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Godel Incompleteness Theorem

 Posts: 114 Add as FriendChallenge to a DebateSend a Message 11/29/2012 12:18:29 AMPosted: 5 years agoIt's additional assertionsG = "G is unprovable". Now we are focusing on "G is unprovable", and what is "G" (because "G" must be a thing)? "G" is proposition, therefore (G = "G is unprovable") is ("Proposition" = "Proposition is unprovable"). ("Proposition" = "Proposition is unprovable") = ("Proposition" = "There is proposition and this proposition is unprovable"). ("Proposition" = "There is proposition and this proposition is unprovable") = ("Proposition" = "unprovable proposition") = (G = "unprovable proposition").OR, ...G = "This (statement) is unprovable". Now we are focusing on "This (statement) is unprovable", and what is "G" (because "G" must be a thing)? "G" is sentence, therefore (Sentence = "This statement is unprovable") is ("Sentence" = "proposition is unprovable"). ("Sentence" = "proposition is unprovable") = ("Sentence" = "There is proposition and this proposition is unprovable"). ("Sentence" = "There is proposition and this proposition is unprovable") = ("Sentence" = "unprovable proposition") = (G = "unprovable proposition").OR, ...(G = "This (statement) is unprovable") = (The following sentence = "This (statement) is unprovable"). Now we are focusing on "This (statement) is unprovable". Which statement? is this "which statement?" pointing to "G" or pointing to "This statement" (without "is unprovable")?- If "which statement?" is pointing to "G", then "which statement?" actually is pointing to "This statement"+"is unprovable"- If "which statement?" is pointing to "This statement" (without "is unprovable"), then actually it's about "This statement" which is unprovable- Meaning, eventually, it has relation with "G is unprovable" which is unprovable "G" (proposition) = "This (statement)"+"is unprovable"Now, where will the system be directed for the test?- The system will try to check "G" or the system will try to check "This (statement) is unprovable"- If the system will try to check "G", then it's the same as checking for "This (statement)"+"is unprovable" or- Meaning whether provable or not, whether it's true or not, but it's directed to "This (statement)"+"is unprovable"An Axiom- Something (without additional assertion) can't transcend beyond something itself (from 1 liter water can't be poured into 1 gallon water).What is G? G is proposition. The proposition has characteristics that will make the "proposition is unprovable". I am not saying that G is an unprovable proposition, but i am asserting that "G" within this "G is unprovable" is considered as unprovable. "G" within "G is unprovable" must be considered as a thing, which is proposition, therefore "G is unprovable" must be related to "unprovable proposition".IOW, before "a proposition (G) could be decided" (whether or not it actually is provable), If LATER "G" is unprovable, then "G" must contain possibility as unprovable. Otherwise "G" transcend beyond G itself. Meaning, "unprovable" is already contained within "G" itself (we just know it later).Therefore "G" as proposition has already "unprovable" or "provable" characteristic, before it (G) could be decided (we just know it later). Meaning, whether from specific or from another different point of views, "G" = "G is unprovable" (eventually) must be considered as "G" = "unprovable proposition", in the sense that "G" has possibility to be unprovable".FURTHER, How to test "This (statement)"+"is unprovable"?- By making test whether "This (statement)" is unprovable? It's the same as checking whether "This (statement)" with possibility to be unproved, is true or false, OR- By making test "This (statement)+is unprovable" is provable or not?, It's the same as checking whether "This (statement)"+with possibility to be unproved, is true or false- Meaning, eventually, we are checking whether unprovable proposition (proposition that has possibility to be unproved) is true or not.An Axiom- Something (without additional assertion) can't transcend beyond something itself (from 1 liter water can't be poured into 1 gallon water).If G is provable or not:- And If G (proposition that has possibility to be unproved") is true ("G" IS PROVABLE BY THE SYSTEM), then ("the possibility to be unproved is actualized to be proved") = INCONSISTENT because it against axiom- And If G (proposition that has possibility to be unproved") is false (THE SYSTEM FAIL TO PROVE "G"), then ("the possibility to be unproved is actualized to be unproved") = CONSISTENT because it doesn't against axiom, but it's incomplete because THE SYSTEM FAIL TO PROVE "G" (there is proposition "G" with possibility to be unproved, which can't be proved by the system)The system is checking proposition:Something is logic because we can trace connection on something. Proposition is logic because there is continuation (connection) within proposition (in between nodes).Proposition has connection or not, within it:- If proposition is provable, then the system can detect whether there is connection or not within proposition- If proposition is unprovable, then the system can't detect whether there is connection or not within propositionCONSEQUENCESThe system to be considered has ability to check whether proposition has connection or not within it. Meaning, the