User blog comment:Emlightened/BIG FOOT is SMALLER than FISH NUMBER 7/@comment-5529393-20161225093559/@comment-27513631-20161225111611

There are two ways of looking at this. First, we can consider this \(V_{\text{ORD}}\) as an elementary substructure of \(V\), in which case you don't actually get a truth predicate for \(V\) as you can only consider the truth predicate for parameters \(\in V_{\text{ORD}}\).

The other way is that \(V_{\text{ORD}}\) is the original \(V\), and that we continue to extend this, but this is easily argued against as invalid from both formalist perspective (obviously \(V\) is the larger one, as it is the domain of the quantifiers etc.) and platonist perspective (\(V\) is absolutely everything, so it makes no sense to argue outside of \(V\)).

Perhaps it's my fault for not specifying the predicate can take parameters. If we have a parametered truth predicate, we can define the class \(C\) of \(\lambda\) such that \(V_\lambda\prec V\), which roasts FOOT.

Admittedly, FOOT does force these correct cardinals to exist in \(V\) whereas \(FOST_n\) doesn't, but we can only actually apply \(\Sigma_n\) correctness in the language of FOOT as we can't define \(\prec_n\) uniformly in \(n\), without a truth predicate (which we only have a non-parametered version of).