Timeline for Minimal subset of axioms for ZFC
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Mar 25, 2020 at 18:26 | comment | added | Emil Jeřábek | Having said that, this formulation of the theory also works over logic without equality, with $x=y$ defined as $\forall z\,(x\in z\leftrightarrow y\in z)$. One only has to check that the theory as written implies axioms of equality under this interpretation; in particular, $\forall z\,(x\in z\leftrightarrow y\in z)\to\forall t\,(t\in x\leftrightarrow t\in y)$ follows from separation and power set. | |
| Mar 25, 2020 at 16:37 | comment | added | Emil Jeřábek | I don’t think the formulation of the axiom of infinity matters much, I suppose what you wrote should work. In usual first-order logic, equality is a primitive symbol. | |
| Mar 24, 2020 at 21:11 | comment | added | user76284 | Thanks for your help. To be sure, are these axioms sufficient to interpret ZFC? What's the proper formulation of the axiom of infinity (the last one)? In particular, how should the equals sign be expanded or interpreted, since we lack extensionality? $x = y \equiv \forall z (x \in z \leftrightarrow y \in z)$? Or $x = y \equiv \forall z (z \in x \leftrightarrow z \in y)$? | |
| Mar 22, 2020 at 17:34 | comment | added | Emil Jeřábek | What you call strong collection does imply replacement, as noted there. The standard axiom of collection is the “weak” one. | |
| Mar 22, 2020 at 17:26 | comment | added | user76284 | I mean so-called “strong collection”, as the question there calls it. Does the same hold true? | |
| Mar 22, 2020 at 9:54 | comment | added | Emil Jeřábek | No, as the answer there tells you. Collection together with separation implies replacement. You can make do with $\Delta_0$ separation, but you cannot drop it altogether. | |
| Mar 22, 2020 at 9:02 | comment | added | user76284 | Doesn’t collection imply replacement, and replacement imply separation? | |
| Mar 22, 2020 at 8:51 | comment | added | Emil Jeřábek | With neither replacement nor separation? This should be a very weak theory. In fact, unless I missed something, $V_{\omega+\omega}\cup\{\infty\}$ (with $\infty$ being a universal set) is a model of this theory, including extensionality. In particular, the presence of a universal set implies collection. | |
| Mar 22, 2020 at 6:22 | comment | added | user76284 | What about infinity, union, powerset, and collection, without extensionality? Is that sufficient to interpret ZFC? | |
| Jun 25, 2013 at 14:31 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix markup?
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| Dec 20, 2012 at 16:52 | comment | added | Emil Jeřábek | I see that the definition of the automorphism is incorrect. It should proceed by induction on rank: $$f(\langle i,x\rangle)=\begin{cases}\langle i,x\rangle&\text{if }\langle i,x\rangle\in M_\alpha,\\\langle1-i,\{f(y):y\in x\}\rangle&\text{otherwise.}\end{cases}$$ | |
| Feb 7, 2011 at 18:26 | history | edited | Emil Jeřábek | CC BY-SA 2.5 |
fix markup, clarification
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| Feb 4, 2011 at 18:21 | history | edited | Emil Jeřábek | CC BY-SA 2.5 |
forgot power set
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| Feb 4, 2011 at 16:41 | history | edited | Emil Jeřábek | CC BY-SA 2.5 |
clarification of H_\kappa
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| Feb 4, 2011 at 16:13 | history | answered | Emil Jeřábek | CC BY-SA 2.5 |
