Timeline for Some questions about Ackermann set theory
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| Apr 21, 2012 at 15:27 | comment | added | wmitt | Btw, note that Muller combines A with Regularity for sets, which gives a conservative extension of A (see Levy & Vaught). Also, it's clear from Reinhardt that any conservative extension of A is, ipso facto, a conservative extension of ZF. (The references for Levy-Vaught and Reinhardt are in the comment by Blass linked to above.) Finally, Muller tries to explicate Ackermann's set/proper class distinction by formalizing the idea of unsharpness; but he only formalizes "X is unsharply distinguished from Y," and not "X is unsharp"- and it's the latter that needs formalization. | |
| Apr 20, 2012 at 21:49 | comment | added | wmitt | There's F.A. Muller's "Sets, Classes and Categories," Brit. Jrnl for the Phil. of Sci. 52 (2001), 539-73. Muller adds a Class Separation Schema ("ClsSep") to Ackermann's theory ("A"), and this yields the desired analogs of Pairing, Union and Powerset (565). Contrary to Muller (564), I believe his theory is a conservative extension of A (and of ZF), since (i) ClsSep entails Ackermann's class existence schema ClsEx, and (ii) ClsEx is just the restriction of ClsSep to classes that contain only sets. Cf. A. Blass's review of Muller at MathSciNet (MR1851712), 3rd paragraph from the end. | |
| Apr 20, 2012 at 15:37 | comment | added | Victor Makarov | @wmitt: Thanks for the answer. Could you provide some references to theorems supporting your answer? Does it follow from your answer that there exists a conservative extension $AZ$ of Ackermann set theory which ($AZ$) is also a conservative extension of $ZF$? | |
| Apr 18, 2012 at 18:49 | history | answered | wmitt | CC BY-SA 3.0 |