# 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:** 10
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 18, 2003, 2:50pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/181 "2003-04-18T14:50:31Z")
> [@](#): > > \*Originally posted by TVAA \* > \*\*To you, perhaps. Others realize that the interactions between the system’s elements necessarily follow some principles, and those principles can be represented by a set of operations. > > I dislike your presentation of GIT, **ultrafilter**. The version I linked to is much better, both in terms of overall clarity and reduced jargon. \*\* The presentation you linked to is so vague that if I didn’t already know the theorem, I wouldn’t learn it from there. You don’t like the jargon? Tough shit. That is exactly what the theorem says, and anything you want to claim that it implies has to follow from that. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 2:58pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/182 "2003-04-18T14:58:53Z")
> [@](#): > > \*Originally posted by Spiritus Mundi \* > \*\*I am. The definition you originally offered was for a noun. As I said, this is one example of your inconsistent use of terms. \*\* And you hadn’t noticed that I appeared to be using a noun as a verb? :rolleyes: The accompaning words almost always make clear whether words that have both noun- and verb-uses are being used in a particular way. > [@](#): > > \*\*I care not at all, so long as you use your chosen term unambiguously and concistently. Thus far, you have not done so. \*\* Yes I have. If I speak of “a fly” and “to fly”, am I being inconsistent? Can I say that the fly flies but flies fly? > [@](#): > > \*\*Why? The behavior of Meta-life(A) is already similar to the behavior of life(B). You need to choose a single consistent definition for model and apply it appropriately in all cases. \*\* I already have, **Spiritus**. I think you _would_ understand the word “emulate”, now that it’s come to that. If life(A) is sufficiently large, and the pattern it contains is appropriate, it can _emulate_ meta-life(A). Meta-life(A) might go on to emulate meta-meta-life(A), and so on. > [@](#): > > \*\*Tell me, does **TVAA** English preserve the relationship between the verb and the noun? Is that which models a system a model of that system? If so, Meta-life(A) can model (verb) life(A) by the definition you have offered for model (noun). Meta-life(A) **manifests properties similar to those of** life(B). \*\* No – in this meaning of the word, the model is the thing being modeled. Active, not passive. In the passive sense, a clay statue can be a model of a posing person. In the active sense, the statue is the model – and it’s being modeled by the clay. Just use emulated. “Modeling” is clearly too confusing for you. > [@](#): > > \*\*Your statement contradicts the definition that you have offered for model (noun). \*\* But not model (verb) (active). > [@](#): > > \*\*I asked you **how is measurement meaningful absent an interpretive agent**. You replied “there don’t have to be any measurements.” > > I don’t care. That wasn’t the question I asked. \*\* [sigh] First, there simply don’t need to be any measurements, so your question is not significant in context of the discussion. Secondly, the nature of the system’s properties come into play when something interacts with it – it doesn’t matter if a “consciousness” is involved or not. A bubble film of water surrounding air takes on a roughly spherical form, even if there are no consciousnesses who are monitoring this or who know that this is a result of minimization. > [@](#): > > \*\*That is not correct. the only way to prove that X is “true in a P. A.” is to use only the tools available to a P. A. to demonstrate X. If you have to use additional axioms, then all you can prove is that X is true in a P. A. +Y. \*\* Ah, I see the problem. I’m not talking about the ways to prove within the PA that the statement is true. I’m talking about the ways to prove that the statement is true within a PA. Consistency requires that the truth of a statement depend on the axioms on the system. Completeness requires that a statement be derivable using only the rules of a limited system. PA is not consistent and complete – there are statements within it that are true, and can be shown to be true within a PA by referencing other axioms that don’t contradict PA’s, but that cannot be shown to be true using only PA’s axioms. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 3:11pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/183 "2003-04-18T15:11:15Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*The presentation you linked to is so vague that if I didn’t already know the theorem, I wouldn’t learn it from there. You don’t like the jargon? Tough shit. That is exactly what the theorem says, and anything you want to claim that it implies has to follow from that. \*\* Yes – _you_ wouldn’t be able to learn it from that presentation, would you? 😃 Rucker’s presentation [here](http://www.miskatonic.org/godel.html) is more than enough for any intelligent person to understand it. And it doesn’t use any jargon at all! (Hey… the exclamation point really _does_ make my statement more meaningful. I’ll have to use it more often! After every sentence! Huzzah!) If you can only understand GIT in a single language, **ultrafilter** , you don’t really understand it – you can’t think about it conceptually and determine whether other concepts are actually implied by it. You can’t evaluate whether a statement in English is actually consistent or not with GIT – so why are you bothering to be in this debate? I don’t recall couching anything I’ve said in mathmatical symbology… Of course, you’re also the person who denies that the universe has any limits or limitations. :smack: ---
