7 ms·

I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. I will just mention my main two quibbles

I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. I will just mention my main two quibbles to this otherwise excellent post and others like it so that other people who embark on introductory posts to Godel's results don't fall into the same trap. 1. Please don't bring the notion of truth into an introductory explanation (such as in the section "Power of Numbers" or in the short intro to "Hilbert's Program"). It makes it very confusing, especially in light of Godel's other landmark result, i.e Godel's Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as "all true things are provable." Just stick to incompleteness and consistency at a syntactic level instead of appealing to truth, that is respectively the ability to always add new axioms and the lack of syntactic contradictions. 2. The lack of a system S's ability to prove its own consistency is not interesting by itself since we would not trust a proof from a potentially inconsistent system. Hence it is not, in a sense, terribly interesting that we require an infinite tower of increasingly stronger systems to prove the consistency of S and themselves. Rather the interesting direction goes the other way. S cannot prove the consistency of stronger systems, which means we cannot have a "minimal" system we use to explore the consistency of all other systems (again subject to some nuance).

xyzzyz 6y ago
> It makes it very confusing, especially in light of Godel's other landmark result, i.e Godel's Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as "all true things are provable." If people want to understand how to reconcile this very vague statement with Goedel incompleteness theorem that appears to say that some "true" statements are in fact not provable, here's the rough idea: we distinguish between "theories", which live in syntactic world, and "models", which are meant to represent semantics. Theories are like specification of an interface of a programing module, and models are actual implementation of said modules. If one can implement an interface to satisfy all requirements of the specification (that is, all the axioms and theorem that the axioms entail), existence of such implementation shows that the specification is self-consistent: in math, we say that theories that have a model are consistent. What Goedel's completeness theorem says is the converse: consistent theories always have a model; in programming talk, non-contradictory specification can always be implemented. Here's why it means that "all true things are provable": if something is supposed to be true fact about the theory, it has be a property of every model of said theory, that's the lowest bar for any notion of "truth" to make sense. But, one can easily show, using Goedel's completeness theorem, that if something is true in every model of the theory, it has to be provable: indeed, suppose that statement f is true in every model of theory T, but is actually not provable from T. We can then extend theory T by statement "not f", and the resulting theory T' = {T, "not f"} is still consistent, because for it to be inconsistent means exactly that T could prove both "not f" and "not (not f)", and the latter is equivalent f, which we assumed T cannot prove. Then, Goedel's completeness theorem says that this extended theory, since it's consistent, must have a model, M'. But, a model of this extend theory T' is also a model of basic theory T; if you implement larger specification, your implementation also implements any subset of it. But f was supposed to be true in every model of T, and so f is true in M', but also "not f" is true in M', which is a contradiction. Hence, f must be provable from T. Why it doesn't conflict with Goedel's in-completeness theorem? After all, the Goedel's sentence constructed in the proof of Goedel's incompleteness theorem is supposed to express something true but not provable. Well, the idea is that it is not true of the theory of natural numbers, but it's true of the particular model of natural numbers that we have in mind when we think of what natural numbers are, the so called standard model of natural numbers, where all of the numbers are of the form 0 and, using OPs notation (next (next (next ... (next 0) ... )). As it turns out, there exist other models of natural numbers, which have extra, "non-standard" natural numbers which are not of that form, numbers you cannot reach by just applying successor operation finite number of times starting from 0. These models are really weird, but they still must abide all provable rules of the behavior of natural numbers. One can still add them, there must be some "prime" non-standard numbers that aren't products of any smaller numbers, the non-standard numbers must also split into unique product of prime numbers (some of which must necessarily also be non-standard), and so on. This is all crazy, and can make your mind twist, so only think about it at your own peril. To sum up, the notion of "truth" with respect to theories coincides with provability, but not so when you start talking about things true in particular models. This means that we cannot describe all true things about specific models, and in particular about standard natural numbers, using standard logic, but if something does follow from the theory and is true in every of its models, it is in fact provable.
- itismonday1 6y ago
  I am trying to follow, but I feel I am hanging Goedel incompleteness theorem -> apply this to "theories" Goedel completeness theorem -> apply this to a specific "model", of the theory So there can be "something true but not provable" in the theory, and "we cannot describe all true things about specific models", but ....what?
  - dwohnitmok 6y ago
    Another example of why I don't like the word "truth" because I think it does more to muddy the situation than clarify it. Let's drop "truth" for a sec. Let's say I come up with a series of axioms about cows. They describe spots on cows, how many legs cows have, etc. Now I come up with a sentence "all cows have spots." There are two ways of proving this sentence. One is a "semantic proof." That is I go and round up every cow and examine it and make sure they all have spots. Another way of proving this sentence is a "syntactic proof." I list out a series of axioms defining what cows are. Then I do a series of string manipulations on my axioms about cows according to an allowed ruleset to arrive at the statement "all cows have spots." Semantic proofs are hard. They're annoying for finite groups of things and impossible to do for infinite groups of things. If possible we would like to use syntactic proofs. But syntactic proofs face two hurdles. The first is how to know we've applied our axioms correctly. For example, maybe one of my axiom is "all cows have 4 legs," and then I just conveniently define the symbol "4" to mean three. You can't really get around that problem. You kind of just have to tell people, don't mess around with the meaning of words. The second hurdle though concerns our ruleset. That is string manipulations can be of an arbitrary form. How do I know, even if I've applied my axioms correctly, that my ruleset isn't crazy? Something like, anytime you see a sentence S, you're allowed to replace it with Not(S) if you want. Godel's completeness theorem says for a lot of reasonable rulesets (including most of the ones we use for mathematics), semantic proofs and syntactic proofs coincide. You can decide to either go out and look at every cow or you can use your axioms about cows to move symbols on a sheet of paper around. As long as your axioms about cows capture enough information about whether cows have spots, with certain popular rulesets, you'll get the same result. Put another way, as long as the result you're looking for is relevant to the axioms you've listed, either a semantic proof or a syntactic proof is valid. Okay, but what about statements that aren't relevant to the axioms I've listed? Things like the weight of a cow? Or what about statements that are under-specified? For example if I just say as an axiom "some cows have spots," does that mean all cows have spots or that some don't? Through the lens of Godel's completeness theorem this just means that our list of axioms is applicable to a lot more situations than we're giving it credit for. Our axioms don't talk about the weight of a cow? Well then they could be used for situations where cows are weightless or when they have weight. Our axioms don't specify whether all cows have spots? Well they can be used in situations where all cows have spots as well as situations where some cows don't have spots. No biggie. That just means the only things relevant to our axioms are those properties in common across all the situations we could apply our axioms. And anything that can be semantically proved in all of these situations is also something that can be syntactically proved. Okay so that concludes Godel's completeness theorem. Let's start going into his incompleteness theorems through the same lens. Cows are really complicated things. They have tons and tons of different properties, so it's no surprise that our list of axioms will probably always under-specify them. What if we strip things down to really really simple things. For example the natural numbers? Can we have a list of axioms that apply to the natural numbers and pin down every property about them? The answer is no. Any list of axioms we come up with will always be "too broad." There are things other than the usual natural numbers (that's already a bit of a minefield to privilege one of them as being the "usual", but we'll ignore it for now) that I can present to you that also fulfill those axioms. In the same way that "some cows have spots" is mum on the issue of whether a certain cow doesn't have spots, and so can be used to describe either a herd of cows that all have spots or a herd of cows split half-and-half with spots and without spots, any system of logic that simultaneously fulfills both Godel's completeness theorem and incompleteness theorem means that it can be applied to more than one unique situation. Moreover those things that the system cannot syntactically prove are exactly those things that have different semantic proofs in different situations. For example, because I can't have a series of string manipulation steps that result in the sentence "all cows have spots" (Godel's incompleteness thereoms) there's gotta be some situations where all my axioms apply and all cows have spots and other situations where at least one cow doesn't have a spot (Godel's completeness theorem). Hence through this lens Godel's incompleteness theorems aren't really about capital-T Truth so much as they are about our ability to completely specify infinite objects. If you and a friend decide to start listing out axioms to try to describe something like the natural numbers, no matter how many axioms you list out there is always something left unspecified.
itismonday1 6y ago
"I will just mention my main two quibbles" -- Poking through your history, why do you quibble on the word truth so often? Do we not agree there are statements that are true? Do we not agree there are provable true statements ? If there are unprovable statements in any consistent set of axioms, might we also conclude there also be an unprovable but true statements?
- dwohnitmok 6y ago
  It's because it's a huge can of worms that leads to really grandiose claims that aren't supported by Godel's statements, of the sort the article is already beginning to make. It also shifts the conversation into becoming fundamentally a philosophical question rather than a logical or mathematical one, which is okay, but considerably changes the table stakes of what background knowledge we need. > If there are unprovable statements in any consistent set of axioms, might we also conclude there also be an unprovable but true statements? That is one potential philosophical position. Another philosophical position is that there are no "true" axioms schemes, but rather that axiom schemes are always indefinitely extendable. Truth lies instead in how we choose to apply the axiom schema to the external world, which is outside the purview of Godel's Incompleteness Theorem. Both positions are supported by Godel's Incompleteness Theorem. For example, do you believe that the Continuum Hypothesis (CH) is true? That is, although it is independent of ZFC, do you believe that objectively we are only allowed to use CH as an axiom or Not(CH) as an axiom. One of those must be objectively wrong. Some people do some people don't. Likewise, do you believe that the imaginary numbers are "true?" That is the axioms that make them up are not artificial statements that somebody can up in pursuit of an abstract theory, but are immutable truths of the universe and that in fact you cannot choose an alternative axiomatization because it would be "wrong?" Some people some people don't. What about the natural numbers? There are many philosophical positions you can take on the notion of mathematical truth and the crucial thing is describing Godel's theorems with reference to truth suggests that Godel's theorems help decide among these notions of mathematical truth, when in fact what happens is that Godel's theorems stretch and shrink to accommodate different philosophical frameworks. The former is why I think a lot of people accord Godel's theorems a mystical status that makes it very difficult to have coherent discussions about them. To pedagogically illustrate why the notion of truth can be confusing, I'll repeat two questions I've listed elsewhere which I think are good things to meditate on for first-time readers of Godel's Incompleteness theorems and explore why the often grandiose