I am learning a lot, just not much from you guys, except in the sense that I’ve been doing a lot of research to study the questions. And I’m not looking to have my preconceived notions validated. From each new position, I try to derive logically what they mean, and I happen to find that what I already knew about how mathematicians do math still applied. And, of course, I’m learning a lot about how to express and defend my position.
And once again, I’ve learned something that none of you could tell me, that gives answers to these questions, from my own study. From the talk of Presburger saying there is no x such that x+x=3, and the similar questions in F, I’ve learned to look closely at the domain of our question. Presburger can only ask questions in the natural numbers. In every extension of Presburger with a different set of numbers, what Presburger says about natural numbers the new system says about natural numbers…
http://en.wikipedia.org/wiki/Domain_of_discourse
“In one interpretation, the domain of discourse could be the set of all real numbers; in another interpretation, it could be the set of natural numbers.”
The field axioms, (F,) only define the limited set, (LS,) consisting of {0,1}. and the only questions that can be asked in F are ones about that limited set. And everything it proves about that limited set is true in every extension of F. (I knew this sounded right when ?Frylock? said it before, but I kept seeing questions that seemed to have different answers in extensions.
F can ask the question, is there a number in LS that is not in LS. And in F, this is equivalent to asking is there a number in our domain that isn’t in LS. But the equivalent question in a system that extends the domain is only the first one, “is there a number in LS that isn’t in LS?” And those always have the same answer. Now, logically, the consequence of this difference is huge.
And just like I said, F is the smallest system that fulfills the axioms of F.
In F, 1+1 is defined to be in our domain, and if that domain is LS, then 1+1 must be a number in LS. (and logically, I showed that it can’t be 0, and so, must be 1.) I had already realized that although 1+1 was in our domain by axiom, the derivation of what number it actually is changes based on our domain. (the value of 1+1 is not a property of F.)
Using just F, and not one of it’s extensions, (and now I hope I can finally show that I don’t actually have to use that caveat,) F[sub]2[/sub] can be constructed. And still, everything F says about LS, every extension also says… about LS
Yes, they do. they certainly answer the question, "is there an x in our limited set, LS, such that xx=1+1? But, the answer to 1+1=? isn’t something that F proves by itself. Logically, if our domain is LS that is true, but that isn’t a property of F. So, xx=1+1 is either true or false, also isn’t something that F proves by itself. Like I said, this isn’t a property of F, has no bearing on its completeness, and isn’t guaranteed to have the same value in every extension. But, it has a truth value in every version of F, including F itself.