**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 18, 2003, 3:58pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/184 "2003-04-18T15:58:20Z")
> [@](#): > > And you hadn’t noticed that I appeared to be using a noun as a verb? I had, thus I pointed to this as a case in which you were incosistent in usage. **THAT WAS WHAT YOU ASKED ME TO DO** , remember? Defining a word as a noun and then using it as a verb is inconsistent. You have also deviated from the specification of that definition in declaring what is and what is not a model. That is also inconsistent, but is a separate issue. > [@](#): > > Yes I have. If I speak of “a fly” and “to fly”, am I being inconsistent? Can I say that the fly flies but flies fly? If you offer only a definition of “a fly” and then use “to fly” in a manner that is not 100% derivable from “A fly” then yes, you are being inconsistent. Really, this is a pretty tired dodge on your part. It would be much simpler to just say, “Oh, I should have given my definition of “model” as an active verb.” :wally > [@](#): > > I already have [offered a single definition and stcuk to it], Spiritus. No, you have offered only one definition and then deviated from it. Nothing in **manifests properties similar to those of** implies the cirteria you are now proposing for “model” (or “emulate”). Look, I don’t really care what words you use. You asked for examples of inconsistent use. I provided them. You can dance around this all you like but you have not been using the word “model” to mean: **a system – a set of interactions – that manifests properties similar to those of another system.** Period. I don’t know why you find it necessary to justify a mistake rather than correcting it, but I am done with this side discussion. You asked. I answered. The music has ended, so I’m going to stop dancing. You can do what you wish. > [@](#): > > If life(A) is sufficiently large, and the pattern it contains is appropriate, it can emulate meta-life(A). Meta-life(A) might go on to emulate meta-meta-life(A), and so on. Fine. Do you have any term to describe the relationship between Meta-life(A) and life(B), since those two systems can be described by identical mathematical rules. > [@](#): > > No – in this meaning of the word, the model is the thing being modeled. Active, not passive. As noted, your original definition made no such distinction, unless you are proposing that the relationship/property “similarity” is not symmetrical. [ul][li]System(A) manifests properties similar to those of system(B).[/li][li]System(B) manifests properties similar to those of system(A).[/ul][/li]Can one be true without the other? (Don’t read this as a return to the definition of model. Here I am really just interested in whether your use of “similar” possesses symmetry.) > [@](#): > > First, there simply don’t need to be any measurements, so your question is not significant in context of the discussion. I gave the context of our discussion. You were the one who objected when I attempted to continue our discussions of “similarity” by explicitely denying the possibility of measurement. **"I never said there’s no way to measure it. It simply does not require human measurement."** The question is in our discussion only because you have made a positive claim about measurement: that it can be meaningful absent an interpretive agent. > [@](#): > > Secondly, the nature of the system’s properties come into play when something interacts with it – it doesn’t matter if a “consciousness” is involved or not. And what, exactly, has that to do with the question that I asked you to answer. Here, I’ll repeat that section of my post for you. > [@](#): > > **Spiritus Mundi** > how is measurement meaningful absent an interpretive agent I’ll repeat this, too. **Why don’t you simply answer the questions that I ask?** > [@](#): > > A bubble film of water surrounding air takes on a roughly spherical form, even if there are no consciousnesses who are monitoring this or who know that this is a result of minimization. Are you saying that “bubble taking a spherical shape” is a **measurement**? > [@](#): > > Ah, I see the problem. I’m not talking about the ways to prove within the PA that the statement is true. I’m talking about the ways to prove that the statement is true within a PA. That is not the problem at all, but I’ve already said that I would let this fall unless you want to continue it for the sake of having something else to fight about. If you specifically reply to this section of my post, I’ll assume that you do. Before I go further, please clarify for me whether you actually have already posted your “draw the lines” proof. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 18, 2003, 4:06pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/185 "2003-04-18T16:06:15Z")
GIT, as it is with all theorems, requires certain axioms to be the case for its truth. Please explain what those axioms are in whatever language you choose. Please follow that up with the assumptions necessary to create the formal mapping between **the laws of physics** , **the initial states of the universe** , and **GIT**. In doing so, be sure to define **key terms**. Is it just me or have you failed to do all of this? ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 5:15pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/186 "2003-04-18T17:15:13Z")
