The positive real numbers are not an extension of presburger arithmetic. For take a look at Presberger’s induction axiom. It says if S is true of 0 and (if S is true of X then S is true of S + 1) then S is true of all Y. But that axiom is not satisfied by the positive real numbers. For example, 0 has remainder 0 when divided by 1, and if any number has remainder 0 when divided by 1 then so does its unit successor–but not all positive real numbers have remainder 0 when divided by 1.
I was wondering if anyone would notice that I defined < for the numbers specified, but didn’t specificly relate it to >. Ha. so, you are claiming that the notion “>” doesn’t exist because i didn’t define it in my axioms?
I was using it as a convention, however. Conventionally, in common usage, when 1<2: 2>1. simply a convention. It’s the conventional way we talk about <, >.
I didn’t read the 5th axiom from the wiki to mean that.
yes, they do. division is defined that way. 1 is the multiplicitive identity. (although, we don’t actually talk about multiplication and division in Presburger.)
x*1=x. so x=x/1. How does that not leave “remainder 0?”
x/1 = x, with **nothing left over **for all x. “remainder” is only a concept in systems such as natural numbers and integers.
in presburger, natural numbers, or integers. “4/3” is “1 remainder 1.” because those systems can’t produce a number between 0 and 1. in the reals, “4/3” is “1.333333 repeating,” but no remainder. the system produces a number which is the answer without a remainder.
what do you mean by unit successor? is 2 the unit successor of 1? and would it still be the unit successor of 1 in the real numbers? isn’t the unit successor of x, x+1?
No, I just didn’t see where you’d defined ‘<’. Sorry about that. I’ll look again.
Well… but that’s what it says. 
Yes, you’re right, I don’t know why I tried to express my point in terms of remainders. That was an error.
Here’s what I meant, though. Zero is an integer. If a positive real number is an integer, then the number you get by adding one to it is also an integer. But it is not true of all positive real numbers that they are integers. This means that the positive real numbers do not satisfy the fifth axiom. This means that the arithmetic of the positive real numbers is not an extension of Presburger arithmetic.
(An integer is a number x such that when you subtract 1 from x, then subtract 1 from the result if the result is not zero, and so on, then eventually the result is zero.)
Yes, by unit successor I meant x+1.
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If the arithmetic of the positive real numbers is an extension of Presberger arithmetic, then in that extension, the Presberger axioms are true of every element in the domain of the positive real numbers.
But the fifth axiom is not true of every element in the domain of the positive real numbers. For the fifth axiom can be used to prove that every element is an integer. But many real numbers are not integers.
Therefore, the arithmetic of the positive real numbers is not an extension of Presberger arithmetic.
ETA: I see you deleted the post I was responding to so I won’t assume you said anything that was in that post. I’ll leave this post here, though since I think it expresses my own view fairly clearly.
So, that’s undecidable, right?
No, that’s flat wrong. As Frylock stated, if an axiom system proves P, any extension of that axiom system also proves P; it may end up also proving not-P, but all that means is that you’ve introduced an inconsistency.
In any case, I’m afraid I lack the stamina to continue this hijack; ch4rl3s has repeatedly been shown wrong, and has stolidly refused to accept this. I have no hope that this is going to change in the foreseeable future.
@Ch4rl3s: Then it is down to you, and it is down to me. 
Or, since you already tried the ad verecundiam route in vain, you might as well proceed to ad populum, sum up your (our) view and have ch4rl3s sum up his, put all of that in an OP in IMHO, and decide the whole thing by poll.
Ok. yup it says that, (sort of). My mistake, sort of.
oops. Is Presburger a bad example as well?.. Presburger is the set of natural numbers, with addition, that satisfies those 5 axioms…
Nope, I was ok.
Because from the 5th axiom of Presburger, x,y are defined as elements of the natural numbers. if I add new numbers other than the natural numbers by adding axioms, that property is not defined to exist for these new numbers. they aren’t natural numbers.
Presburger is defined to include the natural numbers, N. If I add new numbers, they aren’t natural. the 5th axiom is defined for every x,y in N.