system has ability as far as the system itself to check whether proposition has connection or not within it. The system is checking "the proposition with possibility to be unprovable", whether there is connection or not within proposition, and eventually:- The system fails to check "the proposition with possibility to be unprovable" (is unprovable), in the sense, there is no final decision whether there is connection or not within proposition. BUT, it doesn't mean that the system is incomplete, in the sense that, the system still can detect whether proposition has connection or not within it. The failure (incomplete ability) is not on the system, since the failure is beyond capability of the system to handle it (the system has ability as far as the system itself to check whether proposition has connection or not within it).- The system can check "the proposition with possibility to be unprovable" (is provable). BUT, it doesn't mean that the system is inconsistent, in the sense that the system follows consequences. Since if unprovable proposition is provable, it leads to undecideable condition (whether there is connection or not, within proposition). And if the system is halt, then the system follows the consequences, which is not inconsistent, but actually consistent with the consequences.THE POINTS ARE:From possible understanding (different point of view) based on reality, eventually, following Godel's Incompleteness Theorem will lead to this phrase "unprovable proposition is provable/not, which may be understood on different ways, as:- "a thing that can't be functioned is trying to be functioned, whether it can be functioned or not, OR ..."- "undecideable condition is trying to be directed, whether it can be directed or not (halt)"But both understanding can't be related to Incompleteness & Inconsistency.Eventually it asserts that there is no relation with incompleteness and there is no relation with inconsistent. There is no way to make "Godel's Incompleteness Theorem" to be related to Incompleteness and Inconsistency.The system is COMPLETE because the system is functioning as far as its own ability, and the system is CONSISTENT because the system follows the consequences.I am free not because I have choices, but I am free because I rely on God with quality assured!
 Posts: 2,607 Add as FriendChallenge to a DebateSend a Message 11/29/2012 4:04:31 AMPosted: 5 years agoInteresting. I'll have to look through this."Well, that gives whole new meaning to my assassination. If I was going to die anyway, perhaps I should leave the Bolsheviks' descendants some Christmas cookies instead of breaking their dishes and vodka bottles in their sleep." -Tsar Nicholas II (YYW)
 Posts: 3,749 Add as FriendChallenge to a DebateSend a Message 11/29/2012 7:06:22 AMPosted: 5 years agoYou are confusing G"del's proof with the subject matter of the proof. Your syllogisms simply assert contradictions, one tries to state that unprovable means provable, it doesn't, and the other tries to state that incomplete means complete, nope again. The fact that an axiomatic system will always yield propositions that cannot be decided does not render it complete, it means just the opposite. you are postulating that the axiomatic system incompleteness renders it complete and that when it yields a unprovable result that proves something, it doesn't.Kurt G"del's Incompleteness Theorem is analytically perfect and rigidly deductive; therefore it is conclusive as far as logic is concerned. It states categorically that no axiomatic system is, or can be complete without reference to a higher system in which that system must be embedded. Mathematically, G"del proved that even an axiomatic system as simple as arithmetic cannot be internally consistent and logically complete without reference to a higher system. You can't logically just claim the opposite is true with self contradictory syllogisms."It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
 Posts: 114 Add as FriendChallenge to a DebateSend a Message 11/29/2012 8:17:48 AMPosted: 5 years agoAt 11/29/2012 7:06:22 AM, Sidewalker wrote:You are confusing G"del's proof with the subject matter of the proof. Your syllogisms simply assert contradictions, one tries to state that unprovable means provable, it doesn't, and the other tries to state that incomplete means complete, nope again. The fact that an axiomatic system will always yield propositions that cannot be decided does not render it complete, it means just the opposite. you are postulating that the axiomatic system incompleteness renders it complete and that when it yields a unprovable result that proves something, it doesn't.Kurt G"del's Incompleteness Theorem is analytically perfect and rigidly deductive; therefore it is conclusive as far as logic is concerned. It states categorically that no axiomatic system is, or can be complete without reference to a higher system in which that system must be embedded. Mathematically, G"del proved that even an axiomatic system as simple as arithmetic cannot be internally consistent and logically complete without reference to a higher system. You can't logically just claim the opposite is true with self contradictory syllogisms.Can you help me to point on what part of my assertions contradict each other?I am free not because I have choices, but I am free because I rely on God with quality assured!
 Posts: 114 Add as FriendChallenge to a DebateSend a Message 11/30/2012 12:12:43 PMPosted: 5 years agoAdditional assertionsThe first incompleteness theorem states that no consistent system is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.The second incompleteness theorem, shows that such a system cannot demonstrate its own consistency (wikipedia).I am going to start this discussion using axiom. And the consequences of it, and further we can use it to understand Godel Incompleteness Theorem.An axiom:- {Axiom 1} Something can't transcend (without additional assertions) beyond something itself (from 1 liter water can't be poured into 1 gallon water)- {Axiom 2} Something is within momentum that typical for itself until something else is coming to change momentum of something itself (if something is moving straight forward, then something is always moving straight forward with consistency until there is something else changing the consistency on the movement of something itself)-----Consider there is existence with its functions. Me with my functions, human with the functions, system with its functions, a thing with its own possibilities. And axiomatic system with its own possibilities. To simplify understanding, i am going to use "a thing" or "human" or "axiomatic system" with "system". You may call (replace) it later with machine, or any possible systems.There are several possibilities related to "the system" and its functions:- IN PROPER CONDITION WHERE THE SYSTEM CAN ACTUALIZE ALL OF ITS OWN FUNCTIONS (where proper condition support for the functions of the system to be functioning), THEN ALL OF ITS OWN FUNCTIONS MUST BE ABLE TO BE FUNCTIONED.- The system has ability as far as its own functions.- The system can't transcend its ability beyond its own possibilities (functions).- The system is not functions. But the functions is within "the system", or we may assert "functions of the system".- If we consider that the system has ability to show such function beyond itself, then we must consider that there must be additional assertions (insertions, help) from outside the system (if we consider at first there is water and at last there is the same water but with a fish within it, then a fish must come from outside the water)- At the opposite, if the system has inability to show its function, then we must consider that there must be an obstacle from outside which make one or several functions within the system is not functioning. This obstacle could be come from IMPROPER CONDITION (If the system has ability to show its functions, then it can always be like that, until something else changing it, and it will make the system has inability to show its functions as it should be).- Meaning, for a system (whether its axiomatic system, set of something) has functions as far as its own possibilities, and there must be one or several functions can't be decided whether it can be functioned or not.Now, we are going to see the consequences that has relation to this understanding:- If the system can make one or several functions to be functioned, then it's because one or several functions are part of the system.- If the system can't make one or several functions to be functioned, then it's because there is an obstacle from something else outside this system,- If the system can't make decision (undecideable) whether a function can be functioned or not, then it means that the function is out of reach of the system. It asserts that undecideable function (whether it can be functioned or not) is coming from outside of the system. Otherwise, it must be property of the system and there are only two possibilities as: it must (not) be able to be functioned as far as possibilities of the system.- Meaning, if there is no obstacle from outside the system, then the functions of the system must be able to be functioned properly as far as possibilities of the system itself. But, if, one or several functions of the system can't be functioned and there is no obstacle from outside the system, then functions are out of reach of the system, in the sense that functions are not part of the system (owned by outsider, 2nd system, etc). In this case, we can assert that the system is consistent, in the sense that all functions can be functioned by the system but the system has inability to make functions from outside the system to be functioned (validated). Which means, consistent on its own environment. Now, where is "the incompleteness" in this case? It's there, when we are answering by stating that, "we need second system that has relation to undecideable functions (which can't be decided whether it can be functioned or not, by the first system), to make undecideable functions to be functioned, and it can be done, BUT THEN we can judge that the first system is incomplete (since the first system needs the second system to deal with related undecideable functions)CONCLUSIONS- It means that, if the system can decide whether its functions can be functioned or not, then the system is consistent but incomplete. Or IOW, the system can only make functions to be activated as long as functions are owned by related system. Meaning the system can't transcend beyond capabilities of the system itself to validate functions that are placed (related to the) outside the (first) system (which must be related to the second system), otherwise it against axiom 1.- Or, if the system to be considered as the system that can provide completeness, then there is no consistency for the system, in the sense that the system can't activate functions from all of available system (functions that are placed - within another 2nd system, 3rd, etc).QUICK SUMMARIES:- The system is consistent because it can validate as far as it deals with its own possibilities, but incomplete because the system needs help from another system to solve problem which doesn't covered by the first system.- The system is complete because it can validate as far as it deals with its own possibilities, but inconsistent because the system can't validate another out of reach functions ( this can be understood, if in the beginning, there is hope to force the system to validate all kind of functions, which is impossible, in the sense that all kind of functions are segmented and related to the specific system, in the sense that set of axioms can only validate specific functions).I am free not because I have choices, but I am free because I rely on God with quality assured!