> [@](#): > > \*Originally posted by Spiritus Mundi \* > \*\*Defining a word as a noun and then using it as a verb is inconsistent. \*\* But then it’s a different word. It’s not as if I’m using a single word to refer to two different noun concepts, or two different verb concepts. > [@](#): > > \*\*If you offer only a definition of “a fly” and then use “to fly” in a manner that is not 100% derivable from “A fly” then yes, you are being inconsistent. > > Really, this is a pretty tired dodge on your part. It would be much simpler to just say, “Oh, I should have given my definition of “model” as an active verb.” \*\* You didn’t ask for that definition. You demanded that I define “a model”. I did so. If you had asked about “to model” I would have cited the dictionary for that as well… > [@](#): > > [bFine. Do you have any term to describe the relationship between Meta-life(A) and life(B), since those two systems can be described by identical mathematical rules. \*\* There isn’t necessarily a relationship at all. And they can’t be described by the same statements – it’s not meta-life if it’s not a particular configuration manifested by a life grid. The high-level behavior of meta-life obeys the same rules as life. > [@](#): > > \*\*The question is in our discussion only because you have made a positive claim about measurement: that it can be meaningful absent an interpretive agent. \*\* No! I wasn’t the one who made that claim – I _believe_ you were the one who claimed I claimed that first (it may have been someone else). All measurement is interaction, but not all interaction is measurement. Systems have properties regardless of whether they’re measured or not – those properties have “meaning” regardless of whether they’re measured or not. > [@](#): > > \*\*Are you saying that “bubble taking a spherical shape” is a _measurement_? \*\* No. It’s a computation: the shape the bubble takes is a result of the interaction of the water molecules and the air molecules. If you can’t tell, then I haven’t connected the dots sufficiently enough. 😛 I’m still trying to determine what your problem is – I now believe it’s in your ideas about whether the universe is within the category of things GIT applies to instead of problems with GIT itself. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 18, 2003, 5:33pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/187 "2003-04-18T17:33:55Z")
> [@](#): > > \*Originally posted by TVAA \* > \*\*Yes – _you_ wouldn’t be able to learn it from that presentation, would you? 😃 > > Rucker’s presentation [here](http://www.miskatonic.org/godel.html) is more than enough for any intelligent person to understand it. And it doesn’t use any jargon at all!\*\* I don’t even know what you’re thinking at this point. Yes, all of the links you have posted do give you the basic idea of GIT. None of them-not one! Check it in the dictionary you love so much-give **anywhere** enough detail for you to be able to use it. You have demonstrated this fact to my satisfaction. \*\* > [@](#): > > If you can only understand GIT in a single language, **ultrafilter** , you don’t really understand it – you can’t think about it conceptually and determine whether other concepts are actually implied by it. You can’t evaluate whether a statement in English is actually consistent or not with GIT – so why are you bothering to be in this debate? I don’t recall couching anything I’ve said in mathmatical symbology…\*\* Of course not. You don’t know the mathematics behind it. Unsurprisingly, you don’t understand it. I can state the theorem in English, but I’ve passed more than one graduate level class where it was necessary to understand things like this, so I don’t really feel the need to go through such an exercise. \*\* [quoteOf course, you’re also the person who denies that the universe has any limits or limitations. :smack: \*\*[/QUOTE] Please remind me where I said this. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 6:14pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/188 "2003-04-18T18:14:08Z")
> [@](#): > > \*Originally posted by erislover \* > \*\*GIT, as it is with all theorems, requires certain axioms to be the case for its truth. > > Please explain what those axioms are in whatever language you choose. \*\* Haven’t we done this already? We’ve already acknowledged that the Incompleteness Theorem shows that any sufficiently powerful set of consistent axioms will not be complete. In other words, there will be statements that are true relative to the axioms that cannot be proven by those axioms. The proofs lie outside the region the axiom’s operations can reach. What’s a “sufficiently powerful” set of axioms? They’re that powerful if they include the axioms underlying arithmetic (in other words, they contain _operations_, which establish relationships between two or more elements). If the axioms can’t be manipulated to derive more statements, the Theorem doesn’t apply. (There are a variety of systems that are known to be consistent _and_ complete: the definition of the real numbers, for example. That system isn’t very interesting by itself, through.) > [@](#): > > **Please follow that up with the assumptions necessary to create the formal mapping between _the laws of physics_, _the initial states of the universe_, and _GIT_. In doing so, be sure to define _key terms_.** Regardless of what laws of physics actually _are_, the progression of events in time can be represented as the application of operations to statements. (Essentially, an accurate description of the system can be used to determine what it will do next.) > [@](#): > > \*\*Is it just me or have you failed to do all of this? \*\* Honestly? I think these points have gotten lost in the endless bickering going on in this thread. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 6:37pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/189 "2003-04-18T18:37:35Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*I don’t even know what you’re thinking at this point. Yes, all of the links you have posted do give you the basic idea of GIT. None of them-not one! Check it in the dictionary you love so much-give **anywhere** enough detail for you to be able to use it. You have demonstrated this fact to my satisfaction. \*\* What? This paragraph doesn’t make any sense. The third sentence has no obvious referent. Did you splice sentences together out-of-order? Ah, I think I see. You shouldn’t have capitalized ‘check’ – that changes the structure and therefore the meaning of the sentence. What do you mean, “they don’t give enough detail to use it”? The only thing they don’t demonstrate is that the statement “this statement cannot be proven in system X” is well-formed instead of being vague, and enough mathematicians have examined this claim and verified it to my