Timeline for Interpreting peano arithmetic without parameters
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2016 at 21:47 | comment | added | Carl Mummert | It seems to me that if I can interpret a theory into a model without using parameters, then I should be able to interpret the theory into the theory of the model - that is, I can interpret the original theory into every elementarily equivalent model. But if I interpret a theory into a model using parameters, that doesn't seem to tell much about interpreting the theory into other models as well. @Matt Brin | |
| May 28, 2016 at 21:07 | comment | added | Matt Brin | I am confused. arxiv.org/abs/math/0701748 proves that PA can be interpreted in Thompson's group F with two specific "parameters" (two named and specific elements of F, sounding like constants to me) but arxiv.org/abs/0807.1079 does the same "without parameters." I am guessing that the second accomplishes its task not only without parameters, but without constants. I am then curious as to what I can conclude from the second paper that I cannot conclude from the first. | |
| May 27, 2016 at 21:27 | comment | added | Joel David Hamkins | There can't be anything like that, since we can eliminate the need for parameters simply by assigning them names, with constants in the language, and now they don't count as parameters since they are definable. This won't seem to change the strength of anything. | |
| May 26, 2016 at 21:31 | comment | added | Matt Brin | @Joel: you have clarified extremely well the notion (and I sorely needed that clarification), but is there anything you can add about consequences? If I am told that PA can be interpreted in X with parameters, but that it can be interpreted in Y without parameters, does that say something strong about Y as compared to X. | |
| May 25, 2016 at 19:47 | comment | added | Joel David Hamkins | Gerhard, let me encourage you to post your own answer, since I'm not sure that I follow exactly what you have in mind. | |
| May 25, 2016 at 19:37 | comment | added | Gerhard Paseman | Can you expand on this with a brief review of types (towards an Omitting Types theorem)? It also may not answer the question, but it may relate well to the user's purpose. Gerhard "Remembers Some Use Of Parameters" Paseman, 2016.05.25. | |
| May 25, 2016 at 19:06 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |