User blog:Tetramur/My thoughts about functions and numbers

I thought about various googological functions and numbers.

Today I have two conjectures:

1. Rayo's function eventually dominates every function represented in FGH with countable ordinal (doesn't matter whether this function is computable or uncomputable), and

2. Rayo's function is on level with \(f_{\omega_1}(n)\) if the appropriate fundamental sequences are given.

But do they exist? I don't know.

Of course, there are some functions which are represented in FGH with countable ordinal and are themselves uncomputable. For example, take some enumeration of TMs and give value 1 to the function if TM #n halts and 0 if it doesn't halt. This function is uncomputable, but it is nowhere within the range of fast-growing functions!

So, my previous definition of weakly/strongly uncomputable functions must be revised. It was as follows:

1. The weakly uncomputable function is the function, if it may be represented with countable non-recursive ordinal in FGH if the appropriate fundamental sequences are given.

2. The strongly uncomputable function is the function, if it may be represented with uncountable ordinal in FGH if the appropriate fundamental sequences are given.

It is generally supposed that ordinary busy beaver's function grows approximately as \(f_{\omega^\text{CK}_1}(n)\) in the FGH associated to Kleene's \(\mathcal{O}\). Now, oracle order-k function. That is, analogue of busy beaver but with an oracle. I conjecture that the oracle order-k function is growing as \(f_{\omega^\text{CK}_{k+1}}(n)\} - again, if the appropriate fundamental sequences are given.

So, I conjecture the Fish Number 4 is \(f_{{\omega^\text{CK}_{(\omega^{\omega+1})*63}}(63)\).

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