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.
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.
These sentences make a statement about themselves, namely that they cannot be derived.
Solution: Extension of the language. For example, to accept his negation as an axiom.
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._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Gedankenexperimente Weinheim 1999
Wie die Naturgesetze Wirklichkeit schaffen. Über Physik und Realität München 2002