IX 209/10
Consistency/Set Theory/Quine: we have been able to prove it twice as we have a simple model in finite sets - which does not apply if we once have added an axiom of infinity. Consistency is questionable and more difficult and more urgent to prove. And the evidence is even less convincing - problem: the question of whether the methods themselves are consistent.
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Set Theory/Quine.
II 178
The essence of the Corollary of Goedel's incompleteness theorem is that the internal consistency of a mathematical theory can usually only be proved by resorting to another theory based on further premises and it is therefore less reliable than the original one. This has a melancholic connotation.
But this helps us to prove that one theory is stronger than another: This is achieved by proving in one theory that the other is consistent.
II 180
Goedel's third great discovery: the consistency of the continuum hypothesis and the axiom of choice.
II 210
Possible Worlds/QuineVsKripke: possible worlds allow proof of consistency, but no clear interpretation: when are objects equal? For example Bishop Buttler said that any thing is this thing and "no other thing": Problem/QuineVsButler: Identity does not follow necessarily.
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Possible Worlds/Quine.
IX 192
Set theory/Modern Type Theory/Consistency/Quine: we can prove the freedom of contradiction of this version of set theory with cumulative types:
Def Cumulative Types/Set Theory/Quine:
Type 0: only L is of type 0.
Type 1: L and {L} and nothing else.
Type n: should generally include and only include the 2n sets belonging to type n -1.
So each quantification interprets only finitely many cases. Every closed statement can be checked mechanically for truth.
Such a simple proof will no longer work if the infinity axiom is added.
IX 210
Infinite Classes/Consistency: the proof also becomes less convincing if we have to accept infinite amounts.
Problem: whether the methods themselves are consistent (only with infinite classes).
The highest thing we can often strive for is that we prove that such a system is consistent when another, corresponding system is less distrusted.
IX 239
If the consistency of one set theory can be proven in another, the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory.
