Are you saying that in reality every property P has actually three outcomes: true, false, undecidable?
By Godel’s incompleteness theorem yes, unless your theory of arithmetic has a non-recursively enumerable set of axioms or is inconsistent.
And that those always decidable, like e.g. “P(n) <-> (n = 2)” cannot be true for all natural numbers, while those which can be true for all natural numbers, but mostly false otherwise, are always undecidable for… some other values?
I’m having trouble understanding this sentence but I think I know what you are asking about.
There are some properties P(x) which are true for every x in the 0 chain, however, Peano Arithmetic does not include all these P(x) as theorems. If PA doesn’t include P(x) as a theorem, then it is independent of PA whether there exist nonstandard elements for which P(x) is false.
Let’s suppose that for any specific value V in the separated chain it is possible to make such property PV.
What would that prove? Would it contradict the article? How specifically?
I think this is what I am saying I believe to be impossible. You can’t just say “V is in the separated chain”. V is a constant symbol. The model can assign constants to whatever object in the domain of discourse it wants to unless you add axioms forbidding it.
Honestly I am becoming confused. I’m going to take a break and think about all this for a bit.
By Godel’s incompleteness theorem yes, unless your theory of arithmetic has a non-recursively enumerable set of axioms or is inconsistent.
I’m having trouble understanding this sentence but I think I know what you are asking about.
There are some properties P(x) which are true for every x in the 0 chain, however, Peano Arithmetic does not include all these P(x) as theorems. If PA doesn’t include P(x) as a theorem, then it is independent of PA whether there exist nonstandard elements for which P(x) is false.
I think this is what I am saying I believe to be impossible. You can’t just say “V is in the separated chain”. V is a constant symbol. The model can assign constants to whatever object in the domain of discourse it wants to unless you add axioms forbidding it.
Honestly I am becoming confused. I’m going to take a break and think about all this for a bit.