# What does the Incompleteness Theorem imply? **URL:** \ **Category:** Great Debates\ **Created:** [April 12, 2003, 1:16am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157 "2003-04-12T01:16:08Z")\ **Posts on this page:** 20\ **Page:** 8
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 16, 2003, 9:26pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/141 "2003-04-16T21:26:21Z")
You’re welcome, **erislover**. What _is_ it with the people in this thread? Are they all so conditioned by domain-specific terminologies that they can’t use their reason? I find it difficult to accept that the people here don’t understand the meaning of the word “complex”. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 16, 2003, 9:35pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/142 "2003-04-16T21:35:03Z")
I find it difficult to believe that you suppose that in certain contexts a word like “complex” could be ambiguous, undefined, or mean something unexpected from a dictionary definition. No? ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 16, 2003, 9:43pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/143 "2003-04-16T21:43:20Z")
In certain contexts? No. In _this_ context? Yes. ---
**Author:** ![Newton\_meter](https://avatars.discourse-cdn.com/v4/letter/n/57b2e6/32.png) [@Newton\_meter](https://boards.straightdope.com/u/Newton_meter)\ **Post date:** [April 16, 2003, 10:01pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/144 "2003-04-16T22:01:03Z")
> [@](#): > > \*Originally posted by TVAA \* > **No system designed to generate “new” statements according to a finite set of axioms will ever produce Godel’s famous statement. If it could do so, that set of steps would be a proof of the statement.** That’s not exactly what Godel says, though, is it? Are we still talking about Turing Machines? You want a TM that enumerates all true statements of number theory? Here’s one: the machine that enumerates all strings. It generates new strings according to a finite set of rules. It will produce any string you want it to. Perhaps your observation is less earth shaking than you think it is. You observe that a particular TM (the one that tries to enumerate true statements of number theory starting from a set of axioms) will not generate a particular string. And somehow this is a restriction on computation. Here’s another: there are strings that the TM that enumerates all strings containing only zeros will never produce! Astounding, but true. > [@](#): > > **The workings of the universe generate configurations according to basic axioms.** I guess I’m not much of a philosopher, so I’ll stay away from that. But I’m pretty sure that this is the sticking point. Nobody but you seems willing to accept this claim. So you are claiming that, because you interpret the universe as a model of number theory, there is some “state of affairs” that represents a true statement of number theory but can never be reached. If I find a different encoding, say I swap all your protons and neutrons with each other, then does that suddenly make that “unreachable” state of affairs reachable, and render another unreachable? Do you see the problem here? And I’m not really much of a physicist, either, I guess. But I wouldn’t be surprised if there _were_ unreachable states of the universe (say the one where I’m traveling down I-494 at rush hour in a Geo Metro at a velocity of _c_). I just don’t think that has anything at all to do with math or computation. > [@](#): > > \*\*For the last time, **I AM NOT USING COMPUTER SCIENCE TERMINIOLOGY!** \*\* But you are talking about the science of computation, no? ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 16, 2003, 10:38pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/145 "2003-04-16T22:38:34Z")
**TVAA** You have a theorem in mathematics and a construct of computer science. If you want to draw meaningful conclusions from those things it is a good idea to use the terminology appropriate to those fields. For instance, the “common everyday meaning” of “complexity” is of absolutely no use in showing that set of elements that model a system is necessarily bound by the limitations of the system being modeled. There is no hard connection between those two concepts, and saying “complexity” does not draw a line of either mathematical or logical implication. You only have dots. You have drawn no connections. Because you refure to speak in a language specifically designed to make such connections, it is difficult to ever know whether you might have a point. > [@](#): > > You’re not pointing out what statements you think are wrong and then demonstrating that they are wrong – you’re simply denying my argument over and over again. I have shown that your statements are untrue using the terminology appropriate to the field. You say that you have no interest in using appropriate terminology. Okay, then the onus is very much upon you to draw the lines **in specific detail**. The reason that fields of stufy develop specialized usage is so that many things can be communicated subtextually. Since you reject the conventions appropriate to your subject, you don’t have access to any such contextual information. Personally, it seems a silly choice on your part, but if that’s the way you want to proceed then it is certainly your right to do so. > [@](#): > > For example, you claim that the meta-grid that can be constructed with the standard grid of Life is as complex as the grid itself. This is not only incorrect, it’s **stupid.