# Godel: are all undecidable props in consistent math systems subsets of self-referent statements? **URL:** \ **Category:** Factual Questions\ **Created:** [December 9, 2009, 4:02am UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559 "2009-12-09T04:02:12Z")\ **Posts on this page:** 8\ **Page:** 6
**Author:** ![ch4rl3s](https://avatars.discourse-cdn.com/v4/letter/c/c4cdca/32.png) [@ch4rl3s](https://boards.straightdope.com/u/ch4rl3s)\ **Post date:** [December 15, 2009, 3:21pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/101 "2009-12-15T15:21:28Z")
> [@](#): > > 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. 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. > [@Half\_Man\_Half\_Wit](#): > > 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. . 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](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** > [@](#): > > The field axioms do not answer the question: ‘Is there an x such that x\*x=1+1?’ Yes, they do. they certainly answer the question, "is there an x in our limited set, LS, such that x_x=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, x_x=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.** ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 15, 2009, 3:53pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/102 "2009-12-15T15:53:41Z")
> [@ch4rl3s](#): > > [http://en.wikipedia.org/wiki/Domain\_of\_discourse](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.” [spit take] That says something more like what _we’ve_ been saying than like what you’re trying to say. The set of things being represented in a set of axioms is determined by the _interpretation,_ not by the axioms themselves. If you’re modelling the real numbers on some extension of Presburger, then, you are ipso facto _interpreting_ your extended axiom set as representing _the real numbers_. This means you’re _interpreting_ the fifth Presburger axiom (schema) to apply to _the real numbers_. In every axiomatic proof system, each axiom is true of each element in the domain. What you are saying directly contradicts that basic fact about the way axiomatic proof systems work. **Ch4rl3s** , have you ever been in the position to have your understanding of this stuff formally evaluated by an expert? In other words, have you actually taken courses in this material and passed all the exams? I’m assuming not–you appear to be self taught, which is admirable but can be dangerous for reasons you yourself are illustrating right now–and my advice is, when you have the time, go take a course. When you’ve passed the course, you will positively want to _slap_ yourself for the things you’ve said in this thread! I know. I’ve been there myself, though not about this topic. I’ve said twice that I give up, but it’s becoming apparent (as I suspected) that I’m practically unable to give up in discussions like this. So in case we both end up wanting to continue, let’s continue it in a thread in GD. I’ll go start it. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 15, 2009, 4:02pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/103 "2009-12-15T16:02:29Z")
[This](http://boards.straightdope.com/sdmb/showthread.php?p=11890288#post11890288) is the GD thread, which on second thought perhaps should have been in IMHO. Or heck, maybe we should have our first Proof Theory Pitting. 😃 ---
**Author:** ![ch4rl3s](https://avatars.discourse-cdn.com/v4/letter/c/c4cdca/32.png) [@ch4rl3s](https://boards.straightdope.com/u/ch4rl3s)\ **Post date:** [December 15, 2009, 4:30pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/104 "2009-12-15T16:30:26Z")
> [@Frylock](#): > > 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. 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. [QUOTE=Capt. Ridley’s Shooting Party] you seem to recognize that fields other than F2 exist, yet are convinced that the field axioms only describe F2 [/QUOTE] 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. > [@](#): > > You haven’t supplied a single truth table in this thread 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. ---
**Author:** ![Capt.Ridley\_s\_Shooting\_Party](https://avatars.discourse-cdn.com/v4/letter/c/cc9497/32.png) [@Capt.Ridley\_s\_Shooting\_Party](https://boards.straightdope.com/u/Capt.Ridley_s_Shooting_Party)\ **Post date:** [December 15, 2009, 4:39pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/105 "2009-12-15T16:39:44Z")
You _aren’t_ teachable. You’ve argued point blank with people who know what they are talking about. You _don’t_ know what you’re talking about, and, even worse, you show no ability to STFU and learn from people who do. Once again, you’re using terms or art in a completely non-standard way. You _haven’t_ written out a truth table in the whole of this thread, you don’t even seem to know what one is. Furthermore, the issue of truth tables is completely irrelevant, as far as this thread is concerned, to the field axioms. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 15, 2009, 4:46pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/106 "2009-12-15T16:46:18Z")
I’ve responded in [this](http://boards.straightdope.com/sdmb/showthread.php?t=543990) thread. ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [December 15, 2009, 5:54pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/107 "2009-12-15T17:54:02Z")
_A propos_ of nothing, this is how I imagine Indistinguishable at work: [![](https://i.imgur.com/zS2PX.jpg) ](https://i.imgur.com/zS2PX.jpg) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 15, 2009, 9:54pm UTC](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559/108 "2009-12-15T21:54:57Z")
I am a serious man. 🙂 [Previous page](https://boards.straightdope.com/t/godel-are-all-undecidable-props-in-consistent-math-systems-subsets-of-self-referent-statements/520559.md?page=5)