Consider that something has the property of not being something else is not very practical but valid in any way. Nor is practical to consider a number x has the property of being "x + 1 -1" (you can use the rules of mathematics to generalize these properties).
You can also synthesize the infinite properties that has something of not to be something different not considering this as properties but consider it as a rule; cat is not a dog = Property... cat is not a glass = Property... or you can say cat is not nothing different that cat = Rule...
Science sets properties where this is relevant to the study of something; although valid, considering that something has the property of not being something else, it is a misuse for the purpose of establishing properties.
But as is say before... It is not my purpose to discuss the nature of the term "property" or "properties"...
About Universe or Nothing...
The universe cannot start form P.Nothing becuse P.Nothing cannot produce matter or energy, it is logical that there is no evidence that the beginning was from P.Nothing.
For me "Why there is a universe instead of P.Nothing?" it's a reasonable question; and if I have to accept that the universe has always been rather there (in one form or another), why is configured in this way? Why it has the actual amount of matter and energy and not more or less?
Well regardless if they're practical or not, you can make an infinite number of axioms according to the incompleteness theorem which means there are an infinite number of generalizations that could be applied.
And I agree that the universe doesn't start from nothing because nothing isn't a physically valid concept worth considering.
"something has the property of not being something else, it is a misuse for the purpose of establishing properties."
I'm sure you'd agree then that nothing is a misuse or flawed concept because it is defined as the absence of something, or not something if you will.
The better question is why would there be nothing rather than something? The question itself reflects and inherent paradox however because nothing couldn't exist in place of something--if nothing exists then it is a thing and it exists and is therefore something. There couldn't be nothing instead of something.
The incompleteness theorem does not teach you have an infinite number of axioms or generalizations. There never have been an infinite number of either. Nothing is not a physical concept but as long as you continue to insist it is physical, you will keep making th same error.
Why is there existence rather than non existence?
Well you don't really have a good understanding of mathematics.
Many important theories, most significantly first-order Peano arithmetic, and ZFC, the most commonly used axiomatic set theory, have an infinite number of axioms. So does the theory of algebraically closed fields.
If there aren't an infinite number of axioms you can use, and therefore combinations, then how many are there? The incompleteness theorem is then related the fact that there are an infinite number of axioms you can use--no complicated system .
Nothing is indeed a physical concept. It can be quantified as the absence of something. That means it has zero things. If it's quantifiable with numbers it means it's physical. Like roughness, or fractal dimension, or conductivity.
"Why is there existence rather than non existence?"
Again a bad question. Non existence couldn't exist in place of existence--that would be completely illogical. And again non existence still has the property of being separate from existence, which means it would be part of existence by nature of having properties and is thus a contradiction.
WHy would there be non existence instead of existence?
OR
How could there be non existence instead of existence?