>>12526129>I'm not very familiar with second order stuff.You won't like it. Second order logic allows you to quantify over all predicates, even in systems that only allow infinite models. Thus, it includes all of the conceptual difficulties of the meaning of infinite sets, and powersets thereof, right into the basic logic.
>Are you saying that second order arithmetic has a unique model?Yes, it does.
>How does that not contradict Godel?Because in second order logic, you cannot necessarily derive all consequences of the axioms. In other words, most second order logics are incomplete in the completeness theorem sense of the word. This means that for self-referential statements, it's possible that the statement semantically follows from the axioms (it is true in any model), but neither the statement nor its negation can be proven in the logic.
In other words, second order logics have insufficient derivation rules to prove all the valid implications of the axioms.
>This is a highly dubious term. i don't believe it makes sense to talk about "standard naturals.It's not my term.
First-order arithmetic allows for more than one model. The one we all know we mean is called the standard natural numbers, and all the other ones are called nonstandard naturals.
>Yes, I believe Godel proved this.He proved something about definitions of the natural numbers, alright. How that something relates to what you consider satisfactory, I can only guess.
That said, I feel that if you are of the opinion that there is no satisfactory definition of the natural numbers, entering a thread daring people to provide a satisfactory definition of the reals is a bit intellectually dishonest. Your real objection is much deeper, and affects far more basic pieces of mathematics, which means that any discussion about definition of the reals will inevitably get lost in three levels of recursion before they can possibly get anywhere, with endless opportunity for 4chan to get lost along the way.
