<p>Every Turing Machine computing(output) real number is encoded as a natural number, namely, admissible numbering. Then what is the set of computable (partially) transcendental numbers? Is it an immune set? A productive set?</p>

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2024-03-16T04:38:30.227Z
I think this is largely subsumed by https://cstheory.stackexchange.com/questions/33406/oracle-decidability-of-algebraic-independence.
Noah Schweber
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2024-03-16T05:39:43.187Z
@NoahSchweberN， I am not sure
XL _At_Here_There
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2024-03-16T11:11:09.923Z
@XL_At_Here_There: The answer linked by Noah observes right from the start that the set you're asking about is $\Pi^0_2$-complete. That's a direct answer to your question. (Noah goes on, though, because the answer can be improved if we re-interpret the question.)
Andrej Bauer
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2024-03-17T02:40:19.673Z
@AndrejBauer, thanks,   I will read the answer carefully.  And maybe Mr. Schweber is unwilling to go on because of some reason. Could you give an answer  if if we re-interpret the question?
XL _At_Here_There
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2024-03-17T09:13:07.487Z
I am confused. You're asking me to copy Noah Schweber's answer from the other question over here.
Andrej Bauer
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2024-03-17T09:48:14.623Z
@AndrejBauer, if the question is re-interpreted, what answer would you give?
XL _At_Here_There
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2024-03-17T10:13:36.800Z
This one: https://cstheory.stackexchange.com/questions/33406/oracle-decidability-of-algebraic-independence
Andrej Bauer
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2024-03-17T10:31:34.610Z
@AndrejBauer, thank you, I have read the answer carefully, and possibly have understood the post. you say let Mr.Schweber go on improving the answer, so I say possibly he is not willing to, and would you....... anyway, thank you for your comments
XL _At_Here_There
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2024-03-17T22:10:05.367Z
@XL_At_Here_There: I only now saw you're a teacher who is not a mathematician in your profile. I apologize for being so curt, I'll take your non-mathematicianess into account henceforth.
Andrej Bauer
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2024-03-18T03:35:51.100Z
@AndrejBauer, about twenty years ago, some theoritical thinking about what I am doing  had lead me to a new definition of Kolmogorov Complexity,  Since then, having learnt part of logic and it's branch, I have more questions about what I am doing in the view of math. So,
XL _At_Here_There
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2024-03-18T03:43:32.763Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/152091/discussion-between-xl-at-here-there-and-andrej-bauer).
XL _At_Here_There
<computability>
By Gödel numbering, is the set of computable(partially) transcendental numbers an immune set, productive set?
XL _At_Here_There
# By Gödel numbering, is the set of computable(partially) transcendental numbers an immune set, productive set?

(Asked by: username_0 on 2024-03-16)

Every Turing Machine computing(output) real number is encoded as a natural number, namely, admissible numbering. Then what is the set of computable (partially) transcendental numbers? Is it an immune set? A productive set?

**Tags:** computability

> **Comments:**
> **username_1**: I think this is largely subsumed by https://cstheory.stackexchange.com/questions/33406/oracle-decidability-of-algebraic-independence.  
> **username_0**: @NoahSchweberN， I am not sure  
> **username_2**: @XL_At_Here_There: The answer linked by Noah observes right from the start that the set you're asking about is $\Pi^0_2$-complete. That's a direct answer to your question. (Noah goes on, though, because the answer can be improved if we re-interpret the question.)  
> **username_0**: @AndrejBauer, thanks,   I will read the answer carefully.  And maybe Mr. Schweber is unwilling to go on because of some reason. Could you give an answer  if if we re-interpret the question?  
> **username_2**: I am confused. You're asking me to copy username_1's answer from the other question over here.  
> **username_0**: @AndrejBauer, if the question is re-interpreted, what answer would you give?  
> **username_2**: This one: https://cstheory.stackexchange.com/questions/33406/oracle-decidability-of-algebraic-independence  
> **username_0**: @AndrejBauer, thank you, I have read the answer carefully, and possibly have understood the post. you say let Mr.Schweber go on improving the answer, so I say possibly he is not willing to, and would you....... anyway, thank you for your comments  
> **username_2**: @XL_At_Here_There: I only now saw you're a teacher who is not a mathematician in your profile. I apologize for being so curt, I'll take your non-mathematicianess into account henceforth.  
> **username_0**: @AndrejBauer, about twenty years ago, some theoritical thinking about what I am doing  had lead me to a new definition of Kolmogorov Complexity,  Since then, having learnt part of logic and it's branch, I have more questions about what I am doing in the view of math. So,  
> **username_0**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/152091/discussion-between-xl-at-here-there-and-andrej-bauer).  

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## 0 answers