satisfaction. I’ve read about Godel’s numbering system that shows the correspondence between numbers and statements, so although I don’t personally know how to translate any statement into a number or vice versa in any “Godel languages”, I can still reason about the known fact that it’s possible to do so. \*\* > [@](#): > > Of course not. You don’t know the mathematics behind it. Unsurprisingly, you don’t understand it. > > I can state the theorem in English, but I’ve passed more than one graduate level class where it was necessary to understand things like this, so I don’t really feel the need to go through such an exercise. \*\* I’m not familiar with mathematical nomenclature, no. Your point is…? > [@](#): > > \*Of course, you’re also the person who denies that the universe has any limits or limitations. :smack: \*\*\*Please remind me where I said this. \*\* You’re the one who thinks calculations have nothing to do with theorem-proving, correct? You don’t think that operations can be applied to axioms to produce theorems? That’s what a calculation _is_: showing what the consequence of the rules of mathematics are in a specific situation. You’re the one who claims that physical systems and simulations that accurately represent those systems are somehow different, right? You’re the one who says no physical system can be equal to an abstract concept, right? Hasn’t it ever crossed your mind that _you’re_ a physical system that can supposedly handle these abstract concepts? ---
**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 18, 2003, 7:08pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/190 "2003-04-18T19:08:24Z")
> [@](#): > > You didn’t ask for that definition. You demanded that I define “a model”. I did so. If you had asked about “to model” I would have cited the dictionary for that as well… :rolleyes: Once again, I am forced to point out that you really should check the progress of our conversations before making misrepresentations. Here are the interactions leading up to your offered definition: > [@](#): > > **S. M.** I agree. So present a case of “modeling” that does not require an interpretive consciousness. In what sense does “meta-life” model “life”? **verb** > > **TVAA** I already have. What I’m trying to do now is understand why you’re not realizing this. Does the algorithm for the Game of Life has a consciousness? If it doesn’t, haven’t I just presented an example of a non-conscious system modeling another? **gerund (noun form of a verb)** > > **S. M.** That depends upon what you mean by “model”, which is the question I have been asking you to answer. **No specification, but direct reference to your last usage.** > > **TVAA** 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. **noun** I say again. :rolleyes: > [@](#): > > (bolding by **S. M.** ) > > There isn’t necessarily a relationship at all. And [Meta-life(A) and life(B)] \*\*can’t be described by the same statements \*\* > > (earlier) This complicated structure can be considered to have two possible states: “meta-on” and “meta-off”. When placed in a grid with copies of itself, **this pattern reproduces the Game of Life.** > > (in between those two) 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.** You are contradicting yourself. > [@](#): > > No! I wasn’t the one who made that claim – I believe you were the one who claimed I claimed that first (it may have been someone else). This is ridiculous. I quoted the entire progress of our discussion on this point. You made the statement: **"I never said there’s no way to measure it. It simply does not require human measurement."** Deny it if you wish, but the words are yours and the words remain visible. > [@](#): > > All measurement is interaction Well, this is progress. Interaction with what? Are you now saying that an interpretive agent _ **is** _ required for measurement? > [@](#): > > No. It’s a computation: the shape the bubble takes is a result of the interaction of the water molecules and the air molecules. Why would you offer an example of a computation in response to a question about measurement? As I have said before, this would be much easier if you would answer the questions that I actually ask. > [@](#): > > If you can’t tell, then I haven’t connected the dots sufficiently enough. 😛 True. I withheld a detailed critique of your last “proof” because it hardly seemed to “draw the lines” at all. > [@](#): > > I’m still trying to determine what your problem is – I now believe it’s in your ideas about whether the universe is within the category of things GIT applies to instead of problems with GIT itself. I have said so all along. Since it now appears that the latest arguemtn offered was indeed your “draw the lines” proof, I will address it. Rather than taking it as a whole, which will certainly engender many simultaneous lines of contention, I will take the statements one at a time. For reference, I will list the argument here: [ol][li]A system that includes ‘events’ can always be shown to be equivalent to a set of operations. The operations describe how one state of the system leads to another.[/li] [li]If we posit a finite universe, GIT doesn’t apply, because finite systems can only represent a limited number of configurations and require a finite number of statements to describe them. They’re not complex enough to include all of arithmetic.[/li] [li]However, this means that there are fundamental limits on what goes on in these systems. The universe might not include the statement that a given subsystem is vulnerable to; a Godel statement for this subsystem might be too long to be represented by the finite universe, for example.[/li] [li]If we don’t assume that the possible interactions of this system with other things are limited (in other words, if we accept the possibility of an infinite universe), we’re forced to consider everything.[/li] [li]Anything that can be modeled – anything that can be represented as a system of operations – will then be vulnerable to GIT.