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Godel's incompleteness theorem is proved by showing that a formula G with Gödel number g which says: ‘the formula with Gödel number g is not demonstrable’ is true but not provable. However, considering the similar liar paradox : a formula G with Gödel number g which says: ‘the formula with Gödel number g is not true’. Then this liar paradox certainly could neither be true or false. And we know that many paradoxes arise because of self-reference (such as Russell's paradox and Berry paradox). Thus it seems reasonable to make the truth value of all self-referential sentences to be indeterminate. My question is would it be possible to have a complete (and strong enough) logic system if we exclude all these self-referential sentences?

Richard
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  • I am not aware of a proof of Goedel's results that does not somehow use self-reference. Same for the halting problem. – Peter Nov 18 '22 at 10:56
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    The gist of Gödel’s Incompleteness Theorems is that - for certain formal systems satisfying certain assumptions - there is an undecidable statement. Following the original result, other examples of arithmetical" undecidable statements has been found. The proof un the undecidability of these ones does not involve self-referentiality. – Mauro ALLEGRANZA Nov 18 '22 at 11:51
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    "self-referential sentences" are not per se part of the formal system; we consider the formal system of e.g. arithmetic and we use the features of natural numbers to "manage" facts about the formal system itself, like being a formula, a derivation, etc. – Mauro ALLEGRANZA Nov 18 '22 at 11:53
  • Only at the end of the process, when we have proved the GIT, we are left with a sentence $G$, fully compliant with the syntactical specifications of the language such that the rule of logic (first-order predicate calculus) with the axioms of the system (e.g. first-order version of Peano axioms for arithmetic), such that neither %G% nor its negation $\lnot G$ are derivable in the system. – Mauro ALLEGRANZA Nov 18 '22 at 11:55
  • But our "natural expectation" about the axioms of arithmetic and their formal counterpart is that they express "true fact" about the intuitive domain of natural numbers. Thus, having found and undecidable $G$ as above, we are prone to say that - being a statement in the language of formal arithmetic - it express a fact about numbers, and thus either it express a TRUE fact about numbers, or it express a FALSE fact about them. Conclusion (according to the above line of thought): one of $G$ and $\lnot G$ is TRUE about numbers, but neither of them is provable. – Mauro ALLEGRANZA Nov 18 '22 at 11:58
  • Second conclusion: strictly speaking, self-referentiality is not the key feature of GIT result. – Mauro ALLEGRANZA Nov 18 '22 at 12:04
  • Regarding possible approaches self-referential sentences (and others) considered as indeterminate, see e.g Vagueness and Liar Paradox. – Mauro ALLEGRANZA Nov 18 '22 at 12:06
  • Okay, I see that the Godel's sentence could indeed be turned into a sentence about arithmetic, but is there any specific example of this arithmetic statement? If we follow the construction of the godel's sentence, we should be able to obtain a explicit statement about arithmetic; however, I haven't seen any arithmetic statement that is true but couldn't be proved. – Richard Nov 18 '22 at 13:33
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    @Richard See https://math.stackexchange.com/questions/1472769/what-does-a-godel-sentence-actually-look-like/1484929#1484929. Also note that by the (proof of the) MRDP theorem, we can even make do with a Diophantine equation (relatedly, see Jones' paper Universal Diophantine equation). Finally, if you're looking for a natural example of an arithmetic sentence not provable in first-order Peano arithmetic, google "Paris-Harrington theorem." – Noah Schweber Nov 18 '22 at 17:42
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    "I see that the Godel's sentence could indeed be turned into a sentence about arithmetic" No, the Godel sentence is a sentence about (indeed, literally in the language of) arithmetic. The natural-language-ish expression "I am not (PA-)provable" is an informal gloss after the fact, or a blueprint for finding it it, but it is not the sentence itself. This may seem like hair-splitting, but it's crucially important. – Noah Schweber Nov 18 '22 at 17:44
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    @Richard You have it backwards. The Godel sentence is a sentence about arithmetic that could be viewed as asserting its own unprovability. And it’s not that it can’t be proved (otherwise how would we know it’s true?), it’s that it can’t be proved in PA. There are more natural examples of true arithmetic statements not provable in PA, e.g. Goodstein’s theorem – spaceisdarkgreen Nov 18 '22 at 17:46
  • @Noah seems you beat me. – spaceisdarkgreen Nov 18 '22 at 17:47
  • @spaceisdarkgreen I'm everywhere! I'm everywhere! – Noah Schweber Nov 18 '22 at 17:47
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    See the SEP article "Self-Reference" for some discussion of approaches to the paradoxes of self-reference that involve logics with an indeterminate truth value. – jdonland Nov 19 '22 at 03:32
  • Thanks for your guys' corrections and examples. I will check the references you gave. – Richard Nov 19 '22 at 19:28

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