User blog:P進大好きbot/Introduction to the Termination of Pair Sequence System

As I wrote a proof of the termination of PSS (i.e. Pair Sequence System restricted to standard forms) for my own formulation compatible with BM1.1, 2, 2.3, 3.1, and 3.２ with respect to 20/8/2018 version of koteitan's classification, I would like to explain how to verify the termination of PSS.

First, choose a formulation of PSS. In order to write down a mathematical proof without intuition-based arguments, we need a mathematical formulation. Writing down 100 examples of the expansion or a rough sketch of an intuitive expansion is not is not sufficient here. What we should write is an explicit expansion rule \((M,n) \mapsto M[n]\) applicable to all pair sequences \(M\) in the system \(X\) of PSS which we verify the termination. If we verify the termination for all pair sequences, then \(X\) is the set of all pair sequences. If we are only interested in pair sequences of standard form, then \(X\) is the set of all pair sequences of standard form. Indeed, I wrote an expansion rule applicable to all pair sequences and hence to those of standard form.

Secondly, choose a partial ordered set \(Y\) which is provably well-founded under axioms which we use, e.g. Buchholz's ordinal notation system, the set of states of Buchholz's hydra, the set of Rathjen's ordinal notation system, the set of states of Verification/Falsification game, and so on. Choosing a partially ordered set which is not provably well-founded or even is undefined is not sufficient here. That is why (notations associated to) UNOCF and Bashicu's OCF are not suitable. Indeed, I used Buchholz's ordinal notation system.

Thirdly, define a map \(t \colon X \to Y\). Again, writing down 100 examples of the value of \(t\) or an intuitive correspondence is not sufficient here. What we should write is an explicit definition of \(t\) applicable to all pair sequences in \(X\). Indeed, I wrote a definition applicable to all pair sequences of standard form.

Fourthly, verify the strict inequality \(t(M[n]) < t(M)\) for all pair seuqence \(M\) in \(X\) which can be expanded. Again, writing down 100 examples of the inequality or an intuitive similarlity of the expasion rule is not sufficient here. This part is the most non-trivial work in the proof of the termination.

After then, the well-foundedness of \(Y\) immediately implies the termination by transfinite induction on \(t(M)\).

There are essentially three distinct strategies in the formulation of PSS.

One strategy is to formulate PSS in a way compatible with known other versions. Indeed, I formulated in a way compatible with BM1.1, 2, 2.3, 3.1, and 3.2. Then the expansion rule becomes the familiar one, but is completely different from the system of fundamental sequences of Buchholz's ordinal notation system. That is why I need dozens of pages to verify the fourth step above (the strict inequality). Maybe this strategy has the greatest impact for others on the proof of the termination because it is applicable to known existing versions.

Another strategy is to formulate PSS in a non-trivial and completely new way by improving the original definition of a known other version. One natural formulation which I, koteitan, and maybe many others considered is a direct interpretation of the structure of \(Y\). For example, if we interprete the structure of Buchholz's ordinal notation system into a new expansion rule of pair sequences, then \(t\) automatically satisfies the desired strict inequality. Therefore we do not have to suffer from tiresome arguments on the difference of he familiar expansion rule and known other well-founded system. On the other hand, it is not applicable to known other versions, and hence we obtain less impact on the proof of the termination. The benefit to choose this strategy is that we can investigate further directions to improve BMS.

The other strategy is to formulate PSS in a trivial way. Say, define the expansion rule as \(M[n] = (0,0)\). Then we can easily verify the termination, but obtain no impact for others. The benefit to choose this strategy is that we can state "We have verified the termination of PSS!" without spending our life. Great!