Timeline for Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that are weaker than $ Con(T)$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 5, 2018 at 19:41 | vote | accept | Zuhair Al-Johar | ||
| Dec 5, 2018 at 19:40 | comment | added | Zuhair Al-Johar | I see your point, the last formula of yours cannot be deprenexed and since it doesn't contain a proper subformula of it that is a sentence, then it is a simple formula, but by then this would render every formula under the strength of $Con(T)$ being equivalent to a simple sentence, which would be under the strength of $Con(T)$ and of course this kills this method. | |
| Dec 5, 2018 at 13:19 | comment | added | James E Hanson | @ZuhairAl-Johar That was not completely clear to me from the question and your comments. I've edited my answer with a different construction. | |
| Dec 5, 2018 at 13:11 | history | edited | James E Hanson | CC BY-SA 4.0 |
Better example
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| Dec 5, 2018 at 11:57 | comment | added | Zuhair Al-Johar | Simple doesn't simply mean that every proper subformula fails to be a sentence, the other condition must be met also, that upon de-prenxing it you must not get a proper subformula that is a sentence. | |
| Dec 5, 2018 at 11:47 | comment | added | Zuhair Al-Johar | Deprexing means moving in the opposite direction of getting a prenex normal form, so you still can de-prenex a sentence that is not in a prenex normal form! for example the sentence $\forall a (\varphi(a) \wedge \rho)$ this is not in prenex normal form if $\rho$ is a sentence, yet still you can de-prenex it to $\forall a (\varphi(a)) \wedge \rho$. Your sentences can be easily de-prenexed as to get a complex sentence, SO THEY ARE NOT SIMPLE. | |
| Dec 5, 2018 at 7:38 | comment | added | user44143 | @BjørnKjos-Hanssen, the phrase “when de-prenexed” which appears in the question is still not well defined. If you have a clear view of what the question ought to say on this, perhaps you can edit the question appropriately. | |
| Dec 5, 2018 at 3:35 | comment | added | James E Hanson | The sentence I describe isn't in prenex form if the original sentence is not in prenex form. Furthermore every sentence is equivalent to one that is not in prenex form. | |
| Dec 5, 2018 at 3:12 | comment | added | Bjørn Kjos-Hanssen | What about the de-prenexing? Won't that turn your sentence into a complex sentence? | |
| Dec 4, 2018 at 23:08 | history | answered | James E Hanson | CC BY-SA 4.0 |