** Stupid? And what word would you use to describe starting a discussion about mathematics and computer science and then calling mathematicains and computer scientists stupid for using teh terminology appropriate to those fields? If you like to think “having more parts” is the only way to measure “complexity”, then enjoy your little world of irrelevance. You have hit upon yet another thing that is entirely unconneted to GIT. > [@](#): > > The words I’m using all have well-defined meanings in standard English. If you can’t handle that, get off the boards. I’m quite serious – if you can’t rationally consider arguments made in everyday language, you have no place on boards where the arguments are presented in that format. Wow, from **respectfully disagree** to **get off the boards**. Here is some English language for you. [ul][li]The OP is wrong.[/li][li]You misunderstand and misapply GIT with rampant abandon.[/li][li]You attempt to disguise this (perhaps even to yourself) by hiding behind ambiguities in language.[/li][li]When pressed, you not only fail to provide a rigorous argument but turn insulting to those who apply the appropriate language to questions of Godel and Turing Machines.[/li][li]You use words that sound like they belong in mathematics or computer science but specifically deny the meaning appropriate to those contexts. Then you pretend that these “pseudo-terms” actually carry some implication for a theorem of mathematics and a construct of computer science.[/li][/ul] The problem is not the language, though, it is your appalling lack of rigor. Any case that can be made in a formal language can be made in English, but you need to be **very rigorous and very specific in your use of language and your demonstration of implication.** You have failed miserably on both fronts. > [@](#): > > A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word. And how is “similarity” meassured, absent an interpretive consciousness? > [@](#): > > I’m saying that, although the two systems exist on different level of implementation, their high-level behavior is the same. They do the same things. Okay. A model is something that behaves like another thing. So, each is equally a model of the othe, correct? > [@](#): > > It is not an assertion. It’s a consequence. It is not a consequence for which you have provided a valid argument. Thus, it is an assertion. > [@](#): > > The behavior of electrons is such that they can be used as the foundation of a system, on a higher level of implementation, that can perform arithmetic. Therefore their behavior necessarily can demonstrate statements about arithmetic. The assertion was: **the movements of electrons are sufficiently complex that they “contain” arithmetic**. Is this another of your “plain English meanings”? Does “contain” now mean “can be used to model”? Does a pen and paper contain the Mona Lisa? Is it **therfore** valid to say that the Mona Lisa restricts the behavior of pen and paper? > [@](#): > > I have shown you the dots. You should be sufficiently intelligent to be able to connect them yourself. It is not my task to connect the dots of your argument. That task is yours. Nevertheless, if it makes you feel better to imagine that I am simply too stupid to see the obvious connection that make your case, feel free. I care not at all what whims of today rule your image of me. Please, enlighten this poor stupid **Mundi**. Draw the lines. You can use whatever language you wish, but do remember that I am stupid. I shall demand that you make every line very clear and very explicit. Otherwise, I am sure I will lack sufficient intelligence to finish the picture. > [@](#): > > Complexity: the amount of data required to perfectly define a system. Well, since you object to the terminology of mathematics and computer science, you had better define “system”, too. > [@](#): > > > [@](#): > > > > An order N algorithm with 2 pieces of input has exactly the same complexity as an order N algorithm with 2 million pieces of input. > > The implementation of the algorithm is not equally complex in both cases! You are wrong, even under your own definition of complexity. For instance, the psudo code: [ol][li]Read X from tape[/li][li]write X[/li][li]loop to 1 until end of tape[/ol][/li]can be implemented in any numbe rof cimputer languages. The amount of data required to perfectly define that algorithm does not change based upon the scale of input. I know it is too much to ask that you use the language of mathematics of computer theory appropriately. Is it too much to ask that you use your own chosen language consistently? > [@](#): > > I informed you quite a while ago that I’m communicating in the standard language. You’re the one that applies field-specific terminology inappropriately. Actually, I am one among several who have been arguing that field-specific concepts are best addressed with the appropriate terminology. Apparently, your own definition of “appropriate” is “what **TVAA** wants.” Okay, I will stop trying to get you to use the terminology of GIT and TM correctly. Despite what you might think, this will not make your task easier. > [@](#): > > The interaction between the writing implement and the piece of paper can be described by a system of basic rules. There will always be