Psycho-Pass Set Theory

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anime math

Decided the twist in Psycho-Pass is best understood by goedelian incompleteness in axiomatic set theory

In a formal system such as classical arithmetic or human society in Psycho-Pass, goedel’s incompleteness theorem dictates that you encounter statements which may be true but cannot be produced from first principles. As a result, all non-trivial formal systems are incomplete. In these case you can grow your formal system by accepting either this result or its negation as a new axiom

So in the case of classical (say rational) arithmetic, x*x=2 is a statement within the system but cannot be resolved without introducing irrational numbers; x*x=-1 is a statement within the system but cannot be resolved without introducing imaginary numbers; but in either case you could instead accept the negation of the statement and grow the formal system in a different direction

In Psycho-Pass, as with set theory, there are a finite number of axioms (biological processors) and goedelian incompleteness dictates that every so often they encounter a theorem (person) who, although obviously a statement within the system, cannot be categorised by the existing axioms; the only solution is to accept the theorem (or its negation) as a new axiom (by assimilating makishima shougo as a new bioprocessor)

And just as every element in number theory can be expressed in terms of a linear combination of a basis - e.g. the basis [1, i] covers the numbers in the form {A + B.i | A,B ∈ ℤ} - adding the bioprocessor to the Psycho-Pass system allows for the recognition of new people whose behaviour is only understood by makishima (an 'imaginary component' to personality)