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 to Gödel’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 Gödel’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 Gödel’s Jinx.
According to Löb’s Theorem:
“This sentence is provable” is provable.
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 Löb’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!
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