statements allowed by the grammar of the components of those rules that the rule system cannot reach. As a result, there will necessarily be possibilities that the implement-paper system cannot manifest on its own. Prove it. GIT is absolutely no help to you here. GIT says nothing about “possibilities not capable of being manifested within a system”. You appear to be suffering from the misunderstanding that every true statement in a Peano Axiomatization has a “possible proof”, but GIT just won’t let us manifest them. This is incorrect. GIT says that no possible proof exists for some true statements. In other words, the pen and paper can write anything that it is possible to write. Similarly, every possible statement of arithmetic can be generated. GIT stops neithe rof those. It just stops any elements of the first from proving all true elements of the second. > [@](#): > > No system designed to generate “new” statements according to a finite set of axioms will ever produce Godel’s famous statement. If it could do so, that set of steps would be a proof of the statement. > > The workings of the universe generate configurations according to basic axioms. > > Connect the dots. Sorry, I’m too stupid. Why don’t you connect the dots. After all, your the one who imagines that he sees a picture. > [@](#): > > The pencil and paper are being emulated by a deeper level of reality in the same way that a meta-grid is emulated by a grid in Life. They are necessarily limited! You keep saying things like this. You keep forgetting to prove them. > [@](#): > > > [@](#): > > > > What you are saying is analagous to: It’s impossible to write down the method for squaring a circle using geometric construction because the electrons are can’t move in a certain way. > > No. Yes. > [@](#): > > The resources required to represent a specific pattern with the meta-grid are necessarily greater than the resources needed to represent that pattern with the basic grid. By the (inappropriate) definitions you’ve already offered, I conclude that you’re mistaken. By the definitions that I offered, my statement is rigorously correct. Apply your own definitions if you wish, but please stop making ignorant statements about the proper application of terms as used in computer theory. > [@](#): > > For the last time, I AM NOT USING COMPUTER SCIENCE TERMINIOLOGY! I understand. I fear that you have now lost all hope of ever understanding the proper scope of GIT and the limitations of TM, but I confess that it seemd there was little enough hope to be lost on that front. Do go ahead and start drawing nice rigorous and well-defined lines using regular English, though. I would be happy to have my pessimism proved wrong. > [@](#): > > I find it difficult to accept that the people here don’t understand the meaning of the word “complex”. I wish that I could find it difficult to believe that someone would use the word “complex” in a dicussion about computer theory and not only demand that it not be understood in the context of comuter theory but call someone stupid for using it in that context. Unfortunately, I have all too much evidence that some people do indeed embrace that particular juxtaposition of behaviors. Lucky me. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 17, 2003, 4:59am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/146 "2003-04-17T04:59:25Z")
> [@](#): > > \*Originally posted by Spiritus Mundi \* > \*\*If you want to draw meaningful conclusions from those things it is a good idea to use the terminology appropriate to those fields. For instance, the “common everyday meaning” of “complexity” is of absolutely no use in showing that set of elements that model a system is necessarily bound by the limitations of the system being modeled. > > There is no hard connection between those two concepts, and saying “complexity” does not draw a line of either mathematical or logical implication. \*\* A system can be emulated by another only if the elements of the emulating system can represent the elements of the emulated system. The emulated system must therefore contain fewer elements than the emulating system. If it contained as many, it would be identical to the emulating system – or in other words, the system would represent itself, which is the trivial case. If it could consider more, the emulated system would contain elements not represented in the emulating system, which violates our premise. So you don’t like the word “complex”? How about “simple”? The emulated system must be simpler than the emulating system. > [@](#): > > \*\*I have shown that your statements are untrue using the terminology appropriate to the field. You say that you have no interest in using appropriate terminology. \*\* You’ve shown that my statements are invalid in a certain language. I say that I’m not speaking that language. > [@](#): > > \*\*Stupid? > > And what word would you use to describe starting a discussion about mathematics and computer science and then calling mathematicains and computer scientists stupid for using teh terminology appropriate to those fields? \*\* You’re not being stupid because you’re using that terminology. You’re being stupid because you’re acting as if _I’m_ using it, then claiming that my statements are wrong because they’re inconsistent with the terminology. > [@](#): > > \*\*If you like to think “having more parts” is the only way to measure “complexity”, then enjoy your little world of irrelevance. You have hit upon yet another thing that is entirely unconneted to GIT. \*\* The more “parts” the system has, the more relationships exist within it, and the more complex it is. The fewer parts, the fewer relationships, and the simpler it is. These are not difficult concepts to grasp. > [@](#): > > **The problem is not the language, though, it is your appalling lack of rigor. Any case that can be made in a formal language can be made in English, but you need to be _very rigorous and very specific in your use of language and your demonstration of implication._** Where has my use of language been vague? Where has my use of language been inconsistent? > [@](#): > > \*\*And how is “similarity” meassured, absent an interpretive consciousness? \*\* “Similarity” is not measured – it _is_, just as the properties of the system _are_, regardless of whether a mind observes them. Do you think that the Game of Life acts differently when no one’s looking at the computer screen? > [@](#): > > \*\*Okay. A model is something that behaves like another thing. So, each is equally a model of the othe, correct? \*\* No. The meta-grid can easily be disrupted without disrupting the grid – adding or removing a single active square in a crucial location is often sufficient. Disrupting the grid (by interfering with its implementation in the computer, for example) immediately disrupts the meta-grid. The higher-level system is dependent on the lower-level in the same sense that the floors of a tower are dependent on their foundation. Take away the foundation… and the tower falls. > [@](#): > > \*\*The assertion was: **the movements of electrons are sufficiently complex that they “contain” arithmetic**. Is this another of your “plain English meanings”? Does “contain” now mean “can be used to model”? Does a pen and paper contain the Mona Lisa? \*\* No, of course not. Don’t be ridiculous. The possible relationships between atoms contains the Mona Lisa. The Mona Lisa is just a particular configuration (technically a category of configurations… it’s never the same twice, after all). > [@](#): > > \*\*Is it **therfore** valid to say that the Mona Lisa restricts the behavior of pen and paper? \*\* No. The limitations of atomic interactions place inherent limits on the nature of possible Mona Lisas. > [@](#): > > \*\*Nevertheless, if it makes you feel better to imagine that I am simply too stupid to see the obvious connection that make your case, feel free. I care not at all what whims of today rule your image of me. Please, enlighten this poor stupid _Mundi_. Draw the lines. > > You can use whatever language you wish, but do remember that I am stupid. I shall demand that you make every line very clear and very explicit. Otherwise, I am sure I will lack sufficient intelligence to finish the picture. \*\* Very well. > [@](#): > > \*\*_The implementation of the algorithm is not equally complex in both cases!_ > You are wrong, even under your own definition of complexity. For instance, the psudo code: > [ol][li]Read X from tape[/li][li]write X[/li][li]loop to 1 until end of tape[/ol][/li]can be implemented in any numbe rof cimputer languages. The amount of data required to perfectly define that algorithm does not change based upon the scale of input. \*\* > [@](#): > > implementation > > n 1: the act of accomplishing some aim or executing some order; “the agency was created for the implementation of the policy” [syn: execution, carrying out] 2: the act of implementing (providing a practical means for accomplishing something) [syn: effectuation] If X is variable, the larger it becomes, the more cycles the above program will run, and the more memory will be required to represent it. The implementation of the algorithm will change as X changes. > [@](#): > > \*\*Actually, I am one among several who have been arguing that field-specific concepts are best addressed with the appropriate terminology. Apparently, your own definition of “appropriate” is “what **TVAA** wants.” \*\* You’re aware that I’m not using that terminology. Do you feel that it’s appropriate to apply its definitions to statements not made within it? > [@](#): > > \*\*By the definitions that I offered, my statement is rigorously correct. Apply your own definitions if you wish, but please stop making ignorant statements about the proper application of terms as used in computer theory. \*\* If the extent of the Game of Life grid is infinite, the meta-grid can represent any configuration that the grid can. If the grid is finite, the grid will ways be able to take on more configurations than a meta-grid emulated by that grid could. In order for a meta-grid to be able to handle the same configurations as a finite grid, it would need to be emulated on another finite grid **much larger than the first**. It would therefore necessarily require more resources to function. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 17, 2003, 5:02am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/147 "2003-04-17T05:02:02Z")
> [@](#): > > \*Originally posted by TVAA \* > \*\*And this is where you err. > > Computers have been built to generate and verify theorems. > > Therefore, even if we don’t agree on the logic behind the incorrectness of your statement, we are forced to accept that you are incorrect. \*\* Oh, for fuck’s sake. It can be done, therefore it’s necessary? I know better than to ask you to support your assertions, so go on, think you’re right. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 17, 2003, 5:12am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/148 "2003-04-17T05:12:58Z")