[/li] [li]The representation of this system will have statements that it cannot derive but are true anyway.[/li][/ol] So, taking statement (1) first: > [@](#): > > A system that includes ‘events’ can always be shown to be equivalent to a set of operations. The operations describe how one state of the system leads to another. As **ultrafilter** noted, this is not obviously true. In particular, it seems to require a deterministic reality (else the “how” would be inappropriate.) For that matter, the word “operations” also seems to imply a deterministic reality. I can accept that any particular state of a system (“event”) can be mathematically descibed. But the term “operation” implies that a mathematicla tranformation is always possible to move from one state to another. Now, I suppose that one might argue that reducing all elements from the state description of event(A) and then generating all elements of the state description of event(B) can be called an “operation”, but I get the feeling you mean “operation” to imply something more (things that migt justify the word “how”, for instance). So, please explain in some detail exactly how you determine that every system of events can be shown to be equivalent to a system of operations. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 18, 2003, 7:17pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/191 "2003-04-18T19:17:27Z")
> [@](#): > > \*Originally posted by TVAA \* > In other words, there will be statements that are true relative to the axioms that cannot be proven by those axioms. The proofs lie outside the region the axiom’s operations can reach. This is not an obvious or intutitve result. What does “true” mean here when most people would think it means “provable”? Do you think this answer will help, hurt, or prove irrelevant to your claims? Suppose true means a counter-example cannot be found. This means, of course, that there will be statements **we can make that are true** which **cannot be shown to be true**. How does this limit the universe, even supposing your strange insistance on its character as a “model” or “for modelling” or however you’re using the word? Does it say there are states the universe cannot be in? _How?_ Seems to me the universe can be in whatever state it wants. Seems to me like an epistemological limitation, not a real boundry on objective states. > [@](#): > > Regardless of what laws of physics actually _are_, the progression of events in time can be represented as the application of operations to statements. (Essentially, an accurate description of the system can be used to determine what it will do next.) Yes, and I can compare penises to bananas. Nevertheless I do not inspect my penis for spider infestations, nor do I wonder about frying some up in coconut oil to sell in self-serve bins at the supermarket. You’ve made an analogy. It is an interesting analogy. Its power does not extend realistically to the depth and character you wish. If it does, you have failed to convince anyone here of it. Sorry. What does this symbol represent: **2** ? Sometimes, it represents the number of hours before I will consider eating lunch. Sometimes it represents the result of an operation on a calculator. Sometimes it represents the quantity of single dollar bills in my pocket. Fundamentally, we find that given the background necessary to prove GIT that system will be able to make expressions **which are true** but which **cannot be proved**. Since the statement would be true, do you suppose it is or is not the case that I have two dollars in my pocket? How does a funadamental limitation **of knowledge within a system** represent anything about **what actually happens**? ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 7:18pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/192 "2003-04-18T19:18:14Z")
**Spiritus** , _pay attention_. I am explicitly claiming that the _measurement_ of a system has nothing to do with the nature of its properties. Your original question _has nothing to do with this debate_. > [@](#): > > \*\*Interaction with what? Are you now saying that an interpretive agent _is_ required for measurement? INTERACTION WITH SOMETHING ELSE! A truly indeterminate universe couldn’t have any properties – there would be no way for any specific interactions to take place. You’d end up with the superposition of all states, which doesn’t contain any data. Who said anything about an interpretive agent? ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 18, 2003, 7:42pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/193 "2003-04-18T19:42:57Z")
Have I mentioned lately how much I appreciate your presence on these boards, **erislover**? Your arguments are always pleasantly interesting. > [@](#): > > \*Originally posted by erislover \* > \*\*This is not an obvious or intutitve result. What does “true” mean here when most people would think it means “provable”? Do you think this answer will help, hurt, or prove irrelevant to your claims? \*\* That is an excellent question, and a very difficult one. The problem is that “true” doesn’t mean “provable” in mathematics. If a system is complete, that is indeed implied, but it’s possible to be true but not provable. Perhaps true should be interpreted to mean that the system can never generate a result inconsistent with the “true” statement – in other words, it can never be disproved or shown to be incompatible with the system. To be honest, I don’t think this hurts or helps my claim… although I could be mistaken about that. I’ll think about it for a while. > [@](#): > > **Suppose true means a counter-example cannot be found. This means, of course, that there will be statements _we can make that are true_ which _cannot be shown to be true._** Yes, that’s exactly right. > [@](#): > > \*\*How does this limit the universe, even supposing your strange insistance on its character as a “model” or “for modelling” or however you’re using the word? Does it say there are states the universe cannot be in? _How?