 The next theory we present was actually embedded in the set theoretical proposals of Paul Finsler, which were (taken as a whole) incoherent (see the notes on Finsler set theory available in the  Other Internet Resources).  Ackermann later (and apparently independently) presented it again. It is to all appearances a different theory from the standard one (it is our first genuine “alternative set theory”) but it turns out to be essentially the same theory as ZF (and choice can be added to make it essentially the same as ZFC).