the actual case (behind the question). In other words, I don't see why only numbers deduced from the axioms for scalar fields would be the ones to exist in nature. I would think that the proper approach would be to study nature itself, and then infer what numbers "exist in it" from the results of this study. Now at least by ordinary enough intuition, we can note continuous functions in nature, wherefore the numbers of the continuum (including irrationals and complex ones) are all encoded into it. Whether there is a deduction of the continuum from some countable model of logical reasoning seems to be an unnecessary aside.
[In case it's relevant to what you're saying, I'd say you might want to look up the subject of infinitary logic, for which there is an excellent (if esoteric!) article on the Stanford Encyclopedia of Philosophy. I can't testify as to how much it involves the kind of model theory you're referring to, but I can at least testify that it addresses the question of which ordinals characterize the most useful infinitary language, namely L(ω1, ω) (where "ω1" is the first uncountable ordinal, which assuming that the Continuum Hypothesis is true is also the initial ordinal for the cardinality of the continuum); so perhaps the fact that L(ω1, ω) allows logical conjunctions and disjunctions to be < ω1 in length "shows" that the logic of a physical universe, modeled by this L, would be related to the numbers of the continuum in the way you would need to "prove that they exist in nature"; but all this is so vague and programmatic that I don't want to totally commit you to looking for an answer to your question, in this direction.]
Numbers (of any kind) do not exist in nature, but in our mind. Numbers are the representation of boundaries between parts, and such boundaries are always subjective.
Written by a kid :
-The dog has the following: tail, 1; paws, 4; ears, 2; eyes, 2; muzzle, 1; teeth, 32. Total: 42.
Now, think, is this true? Is that false because you are an adult and that is written by a kid? Which is the number that "exists" in nature? For what you would say, you cannot count complex numbers in the dog. But probably, a QM scientific can. Which is the truth?
 Jose Maria Firpo, "Qué Porqería Es El Glóbulo", Ediciones de La Flor, 2005, Argentina.