Timeline for Are there interesting examples of theorems proved using ‘height’ extensions?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 21, 2022 at 3:01 | comment | added | Ali Enayat | Urelements plays a crucial role in the "unorthodox" set theory NFU (The extension of Quine's NF, introduced by (Ronald) Jensen); Jensen showed the consistency of NFU relative to a weak fragment of ZF). As shown by Marcel Crabbé, models of NFU that are not models of NF have an abundant number of urelements (as shown in his paper "On the set of atoms", Log. J. IGPL 8 (2000), no. 6, 751–759). | |
| Apr 27, 2022 at 9:13 | comment | added | Neil Barton | To add to Noah's comment. They're also relevant for certain metamathematical purposes. Vann McGee's result that any two universes of second-order ZFCU + ``there is a set of all urelemente'' have isomorphic pure sets depends on there not being `too many' urelements. | |
| Apr 27, 2022 at 3:44 | comment | added | Noah Schweber | @AlecRhea Interestingly, urelements actually become rather important when one dives into the details around weaker set theories. Barwise's book Admissible sets and structures has a discussion of this at the beginning; ZFC is strong enough to elide the issues that crop up here, but sometimes (e.g. in computability theory) weaker theories (e.g. KP) are important and there urelements may play a more significantly simplifying role. | |
| Apr 26, 2022 at 22:09 | vote | accept | Neil Barton | ||
| Apr 25, 2022 at 17:28 | comment | added | Joel David Hamkins | In our paper, we prove essentially that all of the most common urelement theories are bi-interpretable with pure set theories. For example, ZFC with ZFCU + Ord many urelements or ℝ many, KM with KMU+omega many atoms, etc. A many for any class A of pure sets. We take this to explain on structuralist grounds set theory has largely abandoned urelements. Any mathematical structure to be formed in these urelement theories is isomorphic to one in the pure set universe. | |
| Apr 25, 2022 at 7:02 | comment | added | Alec Rhea | Interesting answer, and nice diagrams; out of curiosity, is there any advantage to set theories with urelements vs pure set theories? Do they ever add height in addition to width, or is the width they add ever useful for concisely proving theorems/making definitions in a way that is cumbersome with pure sets? | |
| Apr 24, 2022 at 21:13 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |