• TheMadFool
    13.8k
    I'm not a mathematician and nor am I logician and so cum grano salis.

    From what I know (not that much I'm afraid), Gödel's Incompleteness Theorems, one of them I suppose, is predicated on a statement called the Gödel sentence.

    Gödel sentence (G): This statement (G) is unprovable.

    The argument, as I understand it, proceeds thus:

    1. Either G is provable or G is unprovable.

    2. If G is provable then G is true i.e. G is unprovable.

    3. If G is unprovable then G is true i.e. G is unprovable.

    4. G is unprovable or G is unprovable (1, 2, 3 CD)

    5. G is unprovable (4 Taut)

    QED

    However,

    The Gödel sentence is a spin-off of the liar sentence (This sentence is false). The assumption that we make with the liar sentence is that it's a proposition and therefore that it has a truth value. Reject that assumption and no contradiction results as there are no truth values that come into opposition.

    Since, the Gödel sentence is the liar sentence in some sense can't we do the same thing we did to the liar sentence: take away its status as a proposition?

    Note here that, drawing from logical nihilism (@Banno), we can reinterpret what we've done here as merely expanding the possibile "states" a proposition can assume. The new "state" we've introduced in dealing with the liar sentence vide infra is a third truth value which is neither true nor false.

    This third "state" for G (the Gödel sentence) would be neither provable nor unprovable i.e. we've slipped between the horns of the dilemma by coming up with a third alternative (trivalent logic)

    I guess I'm accusing Gödel of committing the fallacy of false dichotomy.

    A penny for your thoughts...
  • theRiddler
    260
    I just think everything that exists is true. Lies are truly lies; truths are truly truths. Objectively speaking, it's all just really happening. To take a step further back, you can just objectify the sentence. Whether it deserves more credence is a matter of opinion. It just kind of begs the question of thinking itself.
  • Pantagruel
    3.4k
    Godel's theorems apply to systems that fulfil specific criteria of translatability into facts about natural numbers, as I understand. Without specifying the system - i.e. looking only at the sentence itself - the sentence would not have the same force. So Godel's sentence is true just in case it is true - i.e. it applies to the specific set of systems in which it is true.

    edit: Alternatively, you could consider the sentence itself to be an axiom. E.g. "This is true" has to be taken axiomatically. Likewise, "This is false". If "This is false" is axiomatic, then, axiomatically, it must be pointing to something other than itself, otherwise it is nonsense (unsinn). To claim a different version of the statement "This axiom is false" is true is to deny its status as an axiom. Likewise for provable/unprovable.
  • TheMadFool
    13.8k
    looking only at the sentence itself - the sentence would not have the same force.Pantagruel

    I've focused on those aspects of Gödel's argument that are general:

    1. A proposition is a sentence that has a truth value.

    2. The principle of bivalence (statements can have only two truth values viz. true and false)
  • TheMadFool
    13.8k
    It just kind of begs the question of thinking itself.theRiddler

    What means this?
  • theRiddler
    260


    I mean isn't it really more so the ego's attachment to it's own thoughts. Thinking has always just been useful, but it's not a miracle worker. We work at this problem like we're going to find some underlying wisdom, but what if there is none? What if it's just the illusion of sense created by consonants and vowels? What's the difference between a paradox and gobbledygook?
  • TheMadFool
    13.8k
    I mean isn't it really more so the ego's attachment to it's own thoughts. Thinking has always just been useful, but it's not a miracle worker. We work at this problem like we're going to find some underlying wisdom, but what if there is none? What if it's just the illusion of sense created by consonants and vowels? What's the difference between a paradox and gobbledygook?theRiddler

    Sorry, but how does prove that, and I quote, "It just kind of begs the question of thinking itself"?
  • Hermeticus
    181
    This statement (G) is unprovable.TheMadFool
    This defines the quality of G. "This statement" could be any statement - but if it isn't unprovable, it's not G because we defined it as being unprovable.


    1. Either G is provable or G is unprovable.TheMadFool
    This is always true

    2. If G is provable then G is true i.e. G is unprovable.TheMadFool
    This is never true. We defined G as unprovable. There may be a statement that looks exactly the same like G; but it's not G because G is per definition unprovable.

    3. If G is unprovable then G is true i.e. G is unprovable.TheMadFool
    This is always true because G is defined as being unprovable.

    5. G is unprovable (4 Taut)TheMadFool
    Yes, we said so in the beginning!


    The fallacy here is that
    2. If G is provable then G is true i.e. G is unprovable.TheMadFool
    this assumes something that is impossible. It's an invalid argument. An error in definition.

