# Is CH independent of TG?

Is the continuum hypothesis (CH) independent of Tarski-Grothendieck set theory (TG)?

Own thoughts: I have only very limited knowledge of the different axiom systems. I know that TG implies ZFC. I know CH is independent of ZFC. I know that in TG strongly inaccessible cardinals exist (whatever that means), so I would assume that large cardinals exist, too. This answer on MO states that CH cannot be proven or disproven by large cardinals alone due to the Levy-Solovay theorem (whatever that is). EDIT: As I just found out, inaccessible cardinals come before large cardinals. Together with the other link, this would probably answer my question affirmative.

That is all I could collect about this topic on MSE and MO. Answers are appreciated, explicit references would be a nice bonus.

The method of forcing shows that every model of ZFC has forcing extensions with CH and other forcing extensions with $\neg$CH. Furthermore, the size of the forcing notions can be taken to be at most continuum, and therefore smaller than the smallest inaccessible cardinal. This forcing therefore preserves all inaccessible cardinals to the extension. So the forcing preserves TG theory.