You took the fact that I added numbers to the set to mean that Presburger now spoke of the positive real numbers, and that the 5th axiom now covered more than it covered in Presburger. Presburger doesn’t include the positive reals. But it doesn’t exclude their creation. or the rationals.
(Let’s keep assuming that “positive reals” is the extension we’re talking about, I know I first stated it as “reals” but you were correct to limit it, thanks.)
What do the field axioms, F, prove about 1+1? That it exists in our “number” set. That is true in every field. I already said that I used logic to fill in the truth table. Was it not clear that I could use logic? I know I didn’t state that. But I thought we were dealing with formal systems where our proofs were logically derived from the axioms. Sorry I wasn’t clearer on that.
Sorry, I need logic and a number set as well as the axioms to prove that 1+1=0 in {0,1}. (added: The set defined in F is {0,1})
I know I’m in the minority, but “and decide the whole thing by poll” doesn’t prove the correctness of an argument. I would much prefer the argument from authority.
and you think you’ve been on the correct side of ad verecundiam. I’ve still had an answer for most of your questions. And I think we’re getting somewhere. Even if no one has their mind changed, I’m learning a lot about systems, and yet, everything I learn still confirms the things I already was sure of about math in general.
Have we concluded that F and the axiom “assume no other symbols, operators or axioms,” defines F[sub]2[/sub]? That is progress.
Now, can some one tell me why we need that last axiom? (added: and why that isn’t the default state for mathematical systems? Because if you can’t answer that, then F, [and logic] describes F[sub]2[/sub].)
Math is the kind of inquiry where you can talk about whatever things you would like to talk about, and proofs regarding such can be presented and judged on their own without need of appeal to any external data or authority (though it may help).
All the rest of us mean, by “an interpretation of the language of fields/language of F”, any collection along with a map from {“0”, “1”} into that collection and a map from {"+", “"} into the binary operations upon that collection. All the rest of us mean by “the theory of fields/the proof system F”, a particular collection of properties which or may not hold of any given interpretation of the language of fields. Specifically, those properties are that the interpretations of “+” and "” are commutative and associative, with the interpretations of “0” and “1”, respectively, as their identities, and with every element in the interpretation’s collection having an inverse under the interpretation of “+”, and with those and only those which are not equal to the interpretation of “0” having an inverse under the interpretation of “*”. Not among those selected properties (and again, this just is what the rest of us mean when we say “the theory of fields/the proof system F”) is the property that all elements in the interpretation’s collection are either equal to the interpretation of “0” or the interpretation of “1”. By “a field/model of F”, we mean an interpretation which satisfies all those selected properties. By a statement being “entailed by/provable from the field axioms/the proof system F” we mean that it is satisfied in every model of F.
This happens to be the standard terminology (well, plus the use of the letter “F” for the field axioms now). But it doesn’t actually matter whether or not it’s the standard terminology. It’s the terminology we are using. When we make claims, they are using this terminology; they are correct to the extent that they are correct under this terminology.
Under this terminology, it is manifest that, e.g., “1 + 1 = 0” is not provable within F; as it is not satisfied in every model of F; as it is not satisfied in every interpretation validating the properties mentioned above; as it is not satisfied in the particular interpretation validating the properties mentioned above given by taking the collection of integers, and interpreting “0” as the integer 0, “1” as the integer 1, “+” as integer addition, and “*” as integer multiplication. Accordingly, on our terminology, is is correct to say that “F does not prove 1 + 1 = 0”.
If you mean something else by phrases like these, then you should still be able to understand what we mean, when we spell it out. Similarly, you should be able to spell out what you mean by the things you say (e.g., “F and logic describes F[sub]2[/sub]”), and then we can examine to what extent they are or not true.
That having been said, I don’t care to do so. But since you seem to still have at least one person’s attention, I thought I would make the above remarks.
(In particular, the reason I don’t care to do so is because I don’t think you have a terribly clear or coherent meaning to offer for statements like “F and logic describes F[sub]2[/sub]”, and, given the effort to reward ratio I suspect, I lack the patience to support any altruistic desire to help you sort out your gibberish.)