philosophical proclamations arising from Godel's theorems require very careful footing. I answer them in my post history, but I would recommend you don't look at those before you've come up with answers yourself. ------Repost----- First, Conway's Game of Life: Conway's Game of Life seem like they should be subject to Godel's incompleteness theorems. It is after all powerful enough to be Turing-complete. Yet its rules seem clearly complete (they unambiguously specify how to implement the Game of Life enough that different implementation of the Game of Life agree with each other). So what part of Game of Life is incomplete? What new rule (i.e. axiom) can you add to Conway's Game of Life that is independent of its current rules? Given that, what does it mean when I say that "its rules seem clearly complete?" Is there a way of capturing that notion? And if there isn't, why haven't different implementations of the game diverged? If you don't think that the Game of Life should be subject to Godel's Incompleteness Theorems why? Given that it's Turing complete it seems obviously as powerful as any other system. Second, again, in most logical systems, another way of stating that consistency is unproveable is that consistency of a system S is independent of the axioms of that system. However, that means that the addition of a new axiom asserting that S is either consistent or inconsistent are both consistent with S. In particular, the new system S' that consists of the axioms of S with the new axiom "S is inconsistent" is consistent if S is consistent. What gives? Do we have some weird "superposition" of consistency and inconsistency? Hints (don't read them until you've given these questions some thought!): 1. Consider questions of the form "eventually" or "never." Can those be turned into axioms? If you decide instead to tackle the question of applicability of the incompleteness theorems, what is the domain of discourse when I say "clearly complete?" What exactly is under consideration? 2. Consider carefully what Godel's arithmetization of proofs gives you. What does Godel's scheme actually give you when it says it's "found" a contradiction? Does this comport with what you would normally agree with? An equivalent way of phrasing this hint, is what is the actual statement in Godel's arithmetization scheme created when we informally say "S is inconsistent?" At the end of the day, the philosophical implications of Godel's incompleteness theorems hinge on whether you believe that it is possible to unambiguously specify what the entirety of the natural numbers are and whether they exist as a separate entity (i.e. does "infinity" exist in a real sense? Is there a truly absolute standard model of the natural numbers?).
dang 6y ago
> I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. Sorry for the confusion. We detach comments that are replies to the top comment but don't respond to anything specific that that comment said. This is standard moderation. Sometimes we post that we did this (https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=by%3Adang%20detached&sort=byDate&type=comment https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...), but not always. We do this because users often post replies to the top comment on a page that aren't really replies. Sometimes they do that because they want to get their comment higher on the page (so-called topjacking). Of course that's not always the case. Both of your posts are fine (except for the fact that they duplicate one another) but they are replies to the article, not to anything that the top comment (https://news.ycombinator.com/item?id=25116881 https://news.ycombinator.com/item?id=25116881) specifically says. Indeed your sentence "I will just mention my main two quibbles to this otherwise excellent post and others like it so that other people who embark on introductory posts to Godel's results don't fall into the same trap." makes it quite clear that your reply is to the article, not xyzzyz's comment. That's great! But the appropriate level for such a comment is the top level of the thread. I've therefore done the same again and detached this subthread from https://news.ycombinator.com/item?id=25116881 https://news.ycombinator.com/item?id=25116881.
- dwohnitmok 6y ago
  Ah fair enough. I meant it to be a reply to temper xyzzyz's praise of the article with what I perceived to be minor flaws in it, but such is as it is.
rntksi 6y ago
Thank you, reading this comment and some of yours further down, thinking about the topics you set out, was very interesting. To offer possibly a comment to yours about the notion of "truth", in the article posted, when I read "true" (e.g. in the 1st paragraph of chapter Hilbert's Program) and next to it "false", I immediately made reference in my mind to the true/false duality of programmers. I think that helped in understanding the article, but coupled with a footnote with your details would certainly help the reader think further on the topic. cheers :-)
- dwohnitmok 6y ago
  Well I'm glad it helped!
ProfHewitt 6y ago
Dear dwohnitmok, Actually, powerful strongly-typed theories used in Computer Science can prove their own consistency! See the following article for for how this is done: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 As you pointed out, such a proof does not mean that it is not operationally possible to derive a contradiction using axioms and rules of inference. Instead our confidence in the consistency of a powerful strongly-typed theory is because there is just one model of the axioms of the theory up to a unique isomorphism, which formalizes the concept of "truth" that you mentioned in your post. For example, see the following article: https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802
- unknown_apostle 6y ago
  I looked up your papers... Then I noticed you have also worked on paraconsistent logic. So to just boldly go on a completely new tangent and ask a question I have long sought to ask: do you know a good introduction text for paraconsistent logic, for a graduate level computer science student?
  - ProfHewitt 6y ago
    Hello Unknown_apostle, Thanks for looking at the articles! The most up to date analysis of paraconsistent logic starts on page 11 of the following article: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021
ProfHewitt 6y ago
Model Theory formalizes the notion of "truth" in foundational mathematics of computer science. The axioms of powerful foundational theories of computer science have just one model up to a unique isomorphism. See https://papers.ssrn.com/abstract=3418003 https://papers.ssrn.com/abstract=3418003 and https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 The [Gödel 1930] "completeness theorem" is *false* for the foundational theories of Computer Science because these theories are inferentially incomplete, that is, not every proposition can be proved or disproved.

