Incorrect comments on Logical Pinpointing - Less Wrong
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Comments (338)
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.