    Likewise,
    the liar paradox boils down to exactly this.
    "I am lying" = "G is provable"

    I know that you like yourself some paradoxes - but are paradoxes not just pure fiction? It seems like they exclusively stem either from a lack of knowledge or from assuming statements that are impossible to begin with.
  • TheMadFool
    13.8k
    This defines the quality of G. "This statement" could be any statement - but if it isn't unprovable, it's not G because we defined it as being unprovable.Hermeticus

    It is, I fear, not what you think it is.

    This is always trueHermeticus

    Nothing that wasn't obvious.

    This is never true. We defined G as unprovable. There may be a statement that looks exactly the same like G; but it's not G because G is per definition unprovable.Hermeticus

    Okey dokey.

    This is always true because G is defined as being unprovable.Hermeticus

    Yep.

    Yes, we said so in the beginning!Hermeticus

    It's not a statement that we made so much as it is a statement we discovered.

    this assumes something that is impossible. It's an invalid argument. An error in definition.Hermeticus

    What exactly have we assumed that's impossible?
  • TheMadFool
    13.8k
    Update

    The Liar Paradox

    The liar sentence: This1 sentence is false.

    The negation of the liar sentence: This2 sentence is true.

    For there to be a paradox, This1 = This2 but they're, I discovered, not.

    This1 refers to :point: This1 sentence is false ( L ).

    This2 refers to :point: This2 sentence is true (~L).

    (Notice the difference? False (This1) and True (This2)

    Imagine the following:

    One person saying "I am black and I am not black" (contradiction).

    and

    Two people saying "I am black" and "I am not black" (no contradiction).

    ---

    Gödel's proof

    Gödel sentence = G = This3 sentence (G) is unprovable = G is unprovable.

    The negation of the Gödel sentence = ~G = This4 sentence is unprovable = G is provable.

    Gödel's argument requires that This3 = This4, he needs a contradiction.

    However,

    This3 refers to :point: This3 sentence is unprovable (G)

    This4 refers to :point: This4 sentence is provable (~G)

    Note here too a difference: Unprovable (This3) and Provable (This4).
  • tim wood
    9k
    Since, the Gödel sentence is the liar sentence in some sense can't we do the same thing we did to the liar sentence: take away its status as a proposition?TheMadFool

    Nope, because the English sentence is arguably not a proposition in the sense desired, but Godel's G absolutely is, in that it makes a definite, well-defined statement about a definite, well-defined subject (namely itself), in definite, well-defined terms.
  • TheMadFool
    13.8k
    Nope, because the English sentence is arguably not a proposition in the sense desired, but Godel's G absolutely is, in that it makes a definite, well-defined statement about a definite, well-defined subject (namely itself), in definite, well-defined terms.tim wood

    The liar sentence's problem is precisely because of self-reference (namely itself) which the Gödel sentence also repeats/reenacts.
  • TonesInDeepFreeze
    2.9k
    The Godel sentence is a formula in the language of formal arithmetic. It an exact formula using only the symbols of symbolic logic and the symbols of arithmetic. It is an exact formal sentence about natural numbers. It is a very complicated formal sentence about natural numbers, but still it is a sentence about natural numbers. However, Godel discovered that certain formulas of arithmetic also are interpretable as sentences about formal theories. So the Godel sentence also happens to be interpretable as being true if and only if the Godel sentence is not provable in the formal theory in question. There is nothing suspect whatsoever about the rigor of the Godel sentence as a mathematical formula; not is it suspect whatsoever that the Godel sentence is true if and only if it is not provable in the theory in question.

    Moreover, Godel's proof is purely about syntax and the proof itself does not rely on a notion of truth. That is the important way the Godel sentence is different from the liar statement. Whatever is problematic about the liar statement does not apply to the Godel sentence since the Godel sentence does not mention truth conditions nor is the Godel sentence self referential in the particulary problematic way of the liar statement. (It is only in an addition to the basic proof about syntax that we go on to note the the Godel sentence does happen to be true. And the conclusion from that is not "The Godel sentence is not provable in the system". Rather the conclusion is "If the system is consistent then the Godel sentence is not provable in the system".)

    And Godel's proof can be formulated in primitive recursive arithmetic (purely finitistic combinatoric arithmetic) and with only intuitionistic logic. That would seem to be the minimum platform for mathematical reasoning. If there is any objection to Godel's proof then it would entail objection to even bare finitistic reasoning about natural numbers.

