[FOM] MK |- CON(ZF) ?

[Adam Kolany]

Thanks a lot. Can you show me (links with) more detailed proofs ?


W dniu 15.05.2016 o 02:55, Richard Heck pisze:
> On 05/14/2016 01:44 PM, Adam Kolany wrote:
>> Hallo,
>>
>> is this here:
>>
>>     http://jdh.hamkins.org/km-implies-conzfc/
>>
>> valid ?
> Yes, it is correct that MK |- Con(ZF). The main argument is already in
> Tarski. MK defines truth for the language of first-order set theory, and
> then you can give the trivial argument for ZF's consistency: the axioms
> are true; the rules preserve truth; so all theorems are true; since
> there are untrue sentences, not all sentences are theorems; so ZF is
> consistent.
>
> In fact, much less than MK is needed, but GB is not enough. This is
> because GB will not let you do induction on truth-involving claims,
> since we have only predicative comprehension. But the argument can be
> given in ZF plus Pi-1-1 comprehension for classes.
>
> Such results can be stated in quite general forms, i.e., as applying to
> a wide range of theories.
>
> There are corresponding results for the case where we add to the theory
> a Tarski-style theory of truth. A framework suitable for this kind of
> result is developed (again, following Tarski's original ideas) in my
> paper "Consistency and the Theory of Truth", and very general results of
> this kind are formulated and proved. Moreover, one can establish a
> strong correspondence between the two approaches.
>
> Richard
>