Timeline for Ultrainfinitism, or a step beyond the transfinite
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2018 at 17:14 | comment | added | Qfwfq | "One can say infinitely more about this" :D | |
| Jun 30, 2012 at 18:59 | comment | added | Mirco A. Mannucci | it could be that we need to enlarge the multiverse by considering the category of models of alternative set/class theories. | |
| Jun 30, 2012 at 18:53 | comment | added | Mirco A. Mannucci | , haven't we, in the process, killed other venues to entirely new visions of infinity? My guess is yes. I could -allow me a bit of daydreaming-stipulate a theory where ONLY sets have the comprehension axiom, whereas some strange classes don't , thereby killing the diagonal argument that provides the very basis of cardinal arithmetics. My question to you is then this: are you sure that ZF + "some outrageously big cardinal" could model even theories like the one I just alluded to? Perhaps the answer is yes, in which case you ZF multiverse contains all of my "programme", and much more. Or, | |
| Jun 30, 2012 at 18:49 | comment | added | Mirco A. Mannucci | and richness of large cardinal theory. To me, this family is precisely the same as the family of large integers, down to the finite realm. Fascinating, but still relatively tame. A single example: no kappa is equal to its exponential 2^kappa. Or equivalently, no set is iso to its power set. Fine, I know of course why, but cannot we really conceive an algebraic extension of the cardinal family where such a beast would have citizenship? Cantor's TAV, The Absolute Infinity, would certainly have such a property. Yes, TAV was inconsistent, and that is why we have ZF instead of naive set theory. But | |
| Jun 30, 2012 at 18:41 | comment | added | Mirco A. Mannucci | Joel, before I start, thanks again for your clear answer. Always nice to read your prose, you get down to the meat without wasting a single word! Now, I invite you to read my comment to Andreas: I am quite aware that most of what I advocated in this question can be (and has been ! ) carried out within ZF plus large reflexivity/cardinals axioms. That is why I am such a big fan of your multiverse. However, there is something missing there, to which I have alluded in my 3rd question: is there a RADICALLY NEW concept of higher infinity there? My contention is no, in spite of the incredible beauty | |
| Jun 30, 2012 at 15:03 | comment | added | user10290 | I really like how you describe how M is in a sense both smaller and larger than V in an ultrapower embedding. I haven't heard it described like that before. It is so interesting because it is very different than in the case of taking finite ultrapowers, since a finite set always seems just smaller than its finite ultrapower. | |
| Jun 30, 2012 at 0:42 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 6 characters in body
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| Jun 30, 2012 at 0:36 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |