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Note: a large revision has been done, and I no longer believe the title. However, the first section of the text goes towards illustrating that BIG FOOT is actually not that large a jump and has been preserved. The second section contains clarifications and explanations, after given feed abck from the comments section.

=== 'Proof' === 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.

However, this is practically identical to the Ferferman theory.

And, it is. There are a couple of small differences, such as omitting the limit correct cardinals that are limits of correct cardinals in FOOT, and the cantorsattic entry not specifying that there are \(On\) many such cardinals, but otherwise it's identical (as noted by the creator in the comments, the 'oodleverse' in the reference is the universe). 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.

Questions and accusations of witchcraft welcome.

Notes

As pointed out in the comments, the above is incorrect as \(V_{\text{'Ord'}}\) gives us a parameterless truth predicate for \(V\), so allows us to consider the Rayo function and arbitrary oracles, etc.

The most important thing to note is that the truth predicates given by FOOT lack the ability for us to use arbitary parameters. In fact, we can define the club class \(C\) of correct cardinals (which is essentially equivalent to the [] in FOOT) using a single parametered truth predicate for the language \(\{\in\}\).

To beat BIG FOOT, we define the languages \(\mathcal L_{n+1} = \mathcal L_n+T_n\), where \(\mathcal L_0 = \{\in\}\) and \(T_n\) is a truth oracle for \(\mathcal L_n\). Then, we define \(FOST_n\) analogously to \(FOST\) with the language \(\mathcal L_n\), and beat BIG FOOT with \(FOST_5(5\uparrow\uparrow5)\).

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