xyzzyz 6y ago

> It makes it very confusing, especially in light of Godel's other landmark result, i.e Godel's Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as "all true things are provable." If people want to understand how to reconcile this very vague statement with Goedel incompleteness theorem that appears to say that some "true" statements are in fact not provable, here's the rough idea: we distinguish between "theories", which live in syntactic world, and "models", which are meant to represent semantics. Theories are like specification of an interface of a programing module, and models are actual implementation of said modules. If one can implement an interface to satisfy all requirements of the specification (that is, all the axioms and theorem that the axioms entail), existence of such implementation shows that the specification is self-consistent: in math, we say that theories that have a model are consistent. What Goedel's completeness theorem says is the converse: consistent theories always have a model; in programming talk, non-contradictory specification can always be implemented. Here's why it means that "all true things are provable": if something is supposed to be true fact about the theory, it has be a property of every model of said theory, that's the lowest bar for any notion of "truth" to make sense. But, one can easily show, using Goedel's completeness theorem, that if something is true in every model of the theory, it has to be provable: indeed, suppose that statement f is true in every model of theory T, but is actually not provable from T. We can then extend theory T by statement "not f", and the resulting theory T' = {T, "not f"} is still consistent, because for it to be inconsistent means exactly that T could prove both "not f" and "not (not f)", and the latter is equivalent f, which we assumed T cannot prove. Then, Goedel's completeness theorem says that this extended theory, since it's consistent, must have a model, M'. But, a model of this extend theory T' is also a model of basic theory T; if you implement larger specification, your implementation also implements any subset of it. But f was supposed to be true in every model of T, and so f is true in M', but also "not f" is true in M', which is a contradiction. Hence, f must be provable from T. Why it doesn't conflict with Goedel's in-completeness theorem? After all, the Goedel's sentence constructed in the proof of Goedel's incompleteness theorem is supposed to express something true but not provable. Well, the idea is that it is not true of the theory of natural numbers, but it's true of the particular model of natural numbers that we have in mind when we think of what natural numbers are, the so called standard model of natural numbers, where all of the numbers are of the form 0 and, using OPs notation (next (next (next ... (next 0) ... )). As it turns out, there exist other models of natural numbers, which have extra, "non-standard" natural numbers which are not of that form, numbers you cannot reach by just applying successor operation finite number of times starting from 0. These models are really weird, but they still must abide all provable rules of the behavior of natural numbers. One can still add them, there must be some "prime" non-standard numbers that aren't products of any smaller numbers, the non-standard numbers must also split into unique product of prime numbers (some of which must necessarily also be non-standard), and so on. This is all crazy, and can make your mind twist, so only think about it at your own peril. To sum up, the notion of "truth" with respect to theories coincides with provability, but not so when you start talking about things true in particular models. This means that we cannot describe all true things about specific models, and in particular about standard natural numbers, using standard logic, but if something does follow from the theory and is true in every of its models, it is in fact provable.

