Timeline for Is Extensionality needed for the incompleteness of very weak set theories?
Current License: CC BY-SA 3.0
17 events
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| Mar 15, 2016 at 0:42 | comment | added | Frode Alfson Bjørdal | @Emil Jeřábek An even better account of the result pointed out by Emil is Visser's article CARDINAL ARITHMETIC IN THE STYLE OF BARON VON MÜNCHHAUSEN in RSL, 2(3), pp. 570-589, 2009. | |
| Oct 29, 2015 at 15:06 | comment | added | Frode Alfson Bjørdal | @Emil Jeřábek Thanks, this is very useful to me! | |
| Oct 29, 2015 at 14:52 | comment | added | Emil Jeřábek | It is well known that ST^* (with no additional principles) interprets Robinson's arithmetic, see e.g. Section 3.3 in Albert Visser's "Pairs, Sets and Sequences in First Order Theories". | |
| Oct 29, 2015 at 14:47 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
deleted 12 characters in body
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| Oct 29, 2015 at 13:57 | vote | accept | Frode Alfson Bjørdal | ||
| Oct 29, 2015 at 13:45 | answer | added | Joel David Hamkins | timeline score: 3 | |
| Oct 29, 2015 at 13:32 | history | edited | Frode Alfson Bjørdal | CC BY-SA 3.0 |
The question was made more precise to avoid confusion with incompletenesses that do not appeal to Gödelian style constructions. Cfr. the exchange with Joel David Hamkins below the question.
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| Oct 29, 2015 at 13:29 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Fixed markup for typesetting text in italics.
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| Oct 29, 2015 at 13:28 | comment | added | Frode Alfson Bjørdal | @Joel David Hamkins I see that we crossed each other. Thanks for the further clarifications. | |
| Oct 29, 2015 at 13:27 | comment | added | Frode Alfson Bjørdal | @Joel David Hamkins For instance, a set theory which only has the empty set axiom will obviously be incomplete, but it will not be strong enough to define a Gödelian provability predicate or derive the Gödel-Carnap diagonal lemma. I will edit the question to make these points clear. | |
| Oct 29, 2015 at 13:25 | comment | added | Joel David Hamkins | In that case, your question should be whether the theory supports arithmetic coding, rather than the question of whether it is complete. (The incompleteness theorem is not the only way to prove that a theory is incomplete. For example, we know that ZFC is incomplete, if consistent, because it does not settle CH, and this argument does not use the incompleteness theorem. Similarly, many other theories can be observed to be incomplete in an easy way: the theory of two constant symbols $c,d$, does not settle the question of whether $c=d$, and hence is incomplete.) | |
| Oct 29, 2015 at 13:23 | comment | added | Frode Alfson Bjørdal | @Joel David Hamkins I see that point. However, I want the theory $ST^*$ to be incomplete for Gödelian reasons; more precisely, it is the possibility to define the Gödelian provability predicate and derive the Gödel-Carnap diagonal lemma in $ST^*$ which is the center of my attention. | |
| Oct 29, 2015 at 13:18 | comment | added | Joel David Hamkins | In the usual terminology, a theory $T$ is complete with respect to a language, if for every sentence $\sigma$ in that language, either $T\vdash\sigma$ or $T\vdash\neg\sigma$ (some people also insist that $T$ should be consistent). In this terminology, although Presburger arithmetic is complete in the language of addition, it is not complete in the language of arithmetic, because it neither proves nor refutes the axioms of PA. In your case, are you proposing to change the language of set theory? If not, then your weak theory will be incomplete, whether or not it supports coding. | |
| Oct 29, 2015 at 13:05 | comment | added | Frode Alfson Bjørdal | @Joel David Hamkins The question is whether $ST^*$ is also strong enough to have its own coding so as to have its Gödelian provability predicate. As an example, Presburger Arithmetic is complete, though it will not prove all theorems of the stronger theory Peano Arithmetic. | |
| Oct 29, 2015 at 13:04 | review | Close votes | |||
| Nov 3, 2015 at 3:02 | |||||
| Oct 29, 2015 at 12:52 | comment | added | Joel David Hamkins | Your question confuses me. If ST* is weak, then it will be incomplete, because it will neither prove (nor refute) the axioms of a stronger theory, such as ZFC. | |
| Oct 29, 2015 at 12:41 | history | asked | Frode Alfson Bjørdal | CC BY-SA 3.0 |