_ Seems to me the universe can be in whatever state it wants. Seems to me like an epistemological limitation, not a real boundry on objective states. \*\* To use an anthropomorphic metaphor: can the universe choose what it wants? Claiming that the universe has a particular property is placing a limitation on what it is. If you take away all limitations, you take away all capacities, too – when all statements are considered to apply, _none of them apply_, because each statement is negated by its opposite and vice versa. It’s strange but true: perfect opposites are equivalent. This isn’t directly germane to my point, though. > [@](#): > > **Yes, and I can compare penises to bananas. Nevertheless I do not inspect my penis for spider infestations, nor do I wonder about frying some up in coconut oil to sell in self-serve bins at the supermarket.** The behavior of your penis is (_hopefully_) not very similar to the behavior of a banana. If we view them as collections of atoms, they suddenly become much more similar: they’re atoms of various kinds, positioned in different configurations. They both behave according to similar principles on this level. Does a carbon atom in your penis act differently than one in the banana? Can you even show a clear delineation between either system and the environment? Do the atoms in one system somehow segregate themselves from the rest of the world? > [@](#): > > \*\*You’ve made an analogy. It is an interesting analogy. Its power does not extend realistically to the depth and character you wish. If it does, you have failed to convince anyone here of it. Sorry. \*\* Well, at least you’re honest. I respectfully disagree with your statements (except the part about not convincing anyone here – that’s clearly true). > [@](#): > > \*\*What does this symbol represent: **2**? > > Sometimes, it represents the number of hours before I will consider eating lunch. Sometimes it represents the result of an operation on a calculator. Sometimes it represents the quantity of single dollar bills in my pocket. Fundamentally, we find that given the background necessary to prove GIT that system will be able to make expressions _which are true_ but which _cannot be proved_. \*\* That symbol doesn’t represent anything: it’s not part of a set of relationships. If you don’t specify what system it should be considered in, it signifies nothing. It calls to mind the symbol used to represent “twoness” in certain languages, though. Now, if you’d asked me what the excited phosphorus regions on the screen were representing, I could tell you. They represent this: **2**. And the atoms of which they’re composed represent the phosphorus regions. Do the following patterns of phosphorus excitation have anything to do with felines? “C A T S” For that matter, do patterns of vibrations in air or lines in a variety of mediums (words) have anything to do with reality? > [@](#): > > \*\*Since the statement would be true, do you suppose it is or is not the case that I have two dollars in my pocket? How does a funadamental limitation _of knowledge within a system_ represent anything about _what actually happens_? \*\* Does the system accurately describe how the system acts? If not, what description _does_ accurately describe what the system does? If there’s no description, what properties does the system have? If there is a description, and it’s truly accurate, then the system and the description are equivalent. Many individuals I know loved _The Matrix_, not because they liked the special effects or thought the plot made sense, but because it demonstrated that certain concepts they had thought were obvious and well-accepted were utterly unfamiliar to the majority of people in our society. The things in the Matrix are _real_. When the simulation represents all of the characteristics that humans perceive and make judgments based on, they _are_ those things. All they need to do is match the patterns that the systems in question (human minds) process. The entire simulation is an emulation of a “deeper” level of reality, the one that people’s bodies are in. The people’s minds are being emulated by their brains; they’re on the same ‘level’ of reality as the Matrix. Thus, the interactions between the Matrix and their minds are utterly real, just as the interactions between their bodies and the “real” world are real. But one reality is not the same as the other. ---
**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 18, 2003, 7:55pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/194 "2003-04-18T19:55:55Z")
> [@](#): > > Spiritus, pay attention. I am paying attention. Apparently that is teh problem. You prefer to operate as if each exchange were uninformed by the exchanges that preceded it. > [@](#): > > I am explicitly claiming that the measurement of a system has nothing to do with the nature of its properties. Your original question has nothing to do with this debate. Well, then you should have simply said “yes” when I said: [ul][li]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.[/ul][/li]Instead, you said: [ul][li]I never said there’s no way to measure it. It simply does not require human measurement.[/ul][/li] Now you go all italics at me as if I were the one who had introduced “meaasurement without a human agent” into the debate. :wally > [@](#): > > INTERACTION WITH SOMETHING ELSE! Anything else? Does measurement require no specifications upon the agents doing the interaciont? Alternatively, do you see no contradiction in _going all italics at me to declare that this issue has nothing to do with the debate_ and THEN GOING ALL CAPS AT ME TO SCREAM “MEASUREMENT IS INTERATION WITH SOMETHING ELSE”? Whatever. > [@](#): > > A truly indeterminate universe couldn’t have any properties **Indeterminate?** Who said anything about en indeterminate Universe? (Not that I acept your assertion, there just doesn’t seem to be any point in making it.) What I talked about was an implication for deterministic behavior in your systems of events. The converse of “deterministic” is **“nondeterministic”** (alternatively **“free”** ). > [@](#): > > Who said anything about an interpretive agent? I did, and you did, and I did some more. It appears that you are the one who should be paying attention. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 18, 2003, 8:52pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/195 "2003-04-18T20:52:51Z")
> [@](#): > > \*Originally posted by TVAA \* > Have I mentioned lately how much I appreciate your presence on these boards, **erislover**? Your arguments are always pleasantly interesting. No, you haven’t. 🙂 Was it the penis talk? 😛 > [@](#): > > That is an excellent question, and a very difficult one. The problem is that “true” doesn’t mean “provable” in mathematics. I’m shocked, shocked, I tell you, that common sense understanding of things or dictionary definitions might not apply in certain contexts. I wonder who ever tried to convince me differently? > [@](#): > > To use an anthropomorphic metaphor: can the universe choose what it wants? Sorry, didn’t mean for “wants” to indicate something like that. Just that there is no state the universe can’t be in **even if** GIT limited it, because the statements can still be made. They just can’t be proven. Which, since the universe—not obviously being an intentional agent—doesn’t prove things anyway, is to be expected. > [@](#): > > The behavior of your penis is (_hopefully_) not very similar to the behavior of a banana. If we view them as collections of atoms, they suddenly become much more similar… > > …That symbol doesn’t represent anything: it’s not part of a set of relationships. If you don’t specify what system it should be considered in, it signifies nothing. Exactly, which is why I want to know why I should consider the universe as **the only system** when at one time an atom is just an atom, and at another it is used to represent a binary state (say, in molecular logic chips), and at another… > [@](#): > > Does the system accurately describe how the system acts? What am I using for judging accuracy? > [@](#): > > The things in the Matrix are _real_. When the simulation represents all of the characteristics that humans perceive and make judgments based on, they _are_ those things. All they need to do is match the patterns that the systems in question (human minds) process. They are _perceived_ as those things. Again the limitations of human sensation forbid self-evident statements about things-in-themselves. More accurately phrased, we talk of phenomena. We assume they have a cause lying elsewhere. It seems you want me to accept the primacy of sensation in attributing ontological states to things which are primary or atomic (not atomic theory “atomic” but the original sense of the word). Which is _really_ primary? Why isn’t the Matrix real? How do we know the ship is? It is as if you want me to use the same word to describe them only because they are similar. Of course, I am the one who must sense the similarity, which says something _about me_ not strictly something _about the relationship between the Matrix and the ship._ No? ---
**Author:** ![FranticMad](https://avatars.discourse-cdn.com/v4/letter/f/c68b51/32.png) [@FranticMad](https://boards.straightdope.com/u/FranticMad)\ **Post date:** [April 19, 2003, 8:07pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/196 "2003-04-19T20:07:12Z")
Let me focus on the issue of the “interpretive agent”. It’s not getting resolved, but it’s important and I’ll explain why. The discussion starts with this sequence: 1. **Spiritus** asks **TVAA** what he means by “a model”. 2. **TVAA** replies: \*A model is a system – a set of interactions – that manifests properties **similar** to those of another system. \* 3. **Spiritus** requests clarification on TVAA’s use of the word “similar”. I believe his point is this: if a model requires that two systems manifest _similar_ properties, how does the model define similarity? It’s important to answer this because if you’re going to use a _similarity_ to argue your point, then we must know what you mean. If **TVAA** is using a concept of _similarity_ that has no definition or is wrongly defined, then all of his subsequent reasoning would be invalidated. I agree with **Spiritus** on this. “Similarity” needs to be defined because it is so vague – two things that seem similar to one person can seem different to another person. “Similarity” is not a mathematical concept, nor even an objective concept. Similarity is, at least in part, in the eye of the beholder. By example, TVAA’s model of the universe is in TVAA’s mind – and perhaps nowhere else yet. So, even if you argue that two systems are similar without an observer, how can you possibly **show** that the two systems are, in fact, similar? Why should we believe your model? Without some measure of similarity, your assertions are unprovable. This is why **Spiritus** asks for a definition or measure of similarity. But **TVAA** neglects, forgets, or never understood, the original point. The issue gets lost in the galliarde of unsupported, shifting claims. TVAA’s response to the question (see title of this post) seems to be: models and objects just are similar. Take my word for it, he implies, because it is obvious. No, nothing about a TVAA model is obvious to any other observer here. **TVAA** , you have to define similarity if you are going to use it as one of your axioms. If you’re claiming that you know how the Universe is modeled – possibly the most grandiose claim one can imagine – then you must show us how you know that two systems are “similar” enough for one to model the other. - (I suspect that this whole thread is just TVAA’s attempt to argue that the Universe is deterministic. By using the pseudo-authority invoked by the name of Goedel, he is saying that it is impossible to prove that the Universe is non-deterministic even if it appears that way, and that therefore the Universe is deterministic, or at least, that non-deterministic philosphers are wrong.)\* ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 19, 2003, 11:12pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/197 "2003-04-19T23:12:03Z")
