User blog:Boboris02/Breakthrough! Traditional OCF definition for TON

While I was on holidays I had a lot of time to think about other ways of defining Taranovsky's notation. The problem is that TON is that it's not defined the same way other ordinal collapsing functions are, but rather you are given a lexicographic of strings and ways to compare them. Then you are given requirements for the strings to be valid. Finally, an ordinal function is introduced and is defined from those strings where the lexicographic ordering on a code of strings becomes a normal increasing order on ordinals. The main problem here is that, because the function is defined from lexicographic strings, we cannot be certain that it's well founded - aka that every valid string corresponds to a real ordinal below the limit of TON and vise versa. Since TON may be able to describe ordinals above the PTO of ZFC, it may be impossible to prove that TON is well-founded in ZFC, or any theory it goes beyond.

Here I will intruduce an OCF that may be as well be TON itself, defined without the mention of lexicographic strings and trinary code and all that stuff. Again, for every theory T that TON goes beyond, I cannot prove this OCF to indeed be TON, but there does not seem to be any difference with the functions. Yes, this is a traditional OCF definition, using closure functions and all, but first I need to define some things.

First we need to define standard representations of ordinals standard in TON, without mention of the trinary code. One way to do this is using ordered tuples.

\[\mu((\alpha))=\alpha\iff\alpha\in\text{Ord}\]

\[\mu((\#_1,\#_2,C))=C(\mu((\#_2)),\mu((\#_1)))\]

The first layer of parenthesis is for the function and the inner one is because it's a notation for tuples. That's because it's a function mapping tuples to ordinals.

\[\mu'((\alpha_1,...,\alpha_n))=(\alpha_1,...,\alpha_n)\iff\forall i\leq n: \alpha_i\in\text{Ord}\lor\alpha_i=C\]

Otherwise,

\[\mu'((S_1,...,S_n))=\bigcup_{1\leq i\leq n}\mu'(S_i)\]

Now we define the set of all standard representations with "depth" \(n\) in the \(v\)th system.

\[L^0_v=\{0,\Omega_v\}\]

\[L^{n+1}_v=\{(S_1,S_2,C)|S_{1,2}\in L^n_v\}\]

\[L_v=\bigcup_n L^n_v\]

Finally, we define the unique standard representation of \(\alpha\) in the \(v\)th system as follows:

\[L_v(\alpha)=\mu'(S)\iff S\in L_v\land\mu(S)=\alpha\]

And the standard representation of alpha in the minimal system in which it exists...

\[L(\alpha)=L_{\min\{n|\exists L_n(\alpha)\}}(\alpha)\]

Now a definition of "built from below" based on this definition of standard representation....

\[\alpha<_0\beta\iff\alpha\in\beta\]

\[\alpha<_{k+1}\beta\iff\forall \gamma\in L(\alpha)(\gamma\in\beta\lor\exists\delta(\gamma\in L(\delta)\land\delta<_k\beta))\]

Now we define the set of all standard ordinals in a given system.

\[\text{Stand}_n=\{\alpha|\exists\!\beta\exists\!\gamma(C(\beta,\gamma)=\alpha)\land\Omega_n\in L(\alpha)\}\cup\bigcup_{k<n}\text{Stand}_k\cup\{\Omega_n\}\]

Where \(\Omega_0\) is set to be zero.

\[\text{dom}_n(C)=\{(\alpha,\beta)|\alpha,\beta\in\text{Stand}_n\land\beta<_nC(\alpha,\beta)\land(\beta=C(\gamma,\delta)\iff\alpha\leq\gamma)\}\]

\[\text{dom}(C)=\bigcup_n\text{dom}_n(C)\]

Now we set \(\alpha^E\) to be the least epsilon number above or equal to \(\alpha\)

\[\alpha^E=\min\{\beta|\alpha\leq\beta\land\omega^\beta=\beta\}\]

Finally, once that is done we get to the part you were all waiting for. The closure function for and the definition of this OCF for TON.

\[B^0_0(\alpha)=\alpha\]

\[B^0_{n+1}(\alpha)=\alpha\cap\Omega_n\]

\[B^{n+1}_v(\alpha)=B^n_v(\alpha)\cup\{C(\beta,\gamma)|\gamma\in B^n_v\land\beta\in\alpha\}\]

\[B_v(\alpha)=\bigcup_{n\in\omega}B^n_v(\alpha)\]

\[C(\alpha,\beta)=\min\{\gamma|(\forall v(\Omega_v\leq\alpha):\gamma\notin B_v(\alpha))\land\forall\delta,n(\delta\in\alpha\land n\in\omega\land(\delta,\beta)\in\text{dom}(C)):C^n(\delta,\beta)\in\gamma\}\]

Where \(C^n(\alpha,\beta)\) is defined inductively as \(C^0(\alpha,\beta)=\betaBoboris02 (talk) 14:16, April 14, 2018 (UTC)C^{n+1}(\alpha,\beta)=C(\alpha,C^n(\alpha,\beta))\)

So there you have it! This function should probably make the discussion of whether TON is well-founded irrelevant.