itismonday1 6y ago
I am trying to follow, but I feel I am hanging Goedel incompleteness theorem -> apply this to "theories" Goedel completeness theorem -> apply this to a specific "model", of the theory So there can be "something true but not provable" in the theory, and "we cannot describe all true things about specific models", but ....what?
- dwohnitmok 6y ago
  Another example of why I don't like the word "truth" because I think it does more to muddy the situation than clarify it. Let's drop "truth" for a sec. Let's say I come up with a series of axioms about cows. They describe spots on cows, how many legs cows have, etc. Now I come up with a sentence "all cows have spots." There are two ways of proving this sentence. One is a "semantic proof." That is I go and round up every cow and examine it and make sure they all have spots. Another way of proving this sentence is a "syntactic proof." I list out a series of axioms defining what cows are. Then I do a series of string manipulations on my axioms about cows according to an allowed ruleset to arrive at the statement "all cows have spots." Semantic proofs are hard. They're annoying for finite groups of things and impossible to do for infinite groups of things. If possible we would like to use syntactic proofs. But syntactic proofs face two hurdles. The first is how to know we've applied our axioms correctly. For example, maybe one of my axiom is "all cows have 4 legs," and then I just conveniently define the symbol "4" to mean three. You can't really get around that problem. You kind of just have to tell people, don't mess around with the meaning of words. The second hurdle though concerns our ruleset. That is string manipulations can be of an arbitrary form. How do I know, even if I've applied my axioms correctly, that my ruleset isn't crazy? Something like, anytime you see a sentence S, you're allowed to replace it with Not(S) if you want. Godel's completeness theorem says for a lot of reasonable rulesets (including most of the ones we use for mathematics), semantic proofs and syntactic proofs coincide. You can decide to either go out and look at every cow or you can use your axioms about cows to move symbols on a sheet of paper around. As long as your axioms about cows capture enough information about whether cows have spots, with certain popular rulesets, you'll get the same result. Put another way, as long as the result you're looking for is relevant to the axioms you've listed, either a semantic proof or a syntactic proof is valid. Okay, but what about statements that aren't relevant to the axioms I've listed? Things like the weight of a cow? Or what about statements that are under-specified? For example if I just say as an axiom "some cows have spots," does that mean all cows have spots or that some don't? Through the lens of Godel's completeness theorem this just means that our list of axioms is applicable to a lot more situations than we're giving it credit for. Our axioms don't talk about the weight of a cow? Well then they could be used for situations where cows are weightless or when they have weight. Our axioms don't specify whether all cows have spots? Well they can be used in situations where all cows have spots as well as situations where some cows don't have spots. No biggie. That just means the only things relevant to our axioms are those properties in common across all the situations we could apply our axioms. And anything that can be semantically proved in all of these situations is also something that can be syntactically proved. Okay so that concludes Godel's completeness theorem. Let's start going into his incompleteness theorems through the same lens. Cows are really complicated things. They have tons and tons of different properties, so it's no surprise that our list of axioms will probably always under-specify them. What if we strip things down to really really simple things. For example the natural numbers? Can we have a list of axioms that apply to the natural numbers and pin down every property about them? The answer is no. Any list of axioms we come up with will always be "too broad." There are things other than the usual natural numbers (that's already a bit of a minefield to privilege one of them as being the "usual", but we'll ignore it for now) that I can present to you that also fulfill those axioms. In the same way that "some cows have spots" is mum on the issue of whether a certain cow doesn't have spots, and so can be used to describe either a herd of cows that all have spots or a herd of cows split half-and-half with spots and without spots, any system of logic that simultaneously fulfills both Godel's completeness theorem and incompleteness theorem means that it can be applied to more than one unique situation. Moreover those things that the system cannot syntactically prove are exactly those things that have different semantic proofs in different situations. For example, because I can't have a series of string manipulation steps that result in the sentence "all cows have spots" (Godel's incompleteness thereoms) there's gotta be some situations where all my axioms apply and all cows have spots and other situations where at least one cow doesn't have a spot (Godel's completeness theorem). Hence through this lens Godel's incompleteness theorems aren't really about capital-T Truth so much as they are about our ability to completely specify infinite objects. If you and a friend decide to start listing out axioms to try to describe something like the natural numbers, no matter how many axioms you list out there is always something left unspecified.

