User blog:Emlightened/BIG FOOT is SMALLER than FISH NUMBER 7

The idea behind the creation of BIG FOOT is that we can only define so much in the language of FOST, so we add a new symbol to represent ORD, and then a new one (ORD2) for the new analogue of ORD, and so on, across all of the ordinals.

Most people's reactions to this is "Wow, this is soooo much bigger!!"

My reaction to this is "Wow, this is practically identical to the Ferferman theory!"

And, it is. There are a couple of small differences, such as clubness is not specified in FOOT, and the cantorsattic entry not specifying that there are \(On\) many such cardinals, but otherwise it's identical*. The thing that throws people off is the use and introduction of the logical and new [] symbols, and that few people know of the Ferferman theory.

But! Not only is this language extension quite simple, we can never use all of it anyway! Because we can never do any sort of induction on formula length, we may as well use \(\Sigma_n\)-elementarity for some sufficiently large \(n\), with is definable in the language of FOST (although the formula length will be somewhere between \(n\) and \(2^{2^{2^n}}\), so not small).

However, Fish Number 7 allows us to construct the FOST function, and hence take these sufficiently elementary substructures. And this means that we can interpret FOOT(n) for given n, so we far outmatch BIG FOOT.

We can, however, beat this easily, by defining the languages \(\mathcal L_{n+1} = \mathcal L_n+T_n\), where \(\mathcal L_0 = \{\in\}\) and \(T_n\) is a truth oracle for \(\mathcal L_n\), define \(FOST_n\) analogously to \(FOST\) with the language \(\mathcal L_n\), and submit \(FOST_5(5\uparrow\uparrow5)\).

Questions and accusations of witchcraft welcome.


 * I assume that we always work in the universe, and that the reformulation is all just fancy-talk. Otherwise, we're working outside the universe with \(V\) as an elementary extension of \(V_{Ord}\), and it's not clear that that's even definable (in the model with additional symbols added to the language in a way that don't increase consistency strength) anyway. We need to not increase the consistency strength as otherwise we have no reason to believe the extension and hence number has a chance of being well-defined. Also, taking elementary extensions of the universe would be very involved and not necessarily produce anything stronger than elementary substructures.