    And fundamentally wrongheaded is the argument that Godel uses "false dichotomy" in the manner described in the first post. The proof is in intuitionistic logic, which does not provide "P or not-P" as a basis.

    It is correct though that if one rejects the law of non-contradictions, such as certain systems of paraconsistent logic, then Godel's argument does not hold. Yes, one way out of Godel's theorem is to instead propose that mathematics be formulated paraconsistently. But that still does not impugn Godel's proof as much as it merely says there are alternative contexts in which the argument does not hold.

    And beyond those points, the preponderance of posts above in this thread are an abysmally mangled misrepresentation of any aspect of Godel's proof. This is yet another example of abuse of the Internet to post serial misinformation and confusion. Posters who wish to present objections to Godel's argument should at least study the subject well enough to have informed and coherent notions about it.
  • TonesInDeepFreeze
    2.9k
    The three books mentioned above are well beyond the preparedness of anyone who has not studied at least basic symbolic logic.

    The best book for the layman about Godel's incompleteness proofs is:

    Godel's Theorem - Torkel Franzen
  • EricH
    587

    There are two separate and distinct definitions/usages of the word truth and its derivations.

    The Correspondence Theory of Truth
    This is the plain language and accepted standard usage of the word truth. When you swear to tell the truth, the whole truth, and nothing but the truth? You are asserting that your statements - the sentences that you say, write, sign, etc - will accurately describe events in the real world (AKA existence, AKA the universe, AKA "everything that is the case", etc, etc)

    Usage in Logic & Mathematics
    Propositions in logic/math are true if they are derivable from the basic axioms & rules of the particular system you happen to be working in. I am not an expert in this area, but here is a good starting point if you want to learn: https://en.wikipedia.org/wiki/Peano_axioms

    The so called Liar Sentence "This1 sentence is false" is clearly not formulated in any sort of mathematical/logic framework, so it must be treated as an plain language statement. As such, there is no paradox at all, since it does not make any assertion about any event in the real world. As such it does not take a truth value, it is neither true or false. It is a collection of words that is constructed to make a grammatically correct sentence - you can think of it as a sort of poetry.

    William Hughes Mearns
    Yesterday, upon the stair,
    I met a man who wasn't there
    He wasn't there again today
    I wish, I wish he'd go away...


    As far as your comments about Godel's Theorem go, I suggest you take @TonesInDeepFreeze's comment above to heart. You might also want to read Gödel, Escher, Bach by Douglas Hofstadter. Besides giving an excellent overview of Gödel, it's a great read, very entertaining.
  • TheMadFool
    13.8k
    :up: Thanks a million.
  • TheMadFool
    13.8k
    I'm out of my depths. :grin:
  • TheMadFool
    13.8k
    Update

    1. The Liar Paradox

    The Liar sentence: L = This sentence[1] is false.

    The liar argument:

    1. L is true (assume for reductio ad absurdum)
    2. If L is true then L is false (premise)
    3. L is false (1, 2 MP)
    4. L is true & L is false (1, 2 Conj)
    5. L is false (1 - 4 reductio ad absurdum)
    6. If L is false then L is true (premise)
    7. L is true (5, 6 MP)
    8. L is true & L is false (5, 7 Conj)
    9. L is true ( 5 - 8 reductio ad absurdum)
    10. L is true & L is false (5, 9 Conj) [The liar paradox]

    So far so good.

    The Liar sentence: This sentence[1] is false.

    What does L is false mean? It means ~L is true.

    ~L = This sentence[2] is false.

    The paradox (line 10) = L is true & L is false = L & ~L

    L & ~L = This sentence[1] is false & This sentence[2] is true.

    However, L & ~L, despite how it looks, is not a contradiction. For it to be a contradiction, This sentence[1] = This sentence[2] but they're not:

    This sentence[1] = This sentence[1] is false.

    This sentence[2] = This sentence[2] is true.

    This sentence[1] and This sentence[2] refer to two different sentences as shown above.

    2. Gödel's argument.

    Gödel sentence: G = This sentence[3] is unprovable.

    What is the negation of G? This sentence[4] is provable.

