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5. Theory of Logic / K. Features of Logics / 5. Incompleteness

[some truths of a system evade formal proof]

10 ideas
We can assign a characteristic number to every single object [Leibniz]
The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel ,by Koellner]
If completeness fails there is no algorithm to list the valid formulas [Tharp]
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P]
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
We have no argument to show a statement is absolutely undecidable [Koellner]
The first incompleteness theorem means that consistency does not entail soundness [Horsten]
A deductive system is only incomplete with respect to a formal semantics [Rossberg]
Axioms are ω-incomplete if the instances are all derivable, but the universal quantification isn't [Engelbretsen/Sayward]