 Posts: 114 Add as FriendChallenge to a DebateSend a Message 11/30/2012 12:13:43 PMPosted: 5 years agoConsider that possible behavior of natural numbers are as wide as all kind of functions. In the sense, that all kind of functions can be represented by all kind of behavior of natural numbers.Further, we consider that there is system which is complete. It's complete because just by using this system (set of axiom), we can validate all kind of functions (all kind of behavior of natural numbers). But it's impossible for the system to validate all kind of behavior of natural numbers, since the system has limitation just to handle several behavior of natural numbers. That, all kind of behavior of natural numbers can only be validated by involving not just one system, but there must be more than one systems to be involved. In the sense that, all kind of behavior of natural numbers are segmented to be specifically related to more than one systems.Meaning, that all kind of behaviour of natural numbers are to broad to be handled (validated) by just one system:- The system can only validate for several of all kind of behaviour of natural numbers that specifically for the system and maintain its CONSISTENCY (since there is relation in between the system and specific of several kind of behaviour of natural numbers). But the system can be judged as INCOMPLETE system, since the system can't cover for all kind of behaviour of natural numbers.- Or the system can be judged as COMPLETE system, only if the system to be considered as the system that can validate on its own territory (on its own relevances, for specific of several kind of behaviour of natural numbers). But the system can be judged as INCONSISTENT, IF, we make comparison in between limited abilities of the system with our hope to put the system having abilities to to covers something outside capabilities of the system itself. IOW, Our hope is proven to be wrong, because our hope is inconsistent with reality.Validate consistency of the system:Since, functions are part of existence (system), but functions are not an existence (system). Therefore validating can be provided as far as to validate functions itself. It's because functions are not existence. Otherwise (functions can validate existence), it against axiom 1 = something (functions) can't transcend (without additional assertions) beyond functions (existence) itself.- If we want to validate the consistency of the system (existence), we can't use functions of the system. To validate the system, we must consider that the system actually is functions of another system. Therefore we can only validate the first system by using the second system to validate the first system, through the second system by activating the functions of the second system to make specific validation which is the first system itself.- Meaning, if we want to validate the first system, we need higher system (the second system) which the first system is the function of the second system.- In a short, we have limitation, including all closed (limited) system. And these limitation can't be directed to validate all possibilities. We need system (help) more than as currently available to validate more than currently hoped (needed). IOW, if we consider there is system that has ability to provide validation, bigger and bigger, again and again, then there must be additional help (systems) to support the first system, to handle larger areas of problems.At this point, we may consider this understanding is valid. But an exceptional must be asserted if we expand this understanding to make several assertions:- Since the system can't provide consistency and completeness, then the problem arise. That, we can't provide such consistent and complete validations. But actually it's an affirmation that is too broad beyond as it should be.- The truth is that inability to provide consistency and completeness, it's because we are involving our hopes which somehow an unapplied (irrelevant) hope. But if we put our hope correctly as it should be, then yes, we can provide consistent and complete validations, in the sense that the system must be directed for the purposes relevantly.Godel Incompleteness Theorem must not be broadly accepted as an assertion to discourage our valid judgement. It's too broad applied improperly. Godel Incompleteness Theorem has no relation to inconsistency and incompleteness in a way that it can discourage our valid judgement. It's just a reminder for us to act relevantly, and once we do it (relevantly) then there is nothing to worry about whether we can provide consistent and complete validation.AGAIN, from wherever we are unpacking Godel Incompleteness Theorem, eventually there is no way for Godel Incompleteness Theorem to invalidate our valid judgement, in the sense that we still have ability to provide consistent and complete validation. By act properly as it should be as far as within our own possibilities or, by putting thing to be functioned as it should be as far as within possibilities of something itself. In a short, intuitively speaking: hope relevantly, act properly, because it's relevant, it's properly as it should be. And therefore, it's possible.I am free not because I have choices, but I am free because I rely on God with quality assured!
