Lexicon of Arguments

 

IX 237ff
Stronger/weaker/theory/system/Quine: Problem: Comparability: it fails if both of the two systems have theorems that cannot be found in the other - it also depends on contingencies of interpretation and not on structure.
>Comparisons, >Comparability.
If we can interpret the primitive logic characters (only "ε" in set theory) new so that we can ensure that all theorems of this system are made to translations of the theorems of the other system, then the latter system is at least as strong as the other.
>Systems.
If this is not possible in the other direction, one system is stronger than the other one.
Definition "ordinal strength"/set theory: numerical measure: the smallest transfinite ordinal number whose existence you cannot prove anymore in the system.
The smallest transfinite number after blocking of the apparatus shows how strong the apparatus was.

Relative strength/proof theory: Goedel incompleteness sentence: since the number theory can be developed in set theory, this means that the class of all theorems (in reality all Goedel numbers of theorems) of a present set theory can be defined in this same set theory, and different things can be proven about them.
>Incompletenes/Goedel.
One can produce an endless series of further based on a arbitrary set theory, of which each in the proof-theoretic sense is stronger than its predecessors, and which is consistent when its predecessors were. - One must only add via Goedel numbering a new arithmetic axiom of the content so that the previous axioms are consistent.
Ordinal strength: is the richness of the universe.
>Goedel numbers.
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X 71
Metalanguage/Set Theory/Quine: in the metalanguage a stronger set theory is possible than in the object language. In the metalanguage a set of z is possible so that satisfaction relation z applies. - ((s) A set that is the fulfillment relation (in form of a set of arranged pairs) - not in the object language, otherwise Grelling paradox.
>Meta language, >Set theory,
>Grelling's paradox, >Metalanguage.