Part 6. Elementary Metamathematics
Money is based upon arithmetic; and the study of the logic of arithmetic is called metamathematics. The ‘meta’ is there because arithmetic can study itself; so paradox is possible.
In metamathematics, statements about numbers themselves bear numbers describing their syntactic form; so number statements can refer to each other’s formal properties, including provability, consistency, unprovability, and refutability. It’s also possible to construct statements that refer to their own properties; I call such statements “quanta”.
For instance, I call “this sentence is unprovable” the “quantum of self-doubt”. “This sentence is refutable” is the quantum of self-shame, “This sentence is irrefutable” is the quantum of self-pride, and “This sentence is provable” is the quantum of self-belief.
Call a property of statements “jinxed” if its quantum is false:
“Property P is jinxed”
if and only if
“This sentence has property P” is false;
it does not have property P.
A jinxed property does not apply to its quantum. For instance, “This sentence has six words” does not have six words; therefore “has six words” is jinxed.
Call a property of statements “charmed” if its quantum is true:
“Property P is charmed”
if and only if
“This sentence has property P” is true;
it does have property P.
A charmed property applies to its quantum. For instance, “This sentence has five words” has five words; therefore “has five words” is charmed.
According the Goedel’s First Incompleteness Theorem:
“This sentence is unprovable” is unprovable if your logic system is consistent, otherwise not;
“This sentence is refutable” is refutable if your logic is inconsistent, otherwise not.
So if your logic system is consistent then unprovability is charmed, and refutability is jinxed.
If your logic system is consistent, then self-doubt is true but unprovable, and self-shame is false but irrefutable. On the other hand, if you prove self-doubt or refute self-shame, then your logic system is inconsistent. Self-doubt and self-shame are inherently uncertain; attempts to resolve these paradoxes backfire.
According to Goedel’s Second Incompleteness Theorem:
“This sentence is irrefutable” is refutable.
That’s because the quantum of self-doubt sets a trap. If you assume that self-pride is true, then your logic system must be consistent; so self-doubt must be unprovable; hence self-doubt must be true; but that would be a proof of self-doubt; a contradiction. Assuming self-pride yields a contradiction; this refutes self-pride.
Self-irrefutability refutes itself. I call this Goedel’s Jinx.
According to Loeb’s Theorem:
“This sentence is proveable” is proveable.
This is because the quantum of self-belief is in fact the negation of the quantum of self-pride; true to the extent that the other is false. Since self-pride is refutable, self-belief is provable.
Self-validation validates itself. I call this Loeb’s Charm.
A statement is consistent if and only if there is a model of arithmetic, within which that statement is true; and it is provable if and only if it is true in every model of arithmetic. Therefore ‘true in some model of arithmetic’ is jinxed; and ‘true in every model of arithmetic’ is charmed. In metamathematical terms, existence is jinxed and universality is charmed!