Edwin's FOST taxonomy

**Edwin's FOST taxonomy** (EFOSTT) refers to a way of classifying sentences in first-order set theory devised by Edwin Shade on 18 October 2021.

## Categorization

First order set theory (FOST) sentences can be categorized and listed using this categorization hierarchy, which splits up sentences based on *bivelages*, *subivelages*, *varentages*, *varmutages*, and *quentages*. This affords a quick way to organize all bounded logical sentences in first order set theory similar to animal taxonomies. These groupings are described briefly as follows.

### Bivelages

A *bivel* (**bi**nary + **v**ariable + **el**ement) is a single 'element of' predicate flanked by set variables, or an expression of the form x∈y. Both variables may be the same or different. Bivels are denoted with an italic *b*.

To form *bivelages* (*bivel* + st*ages*), we define the following construction where ↑ is the NAND operator:

* B1 = {*b*}
* Bn = {*b*} U {φ↑ψ:φ∈Bm < n ∧ ψ∈Bm < n}

B1 = {*b*}, B2 = {*b*, *b*↑*b*}, B3 = {*b*, *b*↑*b*, *b*↑(*b*↑*b*), (*b*↑*b*)↑*b*, (*b*↑*b*)↑(*b*↑*b*)},. . .

The bivelage number of a sentence corresponds to the subscript of the *first* bivelage stage which contains it's logical form.

### Subivelages

A *subivelage* (*sub* + b*ivelage*) is an element of a bivelage. We can restrict the bivelage hierarchy by making a new hierarchy B' defined like so:

* B'1 = {*b*}
* B'n = Bn/Bn - 1

B'1 = {*b*}, B'2 = {*b*↑*b*}, B'3 = {*b*↑(*b*↑*b*), (*b*↑*b*)↑*b*, (*b*↑*b*)↑(*b*↑*b*)},. . .

We can enumerate the expressions within these restricted bivelages by realizing that every bivelage can be realized as a set of successive "products" of former expressions then adding in *b* afterwards, like so:

B1

B2

B3
 *b*↑*b* | (*b*↑*b*)↑*b* | (*b*↑*b*)↑(*b*↑*b*)

We can count each expression in these tables left-to-right, top-to-bottom. According to this ordering, the nth φ such that φ is in both Bn and B'n is the nth subivelage of a given bivelage.

### Varentages

*Varentages* (**var**iable ori**ent**ation s**tages**) are the set of instantiations of different variables for every subivelage. Treating set variables as isomorphic to numbers in a radix-v system where v is the number of variables (the bivel count times two), we instantiate variables according to how numbers in such a system would increment. For instance, if we have the elementary bivel *b*, then there are the instantiations of A∈A, A∈B, B∈A, and B∈B. If A = 0 and B = 1 this would be equivalent to binary counting. If the bivel was *b*↑*b* then there are now four unique variables and 44 = 256 instantiations. However, with the addition of quantifiers most of these would behave exactly the same. After all, quantifying the exact way over A∈A would be the same as B∈B, and A∈B would be the same as B∈A. For every set of variable instantiations over a collection of a-bivels we order them by their corresponding base-2a number and then go through the list and remove isomorphic instantiations as they appear. We then relabel the remaining list from 1 and set that as a formula's varentage number.

### Varmutages

*Varmutages* (**var**iable per**mut**ation s**tages**) are collection of all the ways there are to arrange the quantifiers which quantify over variables in a subivelage. Given n unique set variables x1, x2, x3,. . .,xn and 'Q' as an ambiguous quantifier, they are ordered lexicographically according to the number of their subscripts. Eventually all n! permutations are exhausted. The position of a formula in this list is that formula's varmutage number.

### Quentages

*Quentages* (**qu**antifier ori**en**tation s**tages**) are collections of all the ways there are to 'flip' the quantifiers of an expression, either ∃ or ∀. We begin with the right-most quantifier and treat ∃ isomorphic to 0 and ∀ isomorphic to 1, running through all possible binary numbers. We don't need to keep track of every single variation though, since knowing the truth value of one of them is sometimes enough to deduce other ones. For instance, if we know that ∀A∃B∃C∀D(φ(A,B,C,D)) is true, then it follows that ∀A∃B∃C∃D(φ(A,B,C,D)), ∃A∃B∃C∀D(φ(A,B,C,D)), and ∃A∃B∃C∃D(φ(A,B,C,D)) are also true, since every universal quantifier can be replaced by a "there exists" quantifier, as ∀X(φ(X)) ⇒ ∃X(φ(X)). Still, we enumerate every quantifier orientation, but in practice not each one has to be listed out due to this reason.

## Usage

EFOSTT has uses in cataloging set theoretic truths. (See EFOSTT list.)