Vtyjkyuk

I have found the following quotes. Quote $1$ ( source ):

In computability theory, Turing Machines+BB oracles correspond to the same ordinal as ordinary Turing Machines ($omega_1^textCK$). In googology, BB oracles correspond to $ omega_1^textCK times 2 $ to the FGH.

(note that “BB oracles” here denote the oracles that can compute the Busy Beaver function for the lower-order Turing machines).

Quote $2$ ( source ):

With access to the halting oracle, you still cannot compute ordinals greater than $ omega_1^textCK $. The set of computable ordinals is, in fact, still the same. However, given an oracle for $ omega_1^textCK $, we can compute larger ordinals, and in fact the ordinals computable from $ omega_1^textCK $ are exactly the ones below $ omega_2^textCK $.

(in this quote, I don’t understand what “an oracle for $omega_1^textCK$” means).

Quote $3$ ( source ):

Adam Goucher admited he was wrong when he first wrote about strength of $Sigma_2(n)$. It is actually $omega_2^CK$, well over $omega_1^CK times 2$.

(note that $Sigma_2(n)$ here denotes the Busy Beaver function for the second-order oracle Turing machines, that is, Turing machines equipped with an oracle that can compute the Busy Beaver function for the first-order Turing machines).

It seems like Quote $3$ contradicts Quote $1$, and the question is: is it possible (if yes, then how?) to construct a model of Turing machines that correspond to $ omega_n^textCK $ in computability theory, assuming that $n$ can be extended to any natural number greater than $1$? What function would the oracles of such machines compute?

EDIT

Quote $4$ ( source ):

The first two admissible ordinals are ω and $omega _1^mathrm CK $ (the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal.

By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal, but for Turing machines with oracles.

Can anyone explain how exactly such construction is done? I cannot find any accessible explanation online.

There are relatively similar questions, but they do not address the described problem:

• 
  Is there a second Church-Kleene ordinal?


• What classification of countable ordinals above $omega _1^mathrm CK $ exists?

This really should be a comment, but probably too long for it. Regarding [Quote2], I think it follows from a general and rather well-known result. Let $A subseteq mathbbN$ be any set such that $Ain HA$ (HA=hyperarithmethic). Then you can't generate $omega_CK$ with a program that has access to the set $A$. If you denote $H$ as halt-set then because $H in HYP$, one gets the result mentioned in the first half of [Quote2].

I do not have any familiarity with the result personally though (it was mentioned in the first question I asked an year ago).

Also regarding the second half of [Quote2], since you mentioned you don't understand what an "oracle for $omega_CK$ means", here are few comments that might help. I am not good with formal stuff so I hope there isn't an issue in the wording. But formally I think it means having an access to function(or an equivalent set basically) which represents the well-order relation ...... corresponding to the well-ordering of $omega_CK$ in terms of $mathbbN$.

For example, if you defined a function $LE:mathbbN^2 rightarrow mathbbN$ so that:

$LE(x,y)=1$ if and only if $x le y$

then $LE$ represent the well-order relation ..... corresponding to well-ordering of $mathbbN$ with order-type $omega$.

Another example is:

$LE(x,y)=1$ if $x=y$

If $x ne y$ then:

$LE(x,y)=$ truth value of $x<y$ ---- if $x$ is even and $y$ is even

$LE(x,y)=1$ ---- if $x$ is even and $y$ is odd

$LE(x,y)=0$ ---- if $x$ is odd and $y$ is even

$LE(x,y)=$ truth value of $x<y$ ---- if $x$ is odd and $y$ is odd

If you look at it carefully enough, $LE$ here represents the well-order relation corresponding to well-ordering of $mathbbN$ with order-type $omega cdot 2$.

Similarly you can also describe a well-ordering of $mathbbN$ with order-type $omega^2$ using a pairing function (a function describing a 1-1 correspondence between $mathbbN^2$ and $mathbbN$).

Now coming back to the comment in second half of [Quote2]. If you denote $alpha=omega_CK=omega^CK_1$ and denote, for example, $beta$ as the smallest ordinal that can't be generated using a program that has access to the well-order relation describing the well-ordering of $omega_CK$ in terms of $mathbbN$. Then I hope you can easily see why the following should all be true (via a positive demonstration of a program that does it):

$beta > alpha cdot 2$

$beta > alpha ^ 2$

$beta > alpha ^ alpha$

$beta > gamma=supalpha, alpha^alpha, alpha^alpha^alpha,..... $

this goes on...