User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-29915175-20170416151157/@comment-30754445-20170419111206

The drawback is that such notations are of zero practical value. They are of tremendous theoretical importance, but also completely impractical to work with.

Case in point: Can you give me the FS of ε₀ as defined by Kleene's O? Can you even give me the correct m that represents ε₀ in Kleene's O?

We'll first need a way to enumerate the turing machines, but that's not too difficult. We can use following format:

[number of states]-[state 1 instructions]-[state 2 instructions]-....

Where [state x instructions] are formatted like this 1,L,7-1,R,6 (which means "if the tape currently points to '0' then write 1, go left, go to state 7. If the tape currently points to '1' then write 1, go right, go to state 6).

And the order will be lexicographic (with larger numbers coming after smaller ones, R coming after L, and the halting state coming after all other states).

So our enumeration will begin like this:

1: 1-[0,L,1-0,L,1] (for sake of simplicity, we allow machines without a halting state)

2: 1-[0,L,1-0,L,H]

3: 1-[0,L,1-0,R,1]

4: 1-[0,L,1-0,R,H]

5: 1-[0,L,1-1,L,1]

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63: 1-[1,R,H-1,R,1]

64: 1-[1,R,H-1,R,H] (last 1-state machine)

65: 2-[0,L,1-0,L,1]-[0,L,1-0,L,1] (first 2-state machine)

66: 2-[0,L,1-0,L,1]-[0,L,1-0,L,2]

67: 2-[0,L,1-0,L,1]-[0,L,1-0,L,H]

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And so on ad infintum. That's our ordering of the turing machines.

We'll also need a way to define the numerical "output" of a turing machine for a given integer input, but that's easy: An input of n will be represented as n consecutive 1's with the head of the machine pointing to the last "1". The output of the machine will be the total number of (not necessarily consecutive) 1's on the tape when the machine halts.

Given the above, can you tell me what the FS for ε₀ would be? Or even the FS for ω?