Just to note that Tim Button has a trio of linked papers on set theory coming out in the Bulletin of Symbolic Logic, and already downloadable from his website.
The first paper is on ‘Axiomatizing the bare idea of a cumulative hierarchy of set’:
The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: ‘Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets.We find nothing else at S’. Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potter.
The second paper is on ‘Axiomatizing the bare idea of a potential hierarchy’:
Potentialists think that the concept of set is importantly modal. Using tensed language as a heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: ‘Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets’. Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with Level Theory, as developed in Part 1.
The third paper is on ‘A boolean algebra of sets arranged in well-ordered levels’:
On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.
Note the punch at the end of each of the second and third abstracts: set theories we might have thought to be significant rivals to the industry standard turn out to be definitionally equivalent to (versions of) it.
One presentational remark. In Potter’s presentation of his version of Scott-Derrick set theory in his Set Theory and its Philosophy the key definitions of a ‘history’ and the ‘accumulation of a history’ are given in a rather take-it-or-leave-it spirit, just as tricks that work. Button’s definitions at the very outset of a ‘history’ and the ‘potentiation of a history’ likewise will give pause those new to Scott’s trick: perhaps just a little more could be said to win over the reader and carry them through.
But that is a very minor quibble. This trio of papers strike me as taken together a simply terrific achievement.