Timeline for Two equivalent statements about formulas projected onto an Ultrafilter
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22 at 13:32 | answer | added | Stanley sun | timeline score: 0 | |
| Jul 16 at 4:33 | vote | accept | Stanley sun | ||
| Jul 8 at 2:27 | answer | added | Alex Kruckman | timeline score: 5 | |
| Jul 7 at 20:50 | review | Close votes | |||
| Jul 12 at 3:06 | |||||
| Jul 7 at 20:45 | comment | added | Stanley sun | Yes, considering that $𝑥_{x(𝑖)}$ must be a bound variable in $ \forall x_{𝑥(𝑖)} 𝑓(𝑖)$, if it appears. | |
| Jul 7 at 20:34 | comment | added | Alex Kruckman | Let me try again to understand. Does $\mu\equiv_{\forall x_{x(i)} f(i)} \nu$ mean that $\mu$ and $\nu$ agree on all the free variables in $f(i)$ except possibly for $x_{x(i)}$? | |
| Jul 7 at 20:25 | comment | added | Stanley sun | I am sorry that my incorrect translation has caused reading difficulties. I translated the Chinese content using gpt, and the effect is not good. I have modified part of it. $ x_x(i)$ is only a variable in the language. It may not be a free variable of $f(i)$, and it may not even appear in $f(i)$. $∀x_x(i)f(i)$ is only the universalization of a variable in the language. $f(i)$ is only a formula, and $f$ is a function $ \{ (i, f(i))| f(i)\text{ is a formula in the language and } i ∈ X \}$ | |
| Jul 7 at 20:13 | history | edited | Stanley sun | CC BY-SA 4.0 |
added 44 characters in body
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| Jul 7 at 17:49 | comment | added | Joel David Hamkins | Your equivalence $\nu\equiv\mu$ seems to be only about same-value of the valuations, not truth of the formula at those valuations, whereas the conclusion of statement 1 depends on the truth of the formula. So isn't this obviously wrong? Perhaps you are missing a hypothesis about $(M,\nu)$? | |
| Jul 7 at 17:46 | comment | added | Joel David Hamkins | Also, is $f(i)$ a formula, or have you valuated one of the variables of formula $f$ at the individual $i$? And you mention $\mu$ and $\nu$ in the hypothesis of statement 1, but only $\mu$ appears in the conclusion, so I am confused about the intended quantification for that. | |
| Jul 7 at 17:14 | comment | added | Alex Kruckman | I'm confused by your notation. For each $i\in X$, $f(i)$ is a formula which has $x_{x(i)}$ as a free variable? Could it have more free variables? Did you really mean to write "the formulas indexed by $\equiv$"? | |
| Jul 7 at 16:00 | history | asked | Stanley sun | CC BY-SA 4.0 |