I’ll be participating in a workshop at the University of Connecticut, Storrs, philosophy department on October 26-27, 2013, on paraconsistent set theory, organized by Graham Priest and JC Beall. I am given to understand that part of the goal is to develop additional or improved model-construction methods, with which one might expand the range of possible behaviors that we know about.
Have you read Zach Weber’s paper on transfinite cardinals in paraconsistent set theory? I am currently reading this paper along with two of his earlier papers on the same topic. I am particularly interested in his ‘proof’ of the falsity of CH in the paraconsistent set theory (ideal set theory with a paraconsistent logic as rules of inference) he espouses in his paper, in particular how this ‘proof’ relates to forcing arguments–especially in regards to the naturalistic account of forcing. Also, because of the paraconsistent logic, this set theory may allow one to consider cardinals ‘beyond’ the Kunen inconsistency. I would be interested to hear your thoughts on this matter. Thanks in advance.
Dear Thomas, Thanks for your useful link. I hope some day one can find a new foundation which preserves main properties of current ZFC-like set theory and allows us to have an infinitely tall large cardinal tree without any upper bound like Kunen inconsistency theorem. I think any such upper bound theorem shows that we have an unnecessary assumption in our foundation like V=L which restricts us to large cardinals below Zero Sharp. As a mythical example the situation seems like Icarus’s tragic effort to go “beyond” the “sun” which was an upper bound for his flight. Now we know that going beyond the sun is possible but not by plumy wings and we should think about missiles and spaceships. Obviously gods of set theory want some immolation to allow us passing through their realm, the realm of super large cardinals beyond Kunen inconsistency. In our paper Joel, Emil, Mohammad and me offered them the Axiom of Foundation (AF) as our immolation but it seems they want more! Perhaps one should sacrifice the Axiom of Choice (AC) too!
Has the paper been finished yet?
Dear Thomas, Yes. It is almost done. Joel is adding the last editorial notes in order to prepare the submittable version. Are you interested in non-well founded set theory? I am really interested in such weird sets! But unfortunately I had no time (or chance) to explore this realm of set theory before. This paper shows that this kind of foundation can have some surprising effects on main problems of main stream set theory. No body knows what other interesting pictures one can see if he/she looks by a non-well founded paradigm to the main open problems of set theory.
Yes Ali, I most certainly interested in non-wellfounded set theory. I am also interested in whether ZF + ‘some version of Determinateness’ can yield cardinals beyond the Kunen inconsistency. What has been done in this area, if anything? Also, is the preprint on arXiv yet?
Dear Thomas, I didn’t see any related subject yet. Perhaps I cannot see the relevance of the subjects yet! I will announce you about anything which I find useful and interesting for both of us via email. Please let me know about any philosophical or mathematical lecture or paper around the non-well founded set theory and its relevance to Kunen inconsistency theorem or any other deep problem of main stream set theory. I think the “project Icarus” for “going beyond inconsistency” is at least a reasonable “set theoretic dream” and a valuable path to pass.
Please send me an email to the following address:
I haven’t read that paper yet. Would you have a link?
Look at: sites.google.com/site/doctorzachweber. Go to publications and look under the 2012 publications. You will find the article there.
Is Zach Weber going to be at the conference?
I saw his name on a list of participants, so I expect he’ll be there.
How was the conference? Did Weber have anything of interest to contribute?
Professor Hamkins: What was your opinion of the conference? Do you believe that paraconsistent set theory (based on ideal set theory) offers the additional and improved model constructions that the organizers promised (promised might be too strong a term….)?
I enjoyed the conference very much, and it seems that some new model-construction methods came out of it. We are currently writing up an account of them.
I’ll be looking forward to the preprint.
I’m currently thinking a lot about model theory for paraconsistent set theory (of the type described by Weber) and I stumbled across this thread. Did anyone make any progress in writing anything about this topic? I’m particularly interested in algebraic valued models, which seems like a natural approach in this setting. Specifically, it looks like some kind of de morgan monoid valued model might be what’s needed.
I believe that Graham Priest and JC Beale have prepared a document, and also Benedikt Löwe and Sourav Tarafder has an article “Generalized algebra-valued models of set theory” that is forthcoming in the Review of Symbolic Logic, specifically about paraconsistent algebra-valued models of set theory, using a similar method to that which we had discussed at the workshop.
Thanks so much. That’s very helpful.