[FOM] MK |- CON(ZF) ?

[Richard Heck]

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

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