itismonday1 6y ago

"I will just mention my main two quibbles" -- Poking through your history, why do you quibble on the word truth so often? Do we not agree there are statements that are true? Do we not agree there are provable true statements ? If there are unprovable statements in any consistent set of axioms, might we also conclude there also be an unprovable but true statements?

dwohnitmok 6y ago
It's because it's a huge can of worms that leads to really grandiose claims that aren't supported by Godel's statements, of the sort the article is already beginning to make. It also shifts the conversation into becoming fundamentally a philosophical question rather than a logical or mathematical one, which is okay, but considerably changes the table stakes of what background knowledge we need. > If there are unprovable statements in any consistent set of axioms, might we also conclude there also be an unprovable but true statements? That is one potential philosophical position. Another philosophical position is that there are no "true" axioms schemes, but rather that axiom schemes are always indefinitely extendable. Truth lies instead in how we choose to apply the axiom schema to the external world, which is outside the purview of Godel's Incompleteness Theorem. Both positions are supported by Godel's Incompleteness Theorem. For example, do you believe that the Continuum Hypothesis (CH) is true? That is, although it is independent of ZFC, do you believe that objectively we are only allowed to use CH as an axiom or Not(CH) as an axiom. One of those must be objectively wrong. Some people do some people don't. Likewise, do you believe that the imaginary numbers are "true?" That is the axioms that make them up are not artificial statements that somebody can up in pursuit of an abstract theory, but are immutable truths of the universe and that in fact you cannot choose an alternative axiomatization because it would be "wrong?" Some people some people don't. What about the natural numbers? There are many philosophical positions you can take on the notion of mathematical truth and the crucial thing is describing Godel's theorems with reference to truth suggests that Godel's theorems help decide among these notions of mathematical truth, when in fact what happens is that Godel's theorems stretch and shrink to accommodate different philosophical frameworks. The former is why I think a lot of people accord Godel's theorems a mystical status that makes it very difficult to have coherent discussions about them. To pedagogically illustrate why the notion of truth can be confusing, I'll repeat two questions I've listed elsewhere which I think are good things to meditate on for first-time readers of Godel's Incompleteness theorems and explore why the often grandiose philosophical proclamations arising from Godel's theorems require very careful footing. I answer them in my post history, but I would recommend you don't look at those before you've come up with answers yourself. ------Repost----- First, Conway's Game of Life: Conway's Game of Life seem like they should be subject to Godel's incompleteness theorems. It is after all powerful enough to be Turing-complete. Yet its rules seem clearly complete (they unambiguously specify how to implement the Game of Life enough that different implementation of the Game of Life agree with each other). So what part of Game of Life is incomplete? What new rule (i.e. axiom) can you add to Conway's Game of Life that is independent of its current rules? Given that, what does it mean when I say that "its rules seem clearly complete?" Is there a way of capturing that notion? And if there isn't, why haven't different implementations of the game diverged? If you don't think that the Game of Life should be subject to Godel's Incompleteness Theorems why? Given that it's Turing complete it seems obviously as powerful as any other system. Second, again, in most logical systems, another way of stating that consistency is unproveable is that consistency of a system S is independent of the axioms of that system. However, that means that the addition of a new axiom asserting that S is either consistent or inconsistent are both consistent with S. In particular, the new system S' that consists of the axioms of S with the new axiom "S is inconsistent" is consistent if S is consistent. What gives? Do we have some weird "superposition" of consistency and inconsistency? Hints (don't read them until you've given these questions some thought!): 1. Consider questions of the form "eventually" or "never." Can those be turned into axioms? If you decide instead to tackle the question of applicability of the incompleteness theorems, what is the domain of discourse when I say "clearly complete?" What exactly is under consideration? 2. Consider carefully what Godel's arithmetization of proofs gives you. What does Godel's scheme actually give you when it says it's "found" a contradiction? Does this comport with what you would normally agree with? An equivalent way of phrasing this hint, is what is the actual statement in Godel's arithmetization scheme created when we informally say "S is inconsistent?" At the end of the day, the philosophical implications of Godel's incompleteness theorems hinge on whether you believe that it is possible to unambiguously specify what the entirety of the natural numbers are and whether they exist as a separate entity (i.e. does "infinity" exist in a real sense? Is there a truly absolute standard model of the natural numbers?).