Man, I hate it when the board goes out. Well, back to the trenches… [QUOTE] \*Originally posted by Spiritus Mundi \* \*\*Well, then you should have simply said “yes” when I said: [ul][li]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.[/ul][/li]Instead, you said: [ul][li]I never said there’s no way to measure it. It simply does not require human measurement.[/ul][/li][/quote] \*\* But the answer wouldn’t be ‘yes’. **You** said that we have no way to measure it – this is untrue. And you **were** the one to introduce ‘measurement without a human agent’ to the discussion. Measurement is interaction, but interaction is not measurement. > [@](#): > > **Alternatively, do you see no contradiction in _going all italics at me to declare that this issue has nothing to do with the debate_ and THEN GOING ALL CAPS AT ME TO SCREAM “MEASUREMENT IS INTERATION WITH SOMETHING ELSE”?** Do you see no contradiction in calling a square a rectangle but not calling a rectangle a square? I certainly don’t. ---
**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 20, 2003, 1:56am UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/198 "2003-04-20T01:56:27Z")
I see that more post are going by without your even attempting to answer the substantive questions that have been raised about your “proof”. I see that the real issue remains what allows one system to be called similar to another, but you don’t seem to want to address that issue. I see that you really do seem to want “similar” to be a first-order property of objects (or perhaps just “systems”), which makes about as little sense as anything else that I have seen in this thread. I see that you have lost interest in discussing your contradictory statements about the relationship between life and meta-life. I see that you still prefer analogies to detailed development of an argument. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 20, 2003, 11:17pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/199 "2003-04-20T23:17:03Z")
> [@](#): > > \*Originally posted by erislover \* > \*\*No, you haven’t. 🙂 Was it the penis talk? 😛 \*\* [snort] No, I don’t think that was it. > [@](#): > > **Just that there is no state the universe can’t be in even if GIT limited it, because the statements can still be made. They just can’t be proven. Which, since the universe—not obviously being an intentional agent—doesn’t prove things anyway, is to be expected.** No, no, no. First, it doesn’t take an “intentional agent” to prove something. Either the system of rules _can_ generate the particular statement, or it can’t. Secondly, an event in the universe can be considered as a set of operations acting upon initial conditions. There will be statements that this system cannot reach – there will be configurations the universe cannot reach. Reaching the statement would be equivalent to generating a proof for it. > [@](#): > > \*\*Exactly, which is why I want to know why I should consider the universe as **the only system** when at one time an atom is just an atom, and at another it is used to represent a binary state (say, in molecular logic chips), and at another… \*\* It’s not one thing at one time, and another at another. It’s one thing _and_ another. It doesn’t cease being an atom because it’s part of a computer circuit board, does it? > [@](#): > > **What am I using for judging accuracy?** What does it matter what you use? What matters is whether the systems _are_ similar. Does your choice of weight measures make any difference to an object’s mass? > [@](#): > > \*\*They are _perceived_ as those things. Again the limitations of human sensation forbid self-evident statements about things-in-themselves. More accurately phrased, we talk of phenomena. We assume they have a cause lying elsewhere. It seems you want me to accept the primacy of sensation in attributing ontological states to things which are primary or atomic (not atomic theory “atomic” but the original sense of the word). Which is _really_ primary? \*\* > [@](#): > > \*\*Why isn’t the Matrix real? How do we know the ship is? It is as if you want me to use the same word to describe them only because they are similar. Of course, I am the one who must sense the similarity, which says something _about me_ not strictly something _about the relationship between the Matrix and the ship._ No? \*\* The Matrix _is_ real. It’s just not real in the same way that the horrible blasted world and the flying ship are real. They, in turn, aren’t real in the same way that atoms, et cetera, are real. ---
**Author:** ![TVAA](https://avatars.discourse-cdn.com/v4/letter/t/c57346/32.png) [@TVAA](https://boards.straightdope.com/u/TVAA)\ **Post date:** [April 21, 2003, 2:47pm UTC](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157/200 "2003-04-21T14:47:05Z")
The statements aren’t contradictory, **Spiritus**. You’re simply unwilling (or unable?) to interpret the statements properly. What “allows” systems to be considered similar to each other? What’s this “considered” stuff? Two systems _are_ similar if they share properties; the more the properties are shared, the more similar they are. This isn’t rocket science. Having an “interpretive system” identify two systems as similar takes more doing… but this is irrelevant to the (original) discussion at hand. **FranticMad** : You’ve got it backwards. I’m not citing Godel to show that the universe is determinstic – how in the world did you mangle my arguments to product _that_? Besides, if the world were utterly non-deterministic, it wouldn’t have any properties (it would be the superposition of states, a void that contains everything). [Previous page](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157.md?page=9) [Next page](https://boards.straightdope.com/t/what-does-the-incompleteness-theorem-imply/168157.md?page=11)