Timeline for How many variations can be derived from Gödel's fixed-point lemma?
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24 events
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| Sep 22 at 20:10 | vote | accept | Stanley sun | ||
| Aug 9 at 21:37 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 21:25 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 21:16 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 21:11 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 15:44 | answer | added | Joel David Hamkins | timeline score: 7 | |
| Aug 9 at 15:27 | comment | added | Joel David Hamkins | Smullyan has a whole book on these kinds of generalized fixed points, which covers the kind of case in which you are interested. So your question would seem to be part of that theory. | |
| Aug 9 at 15:24 | comment | added | Joel David Hamkins | @Stanleysun I recommend that you use far more words in your question. Write complete English sentences to explain your ideas, with embedded formalism within those sentences, when necessary, for precise reference. For example: "Let us define the function $f$ by $f(x)=\ulcorner x\urcorner$, where $x$ will be any syntactic item in the language." (If that is indeed what you mean.) And that definition shouldn't be part of the defining property of $\mu$ but rather prior to it. | |
| Aug 9 at 15:15 | comment | added | Joel David Hamkins | @AndrejBauer Oh! He was defining the function $f$, and then saying that $f$ is in $\mu$. Totally didn't get that. | |
| Aug 9 at 14:36 | review | Close votes | |||
| Aug 14 at 3:08 | |||||
| Aug 9 at 13:34 | comment | added | Andrej Bauer | It still doesn't make any sense. It might make sense if you deleted those $\forall \sigma \in F_n$ quantifiers, and you wrote your functions in functional notation. So the first clause would read $(\sigma \mapsto \lceil \sigma \rceil) \in \mu$. | |
| Aug 9 at 13:30 | comment | added | Stanley sun | I am sorry for the trouble caused by my statement. The brackets do represent the Gödel numbers. I tried to say that the domain of $ \mu$ is $F_n$ but it seems that I did not translate this point. I noticed that you meant: If the variable I used in the function $f$ is $x$, it may cause a readability conflict because $x$ is already a variable of $PA$, so I changed it. | |
| Aug 9 at 13:27 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 13:18 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 11:19 | comment | added | Joel David Hamkins | Sorry, I still don't get it. What do the brackets mean for you? Are these Gödel numbers? If $\mu$ is a set of functions, then those numbers are constant functions? Or what? And what do you mean by $f(x)=\ulcorner x\urcorner$, where neither $f$ nor $x$ has been quantified? Is $f(x)$ a function, or the value of the function at $x$ or a term? But $\ulcorner x\urcorner$ is the Gödel number of the variable symbol $x$? Or what? I am unsure of almost every single formal statement in your question. | |
| Aug 9 at 8:49 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 5:34 | comment | added | Stanley sun | What I want to express is not that a certain Gödel code belongs to $\mu$, but a function from formula to code. For the formula of $F_n$, mapping its code, this function belongs to $\mu$ | |
| Aug 9 at 5:29 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 9 at 0:28 | comment | added | Joel David Hamkins | You say that $\mu$ is a set of functions, but then make an assertion about a certain Gödel code being an element of $\mu$. I'm not sure what you are trying to say without further explanation. | |
| Aug 8 at 23:03 | comment | added | Stanley sun | I changed it a bit. I don't know if this expression is correct. I wrote $\mu (x)$ probably because the conclusion is too special and I forgot that variables do not all need to be assigned to the same value. | |
| Aug 8 at 22:59 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 8 at 22:50 | comment | added | Andrej Bauer | Why do you write $\mu(x)$ for a family of functions? What is $\mu$ without $x$? I have trouble parsing the question beacause of all those $x$'s. | |
| Aug 8 at 22:45 | history | edited | Stanley sun | CC BY-SA 4.0 |
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| Aug 8 at 22:36 | history | asked | Stanley sun | CC BY-SA 4.0 |