dang 6y ago

> I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. Sorry for the confusion. We detach comments that are replies to the top comment but don't respond to anything specific that that comment said. This is standard moderation. Sometimes we post that we did this (https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=by%3Adang%20detached&sort=byDate&type=comment https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...), but not always. We do this because users often post replies to the top comment on a page that aren't really replies. Sometimes they do that because they want to get their comment higher on the page (so-called topjacking). Of course that's not always the case. Both of your posts are fine (except for the fact that they duplicate one another) but they are replies to the article, not to anything that the top comment (https://news.ycombinator.com/item?id=25116881 https://news.ycombinator.com/item?id=25116881) specifically says. Indeed your sentence "I will just mention my main two quibbles to this otherwise excellent post and others like it so that other people who embark on introductory posts to Godel's results don't fall into the same trap." makes it quite clear that your reply is to the article, not xyzzyz's comment. That's great! But the appropriate level for such a comment is the top level of the thread. I've therefore done the same again and detached this subthread from https://news.ycombinator.com/item?id=25116881 https://news.ycombinator.com/item?id=25116881.

dwohnitmok 6y ago
Ah fair enough. I meant it to be a reply to temper xyzzyz's praise of the article with what I perceived to be minor flaws in it, but such is as it is.

rntksi 6y ago

Thank you, reading this comment and some of yours further down, thinking about the topics you set out, was very interesting. To offer possibly a comment to yours about the notion of "truth", in the article posted, when I read "true" (e.g. in the 1st paragraph of chapter Hilbert's Program) and next to it "false", I immediately made reference in my mind to the true/false duality of programmers. I think that helped in understanding the article, but coupled with a footnote with your details would certainly help the reader think further on the topic. cheers :-)

dwohnitmok 6y ago
Well I'm glad it helped!

ProfHewitt 6y ago

Dear dwohnitmok, Actually, powerful strongly-typed theories used in Computer Science can prove their own consistency! See the following article for for how this is done: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 As you pointed out, such a proof does not mean that it is not operationally possible to derive a contradiction using axioms and rules of inference. Instead our confidence in the consistency of a powerful strongly-typed theory is because there is just one model of the axioms of the theory up to a unique isomorphism, which formalizes the concept of "truth" that you mentioned in your post. For example, see the following article: https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802

unknown_apostle 6y ago
I looked up your papers... Then I noticed you have also worked on paraconsistent logic. So to just boldly go on a completely new tangent and ask a question I have long sought to ask: do you know a good introduction text for paraconsistent logic, for a graduate level computer science student?
- ProfHewitt 6y ago
  Hello Unknown_apostle, Thanks for looking at the articles! The most up to date analysis of paraconsistent logic starts on page 11 of the following article: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021

ProfHewitt 6y ago

Model Theory formalizes the notion of "truth" in foundational mathematics of computer science. The axioms of powerful foundational theories of computer science have just one model up to a unique isomorphism. See https://papers.ssrn.com/abstract=3418003 https://papers.ssrn.com/abstract=3418003 and https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 The [Gödel 1930] "completeness theorem" is *false* for the foundational theories of Computer Science because these theories are inferentially incomplete, that is, not every proposition can be proved or disproved.