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I really should have made this blog post sooner than after two years, but oh well, at least I am making it now. In what follows I will refer to my old blog post.

TL;DR: When I have defined FOOT (and BIG FOOT) I have a somewhat bottom-up language to describe it, starting from the universe of sets and building up. Taking this too literally leads to many serious foundational issues. What I should've done is emphasize the top-down nature of my construction.

What I did

Long story short - I have described FOOT in terms of some unheard-of concept of oodles, only saying that they are analogous to sets, defining in it what I have called "sets", "ordinals", which were ought to resemble structures familiar to all of us, then iteratively extending the language to give it more expressive power.

What I did NOT do

The way I have described the above may strongly suggest that I have done something which resembles the following bottom-up scheme of extending FOST (something like what I did here:

  • We start with the universe \(V\) of all sets.
  • We then consider the family \(V_1\) of all classes of sets.
  • We then consider the family \(V_2\) of all "2-classes" of classes of sets.
  • Iterating this construction, we can construct, say, \(V_\alpha\) for all ordinals \(\alpha\).
  • We then define \(V_{0,1}\) to be the union of all the above.
  • By being clever we define \(V_{\alpha,\beta}\) or what not.
  • Finally we define \(W\) to be "the limit of everything like above", and then we repeat the above construction or similar.

The most significant (foundational) problem with something like above or similar ideas (for example quantifying over larger and larger domains) is that for any of this to work we have to assume there is some "ambient space" in which everything we desire (or more precisely, everything we wish to consider) exists. In above example, this "ambient space" would have to contain all classes of sets, all classes of classes, all of these iterated constructs, and also would have to contain "everything we can get this way".

We are getting onto philosophical grounds here. As far as most would agree about being able to speak about "the" universe of all sets, I doubt many set-theorists would buy it if you were to convince them there is such an "ambient space" in which all these things you describe lie.

I want to emphasise here that what I did does not follow this framework. In my approach, we start with "oodles", which, I admit, do work a bit like an ambient space I talk about above, but either way instead of taking one family ("sets") and iterating "class of all subclasses" operation, it partitions all oodles into a hierarchy, and then selects out certain parts of this hierarchy (the \(V_{\text{Ord}_\alpha}\) things).

What I did, and how I should've worded it

Instead of speaking of these "oodles", whatever the hell they are supposed to be, everything we are working with are just plain old sets, but we are going to do things with them which are neither plain nor old (well, 2 years is kinda old...). I am not going to rewrite all the technical details here.

We have the hierarchy of sets \(V_\alpha\), which is just the standard Von Neumann hierarchy, indexed with ordinals, and its union - the universe \(V\) (sidenote: I still am quite proud of the term "oodleverse" I have coined back then, but I'm afraid we should forget about it for now). Now in \(V\) we will choose a certain class of what I will here call "1-sets". For that, I define the ordinal \(\text{Ord}_1\) the same way as I defined it in oodle blog post (I called it \(\text{Ord}\) there; I add subscript for emphasis sake here). Then 1-sets will be precisely the elements of \(V_{\text{Ord}_1}\). The point here is that \(V\) and \(V_{\text{Ord}_1}\) satisfy the same formulas of FOST. Then we proceed pretty much in the same way as we did with oodles, perhaps defining 2-sets or \(\alpha\)-sets on the way.

My point here is - oodles were just a way for me to call these constructs, so as to give an illusion of extending the class of sets. This brings me to my last point.

Why I did what I did

The reason I have introduced this confusing terminology regarding oodles is, as I have just mentioned, to give an illusion of extending the class of sets. Not much earlier, I have made this post, in which I have defined "higher order set theory" in this bottom-up way I have argued about a few paragraphs earlier. Even at the time of writing I was vaguely aware of the fact I am entering a dangerously non-well-founded zone. FOOT was then an attempt to make things as formal as I thought is necessary to argue these concepts are well-founded.

At this point I must admit that I regret the decision I have made back then.

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