All the quoted sentences are false or, in context, misleading.
But, really, I don’t have the credentials to say so authoritatively, I don’t have the time to explain why they are false all the way from first principles (which is clearly where this is leading) and I don’t have the resources to give a sufficiently complete set of citations of books or of online information sources to show that they are false.
So… with some apologies, I’m afraid I’m going to have to join the ranks of the upgivers.
Are you saying that the wiki is wrong? http://en.wikipedia.org/wiki/Presburger_arithmetic
The basis for Presburger is natural numbers. The 5th axiom is true for natural numbers. that doesn’t change when we extend the system with other numbers. That 5th axiom is still true within the natural numbers. How is that difficult?
I supplied a logically derived truth table for F[sub]2[/sub] based on the field axioms, F, f(rom the wikipedia article http://en.wikipedia.org/wiki/Field_axioms. ) If that table is wrong, then someone should be able to simply show which steps are incorrect. No one has yet refuted the table, or said that it doesn’t describe F[sub]2[/sub], using a logical set of statements.
I even showed which axioms I used to derive each statement. How much clearer can I get?
You don’t understand what the wiki says. That they are a “theory of the natural numbers with addition” doesn’t mean that they can apply only to the natural numbers, and certainly doesn’t mean that they can apply only to the natural numbers even when embedded in an extension. As I said, I don’t have the time or resources to explain to you why this is. It’s just… basic… We’d have to go all the way back to the beginning of metamath 101 and start from scratch. It’d take forever. That’s why I’m giving up.
But the fifth axiom need not be understood as only true about the natural numbers. It’s true of all the elements within the domain of the proof system it’s part of. That’s how axioms work. If you embed the presburger axioms by extending them into an axiom set that’s supposed to be modeled by the real numbers, then the real numbers are now (supposed to be) the elements of the domain the (extended) proof system is part of, and the fifth axiom is supposed to apply to them all. Again, this is just… basic… Arguing over this is pointless. What I need to do is cite a chapter of a metamath 101 textbook and say “look, this is just how it is, you’d fail the first quiz of the course otherwise”–but I have no such books with me at my house so, as I said, I don’t have the resources for this discussion.
The problem is not whether or not your ‘truth table’ is right or wrong, it’s that it’s got nothing to do with the issue at hand, which is, quite simply, this:
The field axioms do not answer the question: ‘Is there an x such that x*x=1+1?’, because different fields answer the question differently; if the field axioms did answer the question, all fields would have to agree on that answer.
It’s a simple and uncontroversial example of manifest undecidability that doesn’t have anything to do with Gödel.
You dragging in F[sub]2[/sub], Presburger arithmetic, ‘logic’ (whatever you may mean by that exactly), etc. is only so much obfuscation brought on by your apparent need to be right about something you frankly don’t seem to understand very well.
You said you were learning a lot about systems; that’s a good thing, and I’m glad at least something came of this otherwise rather pointless exercise. I’m sure if you continue learning, you’ll eventually come round to appreciating what has been laid out for you in this thread. But, if you keep looking only to have your preconceived notions validated (and the fact that your preconceived notions appear to you to be validated is the most certain sign that that’s really what you’re doing), you won’t really learn a thing.
You haven’t supplied a single truth table in this thread, you’ve supplied derivations, where “derivation” is very loosely construed. Furthermore, your use of other terms of art are also non-standard. For instance, your definition of consistency is non-standard, and not exactly useful. Not only that, but you seem to have a catastrophic misunderstanding of the role of the field axioms (although, reading carefully, you seem to recognize that fields other than F2 exist, yet are convinced that the field axioms only describe F2; how you rationalize this internal contradiction is anyone’s guess).
There’s no subtle way to say this: You’re wrong about almost everything in this thread. Furthermore, I’ve read enough ramblings in the likes of sci.logic, on Goedel’s theorems and undecidability, to understand that no amount of persuasion will make you see the light. Now, please, stop hijacking what was a very interesting discussion with your insanity.