User blog:P進大好きbot/Analysis of 非想非非想 notation

This is a survey article on my notation, which I call "非想非非想 notation", submitted to a Japanese googological event. It is precisely introduced here (Japanese). I guess that few are interested in precise definition, and hence I just explain how it works.

I found that my notation is not so easy to compare with Rathjen's standard ordinal notation, and hence I am planning to create a new notation extending 非想非非想 notation.

= Notation =

Terms in 非想非非想 notation is strings consisting of letters "\(非\)", "\(剣\)", "\(想\)", "\(緋\)", the parentheses, and the comma. The letter "\(非\)" plays a role of the constant term \(0\), "\(剣\)" plays a role of a \(2\)-ary function encoding Veblen function \(\varphi_{\alpha}(\beta)\) in arithmetic, "\(想\)" plays a role of a \(3\)-ary function regarded analogous to a sequence of OCFs based on higher regular cardinals, and "\(緋\)" plays a role of a \(2\)-ary function analogous to an OCF based on regular cardinals.

Intuitively, \(想\) generates terms analogous to cardinals below a given weakly inaccessible cardinal. weakly inaccessible cardinals below a given weakly Mahlo cardinal, weakly Mahlo cardinals below a given weakly compact cardinal, and so on. I know that there is no "reasonable" sequence of large cardinal axioms extending \((\Omega_{\alpha},I_{\alpha},M_{\alpha},K_{\alpha})\), and hence I do not intend that "\(想\)" precisely corresponds to a sequence of large cardinal axioms. At least, "\(想\)" theoretically works because 非想非非想 notation admits full expansion rules, unlike UNOCF. It is not an ordinal notation associated to an actual OCF, and hence its well-foundedness is an open problem.

= Analysis =

Since 非想非非想 notation is not a notation associated to an actual OCF, it is very difficult to give a precise analysis. Therefore I simply exhibits a table on how terms are expanded and supposed to correspond to ordinals. Here, \(\psi\) denotes Rathjen's standard OCF based on weakly Mahlo cardinal. In order to shorten expressions, I abbreviate \(剣(a,b)\) to \(剣_a(b)\), \(想(非,非,b)\) to \(想(b)\), and \(緋(a,b)\) to \(緋_a(b)\).

Up to Γ_0
In order to shorten expressions, I abbreviate \(剣_{非}(非)\) to \(一\), \(剣_{非}(一)\) to \(万\), and \(想(非)\) to \(億\). Then \(一\) plays a roles of the constant term symbol \(1\), \(万\) plays a role of the constant term symbol \(\omega\), and \(億\) plays a role of the constant term \(\Omega\).

There is nothing significant, because \(剣\) is defined so that it interprets Veblen function in arithmetic.

Up to ψ_{Ω_1}(Φ_1(0))
In order to shorten expressions, I abbreviate \(想(一)\) to \(兆\), \(想(一一)\) to \(京\), \(想(非,a,b)\) to \(想_a(b)\), \(想(一,非,非)\) to \(垓\). Then \(兆\) plays a role of the constant term \(\Omega_2\), \(京\) plays