You finally admit that Presburger, P, is at once a complete system and has extensions where the truth value of a question appears to be different? I’ve finally realized that Presburger asks the question, “is there an x in the natural numbers such that x+x=3?” has the same value in every extension of P.
Everyone else seems to deny that the field axioms describe F2.
But, what I actually am convinced of is “the field axioms only, … describe F2.” the field axioms without any other axioms, creates a perfectly valid system of their own. F is the smallest system that satisfies the axioms of F; and not coincidentally, they describe the smallest possible field. Everyone else has been seeming to deny that. Even though it should be obvious that that can and should be true.
And if Frylock continues to believe that Presburger assures us that x+x=3 is false for every domain we extend Presburger to, he’s going to have to keep denying that we can create an extension where 1.5 is a number. And that would tend to make him irrelevant for any further advancement of this thread, or my own knowledge.
Capt. Ridley, I think, started irrelevant for my purposes.
Are you complaining that I don’t know how to embed an actual table in this thread, and instead, spelled out the value for every set of elements? Or, that I used the wrong term. I don’t know, you haven’t said. So, you’re not even trying to help. I am teachable if I get the proper information. Even when it comes from people, (like Frylock,) who don’t know what it means themselves, and I have to come up with the answers myself.