 Posts: 1,023 Add as FriendChallenge to a DebateSend a Message 11/30/2012 3:00:05 PMPosted: 5 years agoAt 11/5/2012 12:06:36 AM, Seremonia wrote:I will try to analyze this argument using proposition dependency. But why must dependency of proposition? Because proposition must be associated with existences or it's meaningless, and how an existence related to other existence is through a dependency.Proposition Dependency:- A proposition is constructed to understand realities (existences). Existences can be perceived by us because of their functionality, therefore nodes of a proposition exist as functions.- Anything that exists has functionality. There are two possibilities; dependence upon something else (A->B) or "not" dependence upon something else (A|B).- Therefore, a 'proposition' consists of nodes of functions that form a series of dependencyTerms: Cause = (c) ; Caused = (cd)Liar ParadoxAn example of the use of dependency of proposition can be implemented to analyze this issue, a liar paradox.Liar paradox, "He is telling the truth that He is lying, therefore He is not lying."Syllogism- H then T = (c1) -> (cd1) (If there is him, then, there is telling something)- T then Ac = (cd1) -> (cd2) (If there is telling something, then, there is action from himself)- H then Ac = (c1) -> (cd2) (If there is him, then, there is action from himself - telling the truth)- Ac then Ev = (cd1) -> (cd3) (If there is action from himself, then, there is another event which is never happened as he told - he is lying)- H then Ev = (cd1) -> (cd3) (Therefore, If there is him, then, there is another event which is never happened as he told - he is lying)----- "He is not lying" is not contradict with "He is lying (H then Ev), because "He is not lying" is pointing to (H then Ac).----- H then Ac = (cd1) -> (cd2) is line with H then Ev = (cd1) -> (cd3)Therefore there is no contradiction here.If he is lying, then he is not telling the truth. If H implies both Ac (he is telling the truth) ~Ac (he is not telling the truth) then there's a contradiction according to Aristotle's law of non-contradiction.
 Posts: 114 Add as FriendChallenge to a DebateSend a Message 11/30/2012 9:04:17 PMPosted: 5 years agoHi Enji,Yes you are right, but it happens because we can not get out of semantically problem.Whether we use the same symbols and split it into an opposite one to another ("i am eating and i am not eating, something and not something, i did something and i didn't do something, etc), but it doesn't have to be considered as paradox or contradiction or any kind of ambiguous (in any possible means). It's because terms within an opposite statement has different degree of actualization, where it can't be captured by common reasoning as you already asserted.A proposition was constructed by nodes, words (which must be related to the functions), and a function was constructed by different level of node (which may be another functions). Dependency means, there is continuation from one function to another function. From one function (node) to another function (node) must have priority order.By putting nodes within proposition at the correct priority order, then we can make clear distinction whether there is contradiction or not.In this case, even if H implies both Ac (he is telling the truth) ~Ac (he is not telling the truth), but it's not contradiction, since one to another is located at different placement (at different priority order). And different placement gives us assertion that there is difference in between both. And in this case, H -> Ac (he is telling the truth) IS THE CAUSE for another "he is not telling the truth" (H -> Ev), since both (H-> Ac) & (H -> Ev) are located at different placement and at the same direction "H" (c1)->(cd1)->"Ac" (cd2)-> "Ev" (cd3), which indicates there is no contradiction in between both. By doing this, it helps us much to stay away for being trapped by playing semantically (without being realized by ourselves). And it will make us not to judge easily and this will give us a little bit more patience to see what kind of the relation in between both.Meaning, (he is not telling the truth, he is lying) is part of (he is telling the truth, he is not lying), in the sense, that the truth is "as it is" including "he is lying" as member of the truth. Please, you may refer to this for better understanding on this case http://debate.org...I am free not because I have choices, but I am free because I rely on God with quality assured!
 Posts: 6,071 Add as FriendChallenge to a DebateSend a Message 11/30/2012 10:37:08 PMPosted: 5 years agoThe Fool: it is false, because he is using a logical criteria to make the call of whether or not it is incomplete or complete. Secondly it's also based on first and second order classical,, which is false as a complete system if it is left as IS. Or should I say, as it was left. This reminds me of a time when I was in space" The star that I was closest to spoke to me. It said "those who cannot think for themselves are forever trapped with in the errors of others." As a matter fact, I am pretty sure I was high at the time, but that's the word from the hill, take it for what it's worth.<(8J)"The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL