Timeline for Strengthening Quine's New Foundations with a more flexible stratification criterion?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 17, 2022 at 5:38 | vote | accept | Dylan Pizzo | ||
| Nov 17, 2022 at 5:38 | comment | added | Dylan Pizzo | Nice! My intuition of the answer is that we could define the "same elements" relation $$x \equiv y \leftrightarrow \forall t (t\in x \leftrightarrow t\in y)$$ Notice that this can can be flexibly stratified with any assignment of values for $x$ and $y$. Thus, the flexibly stratified formula $\exists y (y \equiv x \wedge y \not\in x)$ is a form of Russell's Paradox. In fact, with extensionality, it's exactly the same, but it looks like the paradox goes through even without extensionality, which is really cool! | |
| Nov 17, 2022 at 3:48 | comment | added | bof | Very nice. It might be slightly simpler to say that $x\in y$ is logically equivalent to $$\exists z[x\in z\land\forall w(w\in z\to w\in y)],$$ whence $x\notin x$ is logically equivalent to the "flexibly stratified" formula $$\forall z[x\in z\to\exists w(w\in z\land w\notin x)].$$ | |
| Nov 16, 2022 at 19:38 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| Nov 16, 2022 at 19:06 | history | answered | Greg Kirmayer | CC BY-SA 4.0 |