User:Bentley Endeavor/Mini-TON

What if TON had only one space for nesting instead of two?


 * \((0)=1\)
 * \(((0))=2\)
 * \((((0)))=3\)
 * \(\color{#0000ff}{\underbrace{(...(}_{n}0)...)=n}\)
 * \(\color{#0000ff}{(n)=n+1}\)
 * \((\Omega)=\omega\)
 * \(((\Omega))=\omega+1\)
 * \((((\Omega)))=\omega+2\)
 * \(((((\Omega))))=\omega+3\)
 * \((\Omega_2)=\omega2\)
 * \((\Omega_3)=\omega3\)
 * \((\Omega_4)=\omega4\)
 * \((\Omega_5)=\omega5\)
 * \(\color{#0000ff}{(\Omega_{\gamma})=\omega\gamma}\)

The strength of this single-slot notation would be a mere \(\omega^2\)! Compare this to the unknown strength of the normally used double-slot notation and it's seen that just one more place of nesting makes a huge difference.

Extension of ordinal hierarchy
What if we kept the single-slot notation as is and tried to extend it for ordinals equal to and greater than \(\Omega_{\omega}\) by allowing the measure of \(\Omega_n\)'s cardinality \(n\) to be represented by the notation itself?


 * \((\Omega_{(\Omega)})=\omega^2\)
 * \((\Omega_{((\Omega))})=\omega^2+\omega\)
 * \((\Omega_{(((\Omega)))})=\omega^2+\omega2\)
 * \((\Omega_{((((\Omega))))})=\omega^2+\omega3\)
 * \((\Omega_{(\Omega_2)})=\omega^22\)
 * \((\Omega_{(\Omega_2)})=\omega^22+\omega\)
 * \((\Omega_{((\Omega_2))})=\omega^22+\omega2\)
 * \((\Omega_{(((\Omega_2)))})=\omega^22+\omega3\)
 * \((\Omega_{(\Omega_3)})=\omega^23\)
 * \((\Omega_{(\Omega_{(\Omega)})})=\omega^3\)
 * \((\Omega_{(\Omega_{(\Omega_2)})})=\omega^32\)
 * \((\Omega_{(\Omega_{(\Omega_3)})})=\omega^33\)
 * \((\Omega_{(\Omega_{(\Omega_{(\Omega)})})})=\omega^4\)

Now at this point the notation is bounded by \(\omega^{\omega}\), and we cannot choose \((\Omega_{\Omega})\) to be equal to \(\omega^{\omega}\) because then \(\omega^{\omega}+\omega\) would have to equal \((\Omega_{(\Omega)})\) and this expression has already been used for \(\omega^2\). Therefore, a new symbol must be introduced, which denotes a second collapsing of the notation in itself, just as the \(\Omega\) denotes a first collapse, and this second collapsing symbol will be \(K\). We may use this to progress even further.


 * \((\Omega_K)=\omega^{\omega}\)
 * \((\Omega_{(K)})=\omega^{\omega}+\omega\)
 * \((\Omega_{((K))})=\omega^{\omega}+\omega2\)
 * \((\Omega_{(((K)))})=\omega^{\omega}+\omega3\)

Just as \(\Omega_2\) follows \(\Omega\) there is a progression of \(K\) to \(K_2\) to \(K_3\) and so forth.


 * \((\Omega_{K_2})=\omega^{\omega}+\omega^2\)
 * \((\Omega_{K_3})=\omega^{\omega}+\omega^22\)
 * \((\Omega_{K_{(\Omega)}})=\omega^{\omega}+\omega^3\)
 * \((\Omega_{K_{(\Omega_2)}})=\omega^{\omega}+\omega^32\)
 * \((\Omega_{K_{(\Omega_{(\Omega)})}})=\omega^{\omega}+\omega^4\)
 * \((\Omega_{K_{(\Omega_{(\Omega_{(\Omega)})})}})=\omega^{\omega}+\omega^5\)
 * \((\Omega_{K_{K}})=\omega^{\omega}2\)
 * \((\Omega_{K_{K_{K}}})=\omega^{\omega}3\)
 * \((\Omega_{K_{K_{K_{K}}}})=\omega^{\omega}4\)

Now the notation is bounded by \(\omega^{\omega+1}\), and already the \(K\) symbol has exhausted it's usefulness as a collapsing symbol. So naturally we must denote a third collapsing which follows after \(\Omega\) and \(K\) in a progression of collapsing symbols. This is going to to get complicated if we choose letters of the alphabet so a function will be created \(\Theta_{\alpha}(\beta)\) to denote the \(\alpha\)-th collapsing symbol in our extension of single-slot TON, and specifically the \(\beta\)-th symbol in that particular series of collapsing symbols.

