Alright, so someone told me that I cant define things into existence. And I thought to myself, yes I can!
First, I tried defining "box of gold" as box full of gold which appears before me any time when I want. No great suprise that box didnt appear even tho by definition it should have!
But then I thought, what does it mean to appear? What does it mean to exist? Maybe the box appeared and I cannot detect it?
So I changed the definition of box of gold to: box full of gold which appears before me any time when I want, and which I can always detect.
No great wonder, box didnt appear this time either.
Or did it?
I remembered Godel's incompleteness theorems, and Godel's completeness theorem, when applied to arguments.
Completeness theorem says that first order logic and definitions are provable within the system. In other words, the only thing which is provable within the system are basic logical laws, such as A=A, and definitions used, and what is able to be proved by definitions and basic logical laws.
https://en.wikipedia.org/wiki/First-order_logic
So per Completeness theorem, definitions when accepted are used for proof, and this proof is valid everywhere, even outside the system. So the math defines things into existence.
However, what is existence? How do we know that we exist?
And this is where Godel's incompleteness theory comes in. Hilbert's program had a goal to create one universal system for whole math. In simple terms, it had two goals: Create a system which can prove itself to be true, and which can prove all the remaining math as well.
However, Godel's incompleteness theory completely destroyed Hilbert's program. Incompleteness theory gave 3 major problems which program could never solve:
- System used to prove things true cannot prove itself true
- System cannot prove things outside the system
- System cannot prove all true claims in the system
Gödel constructed a statement like:
"This statement is not provable in this system."
If it were provable, the system would be inconsistent.
If it is not provable, it's true, but unprovable ⇒ incompleteness.
How does incompleteness apply to proving existence?
System which is used to prove existence would require ability to prove that such system exists, that such system is consistent. But that is impossible to prove within the system. You cannot prove your senses are consistent or that they even correctly determine what exists.
So while completeness theorem does say definitions are used as proof and are true everywhere, the incompleteness theorem makes any other proof of existence impossible.
In other words, not only that you can define things into existence, but proving existence without definitions is impossible, because senses are flawed.
In simple terms, box of gold exists by definitions, and my senses are just flawed!
More about Hilbert's program:
Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else.