Godel’s incompleteness theorem states that any mathematical system that contains Peano arithmetic is either incomplete or inconsistent. According to the “liar’s revenge” corresponding to incompleteness, incompleteness would be inconsistent. Mathematics could still be incomplete or not incomplete – the revenge statement associated with incompleteness is agnostic on the question of incompleteness.
A revenge statement can be constructed for incompleteness, and the corresponding truth table demonstrates that incompleteness necessitates inconsistency. The “liar’s revenge statement” takes a hypothetical proposed solution to the “liar’s sentence” and shows that proposed solution leads to a paradox.
The “liar’s revenge statement” for a proposed hypothetical solution to the liar’s sentence “L” is written as follows: “this statement is L or false”. So, if someone characterized “this sentence is false” as “ridiculous”, the associated liar’s revenge statement would read “this sentence is ridiculous or false”. The corresponding truth table would show that:
- if the statement were ridiculous it would be true, and ridiculous wouldn’t be true, because “this sentence is false” would be said to be ridiculous, which would differ from truth
- If the sentence were true, it would be either ridiculous, or false. If the sentence were ridiculous, it wouldn’t be true, because “this sentence is false” would be said to be ridiculous, which would differ from truth. If the sentence were false, it wouldn’t be true
- If the sentence were false, it would be either not ridiculous or false. If the sentence were false, and it were described as false, it would be true.
Importantly, the “liar’s revenge statement” can be written in Boolean algebra. Boolean algebra is a formal symbolic language representing classical logic. The expected instantiation of the values “true” and “false” is sufficient to phrase a “revenge sentence”, which necessitates a paradox. Generalizing the revenge sentence to the proposed hypothetical solution to the “liar’s sentence” comprehensively rules out anything other than the paradox.
The case taken up here is incompleteness. Importantly mathematics may or may not be incomplete – but it is definitely inconsistent. A revenge sentence can be constructed for incompleteness – the Boolean algebra formal symbolic logic representation of incompleteness is “both not true and not false”. The formal symbolic logic representation of the sentence “this sentence is incomplete or false” has as the corresponding truth tables:
- if the statement were incomplete it would be true, and incomplete wouldn’t be true, because “this sentence is false” would be said to be incomplete, which would differ from truth
- If the sentence were true, it would be either incomplete, or false. If the sentence were incomplete, it wouldn’t be true, because “this sentence is false” would be said to be incomplete, which would differ from truth. If the sentence were false, it wouldn’t be true
- If the sentence were false, it would be either not incomplete or false. If the sentence were false, and it were described as false, it would be true.
Thus, incompleteness necessitates inconsistency. It is instructive to read the case of the “general liar’s revenge sentence” corresponding to incompleteness in conjunction with Gödel’s incompleteness theorem. Gödel proves that the only alternative to inconsistency is incompleteness. The revenge statement corresponding to incompleteness further narrows solutions to the system of math. Gödel’s work is of monumental importance to mathematics, dooming the aspirational project of Cantor and Frege. The “liar’s revenge statement” corresponding to incompleteness further refines the big-picture expectations we should anticipate from mathematics.
That a revenge statement can be constructed for incompleteness is a major insight. Gödel’s incompleteness theorem already showed that math was incomplete or inconsistent (even if few scholars took that result seriously enough). The “liar’s revenge sentence” corresponding to incompleteness elaborates on Gödel’s incompleteness theorem. The potential solutions to the liar’s paradox, incompleteness and inconsistency, were already known. That a revenge statement can be constructed for incompleteness, in conjunction with Gödel’s incompleteness theorem, shows that math is inconsistent, and that a paradox exists. The revenge sentence corresponding to incompleteness is very important, according to what we already knew, according to Gödel’s incompleteness theorem. Because math is already known to be inconsistent or incomplete, the summary evaluation that incompleteness causes inconsistency shows that math is definitively inconsistent.
Gödel’s incompleteness theorem already substantially narrowed the scope for viable mathematical systems. Further, definitive, definition of mathematical systems as inconsistent is unexpected, difficult to comprehend, literal, and strange. Math is paradoxical, and a paradox exists. The notion that math is inconsistent is wild, and the idea that a paradox literally exists is wondrous.
For math to be inconsistent, another ontological category of the paradoxical would accompany the familiar categories of the “true” and the “false”; proofs would be written in this framework. An “anti-systemic system” would move beyond an axiomatization of nature. Math would be fragmentary, philosophic, aesthetic, relative, subjective, perspectival – “les deux”; some things would be true, some things would be false, and some things would be both. Math would describe itself as paradoxical; math would be a paradox. Language would overflow attempts to list the number of possible sentences limited by the number of words.
The paradox itself defies facile attempts to describe it – it resists definition, and yet, it must exist in the proverbial flesh. Physics may some day be capable of producing a physical paradox in front of us – something visible, tangible. Paradoxical objects would find expression in paradoxical words. Paraodoxical words will signify impossible emotions, plants, and linguistic constructs. Paradoxes will be concrete objects that appear in ways that will be fundamentally different from everything known unto now.
June 2, 2025