User blog:LittlePeng9/First order oodle theory

So in this blog post I'm going to formalize notion of oodles. Because of multitude of paradoxes which appear when we try to think of oodles as "collections of anything", I've decided to build a hierarchial structure out of them.

Note that what I'm describing here isn't going supposed to be subject of generalization, but I still decided (for one or other reason) to call it first order oodle theory - FOOT.

Oodles
Oodles are abstract objects, analoguous to sets, which, in a way, have a broader meaning. The language of FOOT is going to have infinite set of variables, oodle connectives \(\in, =\) and some standard set of logical connectives, say \(\exists,\land,\neg\). We define truth of formula under some variable assignment using standard Tarskian definition of truth: ​We also assume that oodles satisfy all the properties which are considered "natural" properties, e.g. extensionality or existence of full power set.
 * Formula is of the form \(x\in y\) for x,y variables, is true iff x is an element of y under assignment of variables.
 * Formula is of the form \(x=y\) for x,y variables, is true iff x and y have the same oodle assigned.
 * Formula \(\varphi\land\psi\) is true iff both \(\varphi\) and \(\psi\) are true under the assignment.
 * Formula \(\neg\varphi\) is true iff formula \(\varphi\) isn't true.
 * Formula \(\exists x:\varphi(x)\) is true iff we can find oodle X for which formula \(\varphi(X)\) is true.

Oodle ranks
To avoids all possible paradoxes, we will want oodles to be well-founded - there is no infinite chain of oodles such that \(A_1\ni A_2\ni...\). Because of this, we have naturally arising notion of rank of oodle, defined as follows: From now on, we will say that rank \(A\) is smaller than rank \(B\) if \(A\in B\), and that \(A\) is at most \(B\) if it's either smaller or equal. We are going to refer to possible ranks as oodinals, and we are going to denote them with greek letters.
 * If \(A\) is empty oodle, then \(A\) is its own rank.
 * If \(A\) has an element with rank \(B\), such that rank of every element of \(A\) is either \(B\) or its element, then \(A\) has rank \(B\cup\{B\}\).
 * If \(A\) has no elements like the above one, then rank of \(A\) is union of ranks of its elements.

Let denote \(\alpha\cup\{\alpha\}=\alpha+1\), and call such oodinal a successor oodlinal. If \(\beta>0\) is not successor, then it's limit oodinal.

It's quite easy to prove that each oodle has a rank - by contradiction, of some oodle \(A_1\) had no rank, then it would have to have an element \(A_2\) without rank, and continuing we would have infinite descending sequence \(A_1\ni A_2\ni...\) which we claimed do not exist.

Oodle hierarchy
Let's denote the oodle of all oodles of rank smaller than \(\alpha\) as \(V_\alpha\). We can see, from definition above, see that if \(\alpha<\beta\) then \(V_\alpha\subset V_\beta\), and that \(V_{\alpha\cup\{\alpha\}}\) is exactly the oodle of all oodles elements of which have rank at most \(\alpha\). Because of this, we can define these equivalently as a hierarchy: Let's call \(V=\bigcup V_\alpha\) the oodleverse. This structure, while unfortunatelly not an oodle, contains all oodles, because we've shown that each of them has a rank.
 * \(V_\varnothing=\varnothing\)
 * \(V_{\alpha+1}=\cal P(V_\alpha)\)
 * \(V_\alpha=\bigcup_{\beta<\alpha} V_\alpha\)

Sets?
A question arises: how to define sets inside this structure? Turns out, it's impossible. Firstly, because "sets" don't even have definition, secondly, oodleverse could as well work out as a universe of sets.

However, if we had some quite large oodinal, which we'll denote \(\text{Ord}\), we could define sets as oodles which have rank \(<\text{Ord}\). We will also call elements of \(\text{Ord}\) ordinals. But the problem now could be, how to choose \(\text{Ord}\)?

Because we want these sets to satisfy some natural properties, we can't choose it in any way we want. Also, because we know that \(V\) can work as the universe of sets, we want \(V_\text{Ord}\) to satisfy the same statements about sets as \(V\) does about oodles. I propose the following definition: \(\text{Ord}\) is the least oodinal larger than any oodinal which we can define in language of FOOT. Because we have conutably many formulas and we have truly uncountably many oodinals, then such oodinal exists.

We want these sets to satisfy all the formulas which oodles satisfy, except that we will have to limit our quantifiers to range over sets. To check truth, we only need to verify that if oodle satisfying the formula exists, then there exists set which does the same. Indeed, assume that there is oodle \(A\) which is witness for \(\exists x:\varphi(x)\). But now define the following oodinal: "least rank \(\alpha\) of set \(A\) which satisfies \(\varphi(A)\)". Because we could define \(\alpha\), we have \(\alpha<\text{Ord}\), and thus there exists \(A\) with rank \(<\text{Ord}\) which satisfies \(\varphi(A)\), so \(A\in V_\text{Ord}\) is a set witness to the formula.

Rayo's function
Soon.