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Because of Emlightened's recent blog post I was pushed to think again about FOOT, truth predicates and related. The conclusion I came to is:

FOOT is not as strong as I thought it is. Indeed, it is about as strong as FOST with a single truth predicate added. However, it still seems that at the time of definition BIG FOOT was the largest number explicitly defined (in particular, larger than Fish number 7), and possibly still is.

I will address the three parts separately below.

FOOT is not as strong as I thought

When defining FOOT, I first defined \(\text{Ord}\), on order to be able to define the truth predicate with help of it to be able to define Rayo's function. As Emlightened points out, we can't get the full truth predicate this way - we only get it for formulas with parameters taken from \(V_\text{Ord}\). I was vaguely aware of that, but I didn't make much of that because a) we only need to have the truth predicate for parameter-free formulas to define Rayo's function, and b) I thought it doesn't really matter.

Then, when moving to \(\text_{Ord}_2\), I add a constant symbol \(\text{Ord}\) to the language, the reasoning being that \(V_\text{Ord}\) is "essentially" \(V\), and adding it allows us to get pretty much the second-order truth predicate for \(V\), but with restricted parameters again. Enough to define some second-order Rayo's function. And from there you just do the induction.

But I have underestimated the power of parameters. I was completely convinced that adding \(\text{Ord}\) to the language of FOST is something which is far beyond what can be done with just FOST. And, well, this is right. But we have to note that when talking about elementary substructures of \(V\) (the "satisfies exactly the same formulas as"), we have to actually allow formulas with arbitrary parameters from the smaller structure. In particular, the second ordinal which satisfies the same FOST formulas as \(V\) also satisfies the same "FOST + \(\text{Ord}\)" formulas as \(V\), since we can take \(\text{Ord}\) as a parameter. To be fair, this is not what I hoped it to be...

Yet the stronger underestimation from me was about the importance of parameters is in the truth predicates. Above I've stated I thought that inability to feed arbitrary parameters into the truth predicate didn't matter much, and I was very, very wrong. I thought with it we can define \(\text{Ord}\) but not \(\text{Ord}_2\), yet from the discussion above we actually can define it, as \(\text{Ord}\), even though it's not in the langauge, can be used as a parameter.

FOOT vs single truth predicate

The above discussion shows that using the full truth predicate for \(V\) we can define \(\text{Ord}_2\). With the same reasoning, we can define \(\text{Ord}_\alpha\) for any ordinal \(\alpha\), since \(\text{Ord}_\alpha\) turns out to be just the \(\alpha\)th ordinal such that \(V_\gamma\) is an elementary substructure of \(V\) (so FOOT is equivalent to Feferman set theory Emlightened refers to in her blog post), which means FOOT is no stronger than FOST + a truth predicate (\(FOST_1\) in Emlightened's post).

We can also go the other way. Consider any formula with parameters and let \(\alpha\) be such that all the parameters are in \(V_{\text{Ord}_\alpha}\). We can define \(V_{\text{Ord}_\alpha}\) in FOOT, and then we can verify the truth of the formula using the truth predicate with parameters in this set. So we can define the full truth predicate in FOOT.

We conclude FOOT and \(FOST_1\) are equally powerful. In conclusion, FOOT is far weaker than even the second order set theory, destroying an impression which I've had for over two years, that it is stronger than higher order set theories. The reason is that SOST can define the truth predicate for FOST, and more - it can define the second order truth predicate (i.e. the truth predicate for "FOST + truth predicate for FOST"), third order and so on. It can iterate transfinitely along these lines (see also here).

BIG FOOT > Fish number 7

I am going to argue now that we can define Fish number 7 in FOOT. Recalling the definition of \(RR\) from the article Fish number 7, it's enough to show we can define it in FOOT. So let \(f\) be any function (we only need to deal with ones \(\mathbb N\rightarrow\mathbb N\), but it works more generally). Let \(\alpha\) such that \(V_{\text{Ord}_\alpha}\) contains \(f\). Then using this \(f\) as a parameter, we can make sense out of truth in "FOST + \(f\)", which is essentially the system in the definition of \(RR\).

Indeed, I believe just \(\text{Ord}\) (the one with index \(1\)) is enough to beat Fish number 7, since a function \(f:\mathbb N\rightarrow\mathbb N\) is always contained in \(V_\text{Ord}\) (it is essentially contained in \(V_{\omega+1}\), in fact).

So what about the largest number?

Therefore BIG FOOT is larger than Fish number 7. As far as I know, no larger number was created until my definition of BIG FOOT, so BIG FOOT was the record. AFAIK no larger number was really proposed since then, but I feel like the large part of the reason for that is that because of me everyone was convinced going with higher-order theories is not going to beat FOOT. After reading this blog post, I hope the reader is convinced this is not the case. Indeed, pushing to higher-order set theories seems like it could be the way to go. This idea was conceived in the past already, the earliest I know of being probably Keith Ramsay in what he has said in 2004, but also by us here, and by me here.

What really lies out there - I don't know. Are these things well-defined? Well, this is a very important question I was avoiding to this day, but I feel like it is about time it is seriously addressed (Emlightened and Deedlit discuss the topic briefly here, but this didn't attract much attention). The questions like this are philosophical in nature, and whether the systems involved are well-defined can be very much just a matter of one's views. I don't want to really discuss these issues here. I just wanted to mention I am aware of them, and I think I will make a blog post or something to discuss this further, but I am getting quite off-topic from where I wanted to get this blog post.

Thanks to anyone who has read all of this. Now that we are here, I am looking forward to see what the next largest number coined will be, given that (apparently) it is not hard to beat the current record :)

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