> [@](#): > > \*\*You appear to be suffering from the misunderstanding that every true statement in a Peano Axiomatization has a “possible proof”, but GIT just won’t let us manifest them. This is incorrect. GIT says that no possible proof exists for some true statements. \*\* NO. GIT shows that, for any given system of axioms capable of representing arithmetic, there will be statements are true but that the system cannot prove. It does NOT follow that there are statements that are true but cannot be proven in ANY system. It may well be the case that some statements can never be shown to be true, but GIT doesn’t imply this. The statement “This statement cannot be proven in system X” might be provable in system Y, which adds one or more axioms to system X. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 17, 2003, 5:16am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/149 "2003-04-17T05:16:10Z")
**Ultrafilter** : It’s known that systems for generating theorem models can be constructed within the physical world. Specifically, it’s known that it’s possible to make systems that can demonstrate arithmetic. Therefore, we know that the underlying rules of the physical world are capable of demonstrating arithmetic. This is necessarily the case. ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [April 17, 2003, 8:11am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/150 "2003-04-17T08:11:29Z")
> [@](#): > > \*Originally posted by scotandrsn \* > \*\* > > What about the strings of CWL translations that represent TMs that can NOT accept their own CWL translations as input? It has been proven mathematically that the collection (language) of such strings is NOT r.e. It has also been shown that this language is NOT empty. Therefore, there is a class of languages that you CAN NOT build a TM (or by extension a UTM/computer) to accept. > > . . . > > our brains CAN accept these languages, \*\* I’d love to see proof of the above. And it seems to me that if you succeed in proving the above, you will have disproved Church’s hypothesis. But I haven’t studied this stuff in years, so please enlighten me. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 17, 2003, 3:15pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/151 "2003-04-17T15:15:00Z")
**TVAA** I’m still watiing for those lines to be drawn. For the rest, it was obviously not apparent to me (or several others) that you were rejecting the teminology of the fields appropriate to your questions in all cases. Your latest post does so again by using “algorithm” in a manner that is inconsistent with the definition specific to computer theory. Your latest post also continues to “chastise” me for past statements made when I thought that you might wish to apply the terminology standard ot the fields you make broad pronouncements about. So, you have proven nothing except that you like to insult people. Bravo. There are holes in your latest post, too, but until you actually provide the detailed proof that you have promised it seems silly for me to expend any effort refuting hand-waving and broad declarations. Present your proof. Until then , you are simply making unsupported pronouncements. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 17, 2003, 3:35pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/152 "2003-04-17T15:35:13Z")
> [@](#): > > Where has my use of language been vague? Where has my use of language been inconsistent? **Well, the use of the word “model” has been inconsistent:** [ul][li]A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.[/li][li]No. The meta-grid can easily be disrupted without disrupting the grid (In response to “they each model the other”).[/ul][/li]So, here you are adding another requirement to your definition for “model”. I know you like dictionaries, so please provide a dictionary citation that injects a requirement upon “disruption” into the word “model”. Then you can answer the question: **Do life and “meta-life” model each other if the “meta-life” is not being instantiated on the same grid as the game of life?** That removes any potential issues of “disuption” and concentrates on your definition of “model” as it relates to your definition of “complexity”. **The use of the word “model” has also been vague:** [ul][li]A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.[/li][li]“Similarity” is not measured – it is, just as the properties of the system are, regardless of whether a mind observes them. Do you think that the Game of Life acts differently when no one’s looking at the computer screen?[/ul][/li]So a model is a system “similar” to another system, but “similarity” is not measured. It “just is”. Thus, I could say that a system defining the orbital motions of planets models a the digestion of a snickers bar. Hey–similarity “just is”, right. We have no way to measure it, no way to create a cutoff, and no human decision is required to judge one system as “similar” to another. ---
**Author:** ![Newton\_meter](https://avatars.discourse-cdn.com/v4/letter/n/57b2e6/32.png) [@Newton\_meter](https://boards.straightdope.com/u/Newton_meter)\ **Post date:** [April 17, 2003, 4:27pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/153 "2003-04-17T16:27:51Z")
> [@](#): > > \*Originally posted by TVAA \* > \*\ ***Ultrafilter** : It’s known that systems for generating theorem models can be constructed within the physical world. Specifically, it’s known that it’s possible to make systems that can demonstrate arithmetic. Therefore, we know that the underlying rules of the physical world are capable of demonstrating arithmetic. This is necessarily the case. \*\* **TVAA** : It’s known that inconsistent formal systems can be constructed within the physical world. Specifically, it’s known that it’s possible to make a system that can claim both _A_ and ~_A_. Therefore, we know that the underlying rules of the physical world are inconsistent. This is necessarily the case. See, you claim that there must be some unprovable “true” state of the universe (whatever that means). I say “no way”, everything’s provable. So who’s right? I am, because I’m not actually so silly as to think that the physical universe is a formal axiomatic system governed by a finite set of axioms, and ruled by the classical laws of non-contradiction and the excluded middle. Here are some more for you: The universe is capable of “modeling” (via my brain) the axiom of infinity from ZF set theory: there exists a set that contains zero and the successor of each of its element. Does that mean that the physical universe necessarily contains an infinite set? The universe is capable of “modeling” (via Zermelo’s brain) the axiom of choice: that given a set of mutually exclusive non-empty sets there exists a set that contains exactly one element in common with each of the sets. Does that mean that the axiom of choice is necessarily true of the physical universe? But wait! The universe is capable of “modeling” (via Cohen’s brain) that if ZF set theory with the axiom of choice is consistent, then ZF set theory with choice replaced by the negation of the axiom of choice is also consistent. Does that mean that the physical universe is necessarily inconsistent? Or does it mean that it is_not_ necessary at all that either the axiom of choice or its negation is “true” of the physical universe? And we can go on and on and on. The universe is capable of modeling, via a set of _Star Trek_ episodes that people travel faster than light, can be “beamed” through space, and that portals exist that allow travel into the past. The universe is capable of modeling leprechauns, pots of gold at the end of rainbows, and magic genies in lamps. For some reason, you want the limits of a particular sort of _formal axiomatic system_ to have something to do with the physical universe. It is unclear to me that they do, and your only argument has been “it’s obvious”. Well, it’s obvious to me that axiomatic number theory doesn’t limit “the universe” in the way that you think it does. And moreover, it’s obvious that your claims are silly. Don’t you think? ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 17, 2003, 5:30pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/154 "2003-04-17T17:30:48Z")
> [@](#): > > \*Originally posted by Spiritus Mundi \* > \*\ ***Well, the use of the word “model” has been inconsistent:** > [ul][li]A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.[/li][li]No. The meta-grid can easily be disrupted without disrupting the grid (In response to “they each model the other”).[/ul][/li]So, here you are adding another requirement to your definition for “model”. I know you like dictionaries, so please provide a dictionary citation that injects a requirement upon “disruption” into the word “model”. \*\* It’s not “another requirement”. According to [dictionary.com](http://dictionary.com), the primary meaning of the word _disrupt_ is “to throw into confusion or disorder”. If a model of a system is changed so that its properties no longer correspond with some of the properties of the system, it’s no longer a model. The order or pattern that it previously manifested is gone. It has been _disrupted_. 🙂 > [@](#): > > \*\*Then you can answer the question: **Do life and “meta-life” model each other if the “meta-life” is not being instantiated on the same grid as the game of life?** That removes any potential issues of “disuption” and concentrates on your definition of “model” as it relates to your definition of “complexity”. \*\* Life can be used to model meta-Life, but meta-Life can never be used to model Life. To have meta-Life, it’s necessary to have Life in the first place. If the meta-Life is modeled without the underlying system of Life, it’s not meta-Life: it’s just Life. > [@](#): > > \*\*So a model is a system “similar” to another system, but “similarity” is not measured. It “just is”. Thus, I could say that a system defining the orbital motions of planets models a the digestion of a snickers bar. Hey–similarity “just is”, right. We have no way to measure it, no way to create a cutoff, and no human decision is required to judge one system as “similar” to another. \*\* I never said there’s no way to measure it. It simply does not require human measurement. And it’s entirely possible that a system modeling the orbital motions of the planets might also model the digestioni of a Snickers. We know for a fact that the system underlying orbital motion is the same system the underlies digestion. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 17, 2003, 5:42pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/155 "2003-04-17T17:42:30Z")
