In reference to 1961 paper "On Non Computable Functions" by T. Rado.

Motivation - Scott Aaronson's Who Can Name the Bigger Number?.

M is an n-state binary Turing machine. A valid BB-n entry is a set $(M,s)$ where M halts in exactly s steps. $E_n$ is the set of all valid BB-n entries. Since one cannot have both $(M,s_1) \in E_n$ and $(M,s_2) \in E_n$ for $s_1\not=s_2$, Rado concludes that $E_n$ is a subset of all possible n-state binary Turing machines which is finite and hence, $E_n$ is an exceptionally well-defined non-empty finite set. He proceeds to prove that some functions defined over $E_n$ are non-computable.

But because I'm aware of the Halting problem, I am unable to satisfactorily answer the following question : does $E_n$ exist? Let s be the number of steps required by M to halt when started on a blank tape.

Question 1. Can I say s does not exist?

If the answer is yes, then I can say (M,s) does not exist, and hence $E_n$ does not exist. If not, what can I say about s?

Question 2. If I do not want $E_n$ to exist, and in general sets that admit non-computable functions over them to exist, would I need to yank this statement : "every subset of a set is a set" out of my intuition? Is there an axiomatic system where everything that exists is computable?

I was a bit surprised about how easily Rado assumed the existence of $E_n$. When you can't even construct a set, it is not surprising that some functions over it are non-computable. I ask this question in the similar vein as Scott Aaronson asks Succinctly naming big numbers: ZFC versus Busy-Beaver, i.e "mathematical questions that are ultimately about finite processes" which don't include the likes of "CH, AC, the existence of large cardinals".

In reference to 1961 paper "On Non Computable Functions" by T. Rado.

Motivation - Scott Aaronson's Who Can Name the Bigger Number?.

M is an n-state binary Turing machine. A valid BB-n entry is a set $(M,s)$ where M halts in exactly s steps. $E_n$ is the set of all valid BB-n entries. Since one cannot have both $(M,s_1) \in E_n$ and $(M,s_2) \in E_n$ for $s_1\not=s_2$, Rado concludes that $E_n$ is a subset of all possible n-state binary Turing machines which is finite and hence, $E_n$ is an exceptionally well-defined non-empty finite set. He proceeds to prove that some functions defined over $E_n$ are non-computable.

But because I'm aware of the Halting problem, I am unable to satisfactorily answer the following question : does $E_n$ exist? Let s be the number of steps required by M to halt when started on a blank tape.

Question 1. Can I say s does not exist?

If the answer is yes, then I can say (M,s) does not exist, and hence $E_n$ does not exist. If not, what can I say about s?

Question 2. If I do not want $E_n$ to exist, and in general sets that admit non-computable functions over them to exist, would I need to yank this statement : "every subset of a set is a set" out of my intuition? Is there an axiomatic system where everything that exists is computable?

I was a bit surprised about how easily Rado assumed the existence of $E_n$. When you can't even construct a set, it is not surprising that some functions over it are non-computable. I ask this question in the similar vein as Scott Aaronson asks Succinctly naming big numbers: ZFC versus Busy-Beaver, i.e "mathematical questions that are ultimately about finite processes" which don't include the likes of "CH, AC, the existence of large cardinals".

In reference to 1961 paper "On Non Computable Functions" by T. Rado.

Motivation - Scott Aaronson's Who Can Name the Bigger Number?.

M is an n-state binary Turing machine. A valid BB-n entry is a set $(M,s)$ where M halts in exactly s steps. $E_n$ is the set of all valid BB-n entries. Since one cannot have both $(M,s_1) \in E_n$ and $(M,s_2) \in E_n$ for $s_1\not=s_2$, Rado concludes that $E_n$ is a subset of all possible n-state binary Turing machines which is finite and hence, $E_n$ is an exceptionally well-defined non-empty finite set. He proceeds to prove that some functions defined over $E_n$ are non-computable.

But because I'm aware of the Halting problem, I am unable to satisfactorily answer the following question : does $E_n$ exist? Let s be the number of steps required by M to halt when started on a blank tape.

Question 1. Can I say s does not exist?

If the answer is yes, then I can say (M,s) does not exist, and hence $E_n$ does not exist. If not, what can I say about s?

Question 2. If I do not what $E_n$ to exist, and in general sets that admit non-computable functions over them to exist, would I need to yank this statement : "every subset of a set is a set" out of my intuition? Is there an axiomatic system where everything that exists is computable?

I was a bit surprised about how easily Rado assumed the existence of $E_n$. When you can't even construct a set, it is not surprising that some functions over it are non-computable. I ask this question in the similar vein as Scott Aaronson asks Succinctly naming big numbers: ZFC versus Busy-Beaver, i.e "mathematical questions that are ultimately about finite processes" which don't include the likes of "CH, AC, the existence of large cardinals".

In reference to 1961 paper "On Non Computable Functions" by T. Rado.

Motivation - Scott Aaronson's Who Can Name the Bigger Number?.

M is an n-state binary Turing machine. A valid BB-n entry is a set $(M,s)$ where M halts in exactly s steps. $E_n$ is the set of all valid BB-n entries. Since one cannot have both $(M,s_1) \in E_n$ and $(M,s_2) \in E_n$ for $s_1\not=s_2$, Rado concludes that $E_n$ is a subset of all possible n-state binary Turing machines which is finite and hence, $E_n$ is an exceptionally well-defined non-empty finite set. He proceeds to prove that some functions defined over $E_n$ are non-computable.

But because I'm aware of the Halting problem, I am unable to satisfactorily answer the following question : does $E_n$ exist? Let s be the number of steps required by M to halt when started on a blank tape.

Question 1. Can I say s does not exist?

If the answer is yes, then I can say (M,s) does not exist, and hence $E_n$ does not exist. If not, what can I say about s?

Question 2. If I do not want $E_n$ to exist, and in general sets that admit non-computable functions over them to exist, would I need to yank this statement : "every subset of a set is a set" out of my intuition? Is there an axiomatic system where everything that exists is computable?

I was a bit surprised about how easily Rado assumed the existence of $E_n$. When you can't even construct a set, it is not surprising that some functions over it are non-computable. I ask this question in the similar vein as Scott Aaronson asks Succinctly naming big numbers: ZFC versus Busy-Beaver, i.e "mathematical questions that are ultimately about finite processes" which don't include the likes of "CH, AC, the existence of large cardinals".