You are making logical mistakes in your claim#1, as follows:
You are assuming the universe is a machine that can be modeled by a mechanical machine and you applied Church-Turing thesis and claimed that the universe can be expressed in arithmetic axioms, then you applied Gödels incompleteness theorem to arrive that there must be a true axiom that cannot be proven. Then you moved that axiom outside of the system, to make it your God I presume.
Please show us that the universe can be treated as a Turing machine. Where is your proof of that? Are you familiar with quantum effects, or you continue to live in the 1930s?
You are twisting the existing Math theorems to fit your conclusions. Why do you even bother when you skip the fundamental steps along the way?
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You are applying Math incorrectly and connecting the dots to fit your desired outcome.
This is a fair challenge.
"Gödels Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Incompleteness is true in math; its equally true in science or language or philosophy.
And:
If the universe is mathematical and logical, Incompleteness also applies to the universe."
My argument assumes apriori that the universe is mathematical and logical. I take the position that the totality of human experience including science backs that claim.
Nevertheless I concede that I cannot prove it.
All systems of belief can be traced back to assumed axioms. Belief in a mathematical and logical universe is one of mine.
I would note that quantum mechanics in no way shows that the universe is illogical or that it cannot be described with mathematics.
You should re-phrase the description of your claim#1 and state that you cannot prove it. Then you can stop there.
Doing otherwise, you are just fooling yourself. If you care about the truth, you should double check each step in your logic before you proceed further.
BTW, the universe as we know it has infinities and probabilistic representations that are hard to represent by logic or arithmetic.