Which are the two good Turing Machines that don't have infinitely many Turing Machine Numbers?

On page 70 of his book "The Emperor's New Mind", Sir Roger Penrose lists the first 13 Turing Machines and he gives them the names T₀ through T₁₂. He calls the Turing Machine T₃ the "first respectable machine", because that's the first one that has a STOP instruction. The only other Turing Machine on that list with the STOP instruction is T₁₁. Penrose wrote "Only T₃ and T₁₁ are working Turing Machines, ..."

It turns out that these two Turing Machines--the ones with Turing Machine Numbers 3 and 11--are the only good Turing Machines that don't have infinitely many Turing Machine Numbers.

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You know what! Rather than have me go into some long boring explanation as to why these are the only two good Turing Machines that don't have infinitely many Turing Machine Numbers, how about if you discover it for yourself!

Just remember that to generate more Turing Machine Numbers for the same Turing Machine, all you have to do is insert Optional Leading Zeros into a Turing Machine Number. If you need a review of how to do that, just read this FAQ and this FAQ.

Ok. Before you continue reading, try to insert some Optional Leading Zeros into the Turing Machine Numbers for T₃ and T₁₁. See if you can do it!

Ok. If you just tried to do what I asked you to try, then you found out that you can't insert any Optional Leading Zeros. The reason you can't is because you can only insert Optional Leading Zeros between instructions. But, both of these Turing Machine Numbers have only one instruction encoded into the Turing Machine Number!

For example, T₃ consists of the two instructions 00R and 00STOP. However, the 00R, which is at the start of every Turing Machine, never gets encoded into the Turing Machine Number, so only the 00STOP is encoded (and even then, not the entire instruction). So, the Turing Machine Number for T₃ only has one instruction encoded into it.

The same sort of thing is true for T₁₁

So, for both of these Turing Machines, there is no way you can insert Optional Leading Zeros between instructions because there is only one instruction encoded into the Turing Machine Number.

Furthermore, any good Turing Machine must have a STOP instruction!

So, I guess the only good Turing Machines, for which you can't insert any Leading Optional Zeros, are 00R followed by either 00STOP or 01STOP. Any other good Turing Machine must have one or more additional instructions, and that will allow you to insert Leading Optional Zeros between instructions.

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Now, I am sure that the most astute among you may be asking:

How about if, in place of 00STOP we use instructions such as 10STOP, or 100STOP, or 110STOP, or 1000STOP, or 1010STOP, or 1100STOP, etc? All of these effectively do the same thing as 00STOP.

Or, how about if, in place of 01STOP we use instructions such as 11STOP, or 101STOP, or 111STOP, or 1001STOP, or 1011STOP, or 1101STOP, etc? All of these effectively do the same thing as 01STOP.

Can't we have a Turing Machine consisting of just 00R and one of these STOP instructions? If so, then I guess there are infinitely many Turing Machines that do not have infinitely many Turing Machine Numbers.

I must admit, that you may have a very valid point here.

However, do you think that any Turing Machine having any one of those STOP instructions is a good Turing Machine?

Whether it is a good Turing Machine or not may be a good subject for a debate. However, I myself would argue that they are not good Turing Machines.

The reason is that all of the STOP instructions that you gave are examples of instructions that have something that I decided to call the Non-Zero Ignored Stop State.

When composing any Turing Machine, there is never any good reason to use an instruction with a Non-Zero Ignored Stop State in place of 00STOP or 01STOP.