Timeline for NF and incompleteness
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| May 23, 2020 at 8:04 | comment | added | Emil Jeřábek | @ZuhairAl-Johar Yes, the adjunctive set theory is mutually interpretable with $Q$. However, the result about $Q$ is more fundamental. In particular, the second incompleteness theorem refers to $\mathrm{Con}_T$, which is an arithmetical formula: it says that no consistent r.e. theory $T$ can interpret $Q+\mathrm{Con}_T$ (more precisely, $Q+\mathrm{Con}_\tau$ for any $\Sigma_1$ formula $\tau$ that defines an axiom set for $T$ in $\mathbb N$). You cannot even formulate this properly for the adjunctive set theory without first fixing a specific interpretation of $Q$ inside the theory. | |
| May 22, 2020 at 18:48 | comment | added | Zuhair Al-Johar | @EmilJeřábek, NoahSchweber, I think since we are speaking about a set theory, its more relevant to refer to the result that if any set theory can interpret Adjunctive set theory (i.e. Empty set + Adjunction) then it would be subject to Godel's incompleteness theorems!!! And of course NF satisfies those. [Note: adjunction is the axiom: $\forall x \forall y \exists z \forall m (m \in z \iff m \in x \lor m=y)$] | |
| May 20, 2020 at 5:59 | comment | added | Emil Jeřábek | @NoahSchweber Yes, it is enough. See Pudlák, Cuts, consistency statement and interpretations, JSL 50 (1985), 423–441. See also Visser, Can we make the second incompleteness theorem coordinate free?. | |
| May 20, 2020 at 3:25 | review | Close votes | |||
| May 26, 2020 at 3:10 | |||||
| May 19, 2020 at 23:18 | answer | added | Zuhair Al-Johar | timeline score: 6 | |
| May 19, 2020 at 19:26 | comment | added | Noah Schweber | @EmilJeřábek Interpreting $Q$ is not enough for the second incompleteness theorem, is it? | |
| May 19, 2020 at 18:09 | comment | added | Emil Jeřábek | NF interprets $Q$, and as such it is subject to Gödel’s first and second incompleteness theorems. | |
| May 19, 2020 at 17:41 | review | First posts | |||
| May 19, 2020 at 18:30 | |||||
| May 19, 2020 at 17:40 | history | asked | PaleChaos | CC BY-SA 4.0 |