> [@](#): > > \*Originally posted by Newton meter \* > \*\ ***TVAA** : It’s known that inconsistent formal systems can be constructed within the physical world. Specifically, it’s known that it’s possible to make a system that can claim both _A_ and ~_A_. Therefore, we know that the underlying rules of the physical world are inconsistent. This is necessarily the case. \*\* It’s entirely possible for a consistent system to emulate an inconsistent system. **Godel, Escher, Bach** points out several examples of this, in fact. However, an inconsistent system can’t emulate a consistent one. > [@](#): > > \*\*See, you claim that there must be some unprovable “true” state of the universe (whatever that means). I say “no way”, everything’s provable. So who’s right? \*\* Everything’s provable? Fine. There is therefore a proof that everything isn’t provable. > [@](#): > > \*\*I am, because I’m not actually so silly as to think that the physical universe is a formal axiomatic system governed by a finite set of axioms, and ruled by the classical laws of non-contradiction and the excluded middle. \*\* Then you’ve just claimed that the concept of truth doesn’t apply to the universe. Nothing can be either true or false, because anything can be proven true and proven false. Brilliant, Einstein. I refuse to repeat the reasoning that falsifies the examples you offer – I’ll only state it once. It’s certainly true that the system underlying the universe is capable of emulating episodes of Star Trek, and systems that state the axiom of choice, stories of leprechauns, etc. We know this because these things exist in the universe. Just because episodes of Star Trek exist does not imply that the people and events described in those episodes exist in the universe. > [@](#): > > \*\*For some reason, you want the limits of a particular sort of _formal axiomatic system_ to have something to do with the physical universe. It is unclear to me that they do, and your only argument has been “it’s obvious”. > > Well, it’s obvious to me that axiomatic number theory doesn’t limit “the universe” in the way that you think it does. And moreover, it’s obvious that your claims are silly. Don’t you think? \*\* [sigh] I’ll deal with this when I “connect the dots” for **Spiritus** … I see no reason to waste time dealing with the same objection twice. ---
**Author:** ![Newton\_meter](https://avatars.discourse-cdn.com/v4/letter/n/57b2e6/32.png) [@Newton\_meter](https://boards.straightdope.com/u/Newton_meter)\ **Post date:** [April 17, 2003, 8:02pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/156 "2003-04-17T20:02:19Z")
> [@](#): > > \*Originally posted by TVAA \* > **It’s entirely possible for a consistent system to emulate an inconsistent system. Godel, Escher, Bach points out several examples of this, in fact. However, an inconsistent system can’t emulate a consistent one.** But you have not demonstrated in what way the physical universe is emulating _anything_. I had some people over last night to watch hockey. At one point during the evening, I observed that there were three people in the living room, and that two people were in the kitchen. When those two people entered the living room, I counted five people in the living room. Does this mean that my living room is a formal system capable of modeling arithmetic? That my living room is necessarily inconsistent or incomplete, because it is the sort of formal system GIT talks about? No, it does not. > [@](#): > > **Everything’s provable? Fine. There is therefore a proof that everything isn’t provable.** And a proof that the moon is made of green cheese! And one that it’s not! Big whup. > [@](#): > > **Then you’ve just claimed that the concept of truth doesn’t apply to the universe. Nothing can be either true or false, because anything can be proven true and proven false.** Ummm, no, I did not claim that. Notice how wishy-washy you are about “proven true” and “being true” in everything you write? That’s a pretty big problem, when you want to talk about GIT. Here’s the nub. I think you want to claim that the universe is “doing arithmetic”. Well, it’s not. What is the universe doing? It’s just doing whatever universes do. You know, universe stuff. I don’t think you can construct any kind of Godel sentence about the universe–something that’s “true about the universe” but can’t be “proven by the universe”, because the universe lacks the sort of self reference you need to make the stuff that happens _in_ the universe be _about_ the universe. Without that, you’ve got dick. > [@](#): > > \*\*I’ll deal with this when I “connect the dots” for **Spiritus** … I see no reason to waste time dealing with the same objection twice. \*\* I’m eagerly awaiting that. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 17, 2003, 8:12pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/157 "2003-04-17T20:12:20Z")
Hey, **TVAA** , I’ve got a question for you. In your understanding, what does GIT **not** apply to? ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 17, 2003, 8:24pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/158 "2003-04-17T20:24:08Z")
> [@](#): > > It’s not “another requirement”. According to [dictionary.com](http://dictionary.com), the primary meaning of the word disrupt is “to throw into confusion or disorder”. If a model of a system is changed so that its properties no longer correspond with some of the properties of the system, it’s no longer a model. Well, this is another example of your inconsistent usage. Or perhaps inconsistent reading. The question I asked was: **does the “Meta-life” game model the life game** , not _will the “meta-life” game be forever and ever a model of the life game_. So, either you misread the question and anwered whether some future, post-disruption, Meta-life game would be a model of the life game, or else you have added another requirement to “model” that considers the possibility of future disruption. > [@](#): > > Life can be used to model meta-Life, but meta-Life can never be used to model Life. To have meta-Life, it’s necessary to have Life in the first place. If the meta-Life is modeled without the underlying system of Life, it’s not meta-Life: it’s just Life. This one I am certain is a discrepancy in reading. I explicitely asked about a “Meta-life” that is not being instantiated on the same grid. I thought that was clear, but obviously not. Consider the following: [ul][li]Comuter(A) -\> life(A) -\> Meta-life(A)[/li][li]Computer(B) -\> life(B) -\> Meta-life(B)[/ul][/li]Can Meta-life(A) model Meta-life(B)? Can life(A) model life(B)? **Can Meta-life(A) model life(B)?** The bolded question, of course, is the one I asked in my last post. > [@](#): > > I never said there’s no way to measure it. It simply does not require human measurement. And it’s entirely possible that a system modeling the orbital motions of the planets might also model the digestioni of a Snickers. We know for a fact that the system underlying orbital motion is the same system the underlies digestion. Ooops. There you go being inconsistent in your use of “model” again. We know for a fact that the same system underlies a “disrupoted” game of Meta-life and a game of life, and just a few lines ago you said that Meta-life could never model life. As I said some time ago, you do yourself no favors when you reject the well-defined terminology of the field(s). And, for the record, what you said was: > [@](#): > > “Similarity” is not measured – it is Now, if you meant that statement to include the possibility that “similarity” **can be measured** , then I’m afraid I find your word choice to be misleading. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 17, 2003, 8:30pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/159 "2003-04-17T20:30:49Z")
> [@](#): > > It’s not “another requirement”. According to [dictionary.com](http://dictionary.com), the primary meaning of the word disrupt is “to throw into confusion or disorder”. If a model of a system is changed so that its properties no longer correspond with some of the properties of the system, it’s no longer a model. Well, this is another example of your inconsistent usage. Or perhaps inconsistent reading. The question I asked was: **does the “Meta-life” game model the life game** , not _will the “meta-life” game be forever and ever a model of the life game_. So, either you misread the question and anwered whether some future, post-disruption, Meta-life game would be a model of the life game, or else you have added another requirement to “model” that considers the possibility of future disruption. > [@](#): > > Life can be used to model meta-Life, but meta-Life can never be used to model Life. To have meta-Life, it’s necessary to have Life in the first place. If the meta-Life is modeled without the underlying system of Life, it’s not meta-Life: it’s just Life. This one I am certain is a discrepancy in reading. I explicitely asked about a “Meta-life” that is not being instantiated on the same grid. I thought that was clear, but obviously not. Consider the following: [ul][li]Comuter(A) -\> life(A) -\> Meta-life(A)[/li][li]Computer(B) -\> life(B) -\> Meta-life(B)[/ul][/li]Can Meta-life(A) model Meta-life(B)? Can life(A) model life(B)? **Can Meta-life(A) model life(B)?** The bolded question, of course, is the one I asked in my last post. > [@](#): > > I never said there’s no way to measure it. It simply does not require human measurement. And it’s entirely possible that a system modeling the orbital motions of the planets might also model the digestioni of a Snickers. We know for a fact that the system underlying orbital motion is the same system the underlies digestion. Ooops. There you go being inconsistent in your use of “model” again. We know for a fact that the same system underlies a “disrupoted” game of Meta-life and a game of life, and just a few lines ago you said that Meta-life could never model life. As I said some time ago, you do yourself no favors when you reject the well-defined terminology of the field(s). And, for the record, what you said was: > [@](#): > > “Similarity” is not measured – it is Now, if you meant that statement to include the possibility that “similarity” **can be measured** , then I’m afraid I find your word choice to be misleading. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 17, 2003, 8:37pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/160 "2003-04-17T20:37:27Z")
Also, would you mind taking my statement of GIT on page 2 and translating it into plain English? [Previous page](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157.md?page=7) [Next page](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157.md?page=9)