Lexicon of Arguments

 

II 213
Goedel/incompleteness/Hilbert/Genz: in 1917 Hilbert had drawn up the program to summarize all mathematics in a scheme in the logic of the 1st level, Goedel proved in 1931 that this was not possible. It works well for Euclidean and non-Euclidean geometry, but not for addition and multiplication, if you take their derivation rules together.
It is always about sentences that are formulated in a language, but cannot be derived or refuted.
>Incompleteness.
Abundance/Genz: in poor languages, all statements that can be formulated in them can either be derived or disproved. The richer they are, the more statements can be formulated that do not succeed.
>Semantic closure.
II 214
These sentences make a statement about themselves, namely that they cannot be derived.
Solution: a solution is the extension of the language. For example, to accept his negation as an axiom.
>Extension, >Levels (order), >Description levels.
Problem: in every extension there are new non-derivable sentences.
Deductibility: a language in which any sensible phrase at all could be derived would allow to derive contradictions.
>Derivation,
>Derivability. >Contradictions.