Here are some examples of \(\Theta_{\alpha}(\beta)\):


 * \(\Theta_1(2)=\Omega_2\)
 * \(\Theta_1(5)=\Omega_5\)
 * \(\color{#0000ff}{\Theta_1(\gamma)=\Omega_{\gamma}}\)
 * \(\Theta_2(2)=K_2\)
 * \(\Theta_2(\Omega_3)=K_{\Omega_3}\)
 * \(\color{#0000ff}{\Theta_2(\gamma)=K_{\gamma}}\)
 * \(\Theta_3(1)=K_{K_{K_{._{._{.}}}}}\)

Now that sample evaluations have been shown the continuation of mini-TON is as follows:


 * \((\Omega_{\Theta_3(1)})=\omega^{\omega+1}\)
 * \((\Omega_{\Theta_3(2)})=\omega^{\omega+1}+\omega^2\)
 * \((\Omega_{\Theta_3((\Omega))})=\omega^{\omega+1}+\omega^3\)
 * \((\Omega_{\Theta_3(K)})=\omega^{\omega+1}+\omega^{\omega}\)
 * \((\Omega_{\Theta_3((K))})=\omega^{\omega+1}+\omega^{\omega}+\omega^2\)
 * \((\Omega_{\Theta_3(K_2)})=\omega^{\omega+1}+\omega^{\omega}+\omega^3\)
 * \((\Omega_{\Theta_3(K_K)})=\omega^{\omega+1}+\omega^{\omega}2\)
 * \((\Omega_{\Theta_3(K_{K_K})})=\omega^{\omega+1}+\omega^{\omega}3\)
 * \((\Omega_{\Theta_3(\Theta_3(1))})=\omega^{\omega+1}2\)
 * \((\Omega_{\Theta_3(\Theta_3(\Theta_3(1)))})=\omega^{\omega+1}3\)
 * \((\Omega_{\Theta_4(1)})=\omega^{\omega+2}\)
 * \((\Omega_{(\Theta_4(1))})=\omega^{\omega+2}+1\)
 * \((\Omega_{\Theta_4(\Theta_4(1))})=\omega^{\omega+2}2\)
 * \((\Omega_{\Theta_4(\Theta_4(\Theta_4(1)))})=\omega^{\omega+2}3\)
 * \((\Omega_{\Theta_5(1)})=\omega^{\omega+3}\)
 * \((\Omega_{\Theta_6(1)})=\omega^{\omega+4}\)

If expressions in mini-TON are allowed to appear in the subscript of the Theta function then the notation's limit is not \(\omega^{\omega2}\) but a bit higher:


 * \((\Omega_{\Theta_{\Theta_1(1)}(1)})=\omega^{\omega2}\)
 * \((\Omega_{\Theta_{\Theta_{\Theta_1(1)}(1)}(1)})=\omega^{\omega^{\omega2}}\)

The limit of this extension of Mini-TON is likely \(\varepsilon_0\). Beyond this various schemes to extend it could be devised, but there is double-slot notation so there's really no need.

The Conclusion
Why make a needlessly complicated notation which only gets to \(\varepsilon_0\)? It is an exercise, and also a lesson. The lesson is that sometimes when working in a particular notation you can create however many extensions you want but it will not get you any farther than if you take a step out of everything as a whole and look for the "supreme diagonalization", which in this case would be to add another slot to get normal TON.

Now, if Taranovsky's C (if I'm correct) also has been defined for a higher number of slots then you can see why triple-slotted TON would be so much more powerful than double-slot TON, and quadruple-slotted TON and so on.