    If, in Gödel's proof, it's necessary that This sentence[3] = This sentence[4] (perhaps to arrive at a contradiction) then Gödel's proof fails because:

    This sentence[3] = This sentence[3] is unprovable

    This sentencea[4] = This sentence[4] is provable

    This sentence[3] and This sentence[4] are not the same sentence.
  • TonesInDeepFreeze
    2.9k
    He says he's out of his depth but persists by positing an "update". The update is nonsense updating nonsense. Meanwhile, he's not the least bit interested in actually studying the subject so that he could understand it and thus not serially broadcast his confusions and ignorance to other people.
  • EricH
    587
    1. L is true (assume for reductio ad absurdum)TheMadFool
    As I (along with several other people) explained, this is wrong. L cannot take a truth value since it has no semantic content that can possibly be verified or denied.

    I'm out of my depths.TheMadFool
    Yes. This stuff is difficult. I am not an expert in this, but you can become better educated. Both @TonesInDeepFreeze & I gave you several excellent books that can point you in the right direction.
  • Newberry
    5
    You are right. I have written papers about this. Here is one.
    http://arxiv.org/abs/1510.04393
  • ssu
    8.3k
    I guess I'm accusing Gödel of committing the fallacy of false dichotomy.

    A penny for your thoughts...
    TheMadFool

    Some make similar arguments like you. Haven't been successful.

    You see, even if it comes close to be like the liar paradox, it isn't a paradox. It just clearly shows the limits of provability. And that's it.

    That's the whole point of Gödel's incompleteness theorems.
  • Shwah
    259

    I think an issue is with treating Truth/Falsity like an objest as Frege did. It creates contradictions by asserting the existence of false things (and if false things exist then what is truth aptness really doing).
  • PL Olcott
    626
    The Gödel sentence is a spin-off of the liar sentence (This sentence is false). The assumption that we make with the liar sentence is that it's a proposition and therefore that it has a truth value. Reject that assumption and no contradiction results as there are no truth values that come into opposition.

    Since, the Gödel sentence is the liar sentence in some sense can't we do the same thing we did to the liar sentence: take away its status as a proposition?
    TheMadFool


    That is my thoughts exactly, yet so far we have not even done this
    to the actual Liar Paradox itself: "This sentence is not true."

    When we formalize the exact same structure as the Liar Paradox
    in Prolog we see that Prolog rejects it as unsound because it has
    a cycle in the directed graph of its evalulation sequence.

    ?- LP = not(true(LP)).
    LP = not(true(LP)).

    ?- unify_with_occurs_check(LP, not(true(LP))).
    false.

    We get the exact same thing with your simplified Gödel sentence.
    He sums up his own work this same way:

    ...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
    (Gödel 1931:43-44)

    Gödel, Kurt 1931.
    On Formally Undecidable Propositions of Principia Mathematica And Related Systems
  • Count Timothy von Icarus
    2.3k


    This :up: . It's a statement about provability for statements in a certain class of consistent systems (those than can encompass arithmetic), using "effective procedures," roughly those that can be described with a set of instructions where the results should be the same each time and are completed in a finite number of steps and which don't require any creativity (i.e. introducing things outside the instructions).
  • PL Olcott
    626


    x := y means x is defined to be another name for y
    https://en.wikipedia.org/wiki/List_of_logic_symbols
    This seem to be the only way that we get actual self-reference
    all of the textbooks merely approximate self-reference with ↔

    G := ¬(F ⊢ G) // G asserts its own unprovability in F

    Proving G in F requires a sequence of inferences steps in F that proves there is no such sequence of inference steps in F.

    Copyright 2023 PL Olcott

    ...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...
    (Gödel 1931:43-44)

    Gödel, Kurt 1931.
    On Formally Undecidable Propositions of Principia Mathematica And Related Systems
  • Count Timothy von Icarus
    2.3k


    To the original question, while I'm not super familiar with paraconsistent logic, I do think this changes things. Paraconsistent logics can retrieve naive set theory. They aren't susceptible to the principle of explosion, that anything can be proved from a contradiction.

    The axiom of choice also becomes trivial in this case. Disjunctive syllogism also goes out the door though.

    Such systems aren't necessarily complete, but I do believe they can be complete in cases where a classical version would not be. Here is a dialetheist (allowing for "true" contradictions) arithmetic that and can prove itself, but which is inconsistent.

    Consistency is less important when explosion isn't a factor (to some degree) and the undefinability of truth is also less of an issue.

    https://academic.oup.com/mind/article-abstract/111/444/817/960902

    The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness.

    Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow.

    Reading some out there stuff on category theoretic attempts to formalize Hegel that I didn't understand, apparently that sort of system gets at these "perks" of not having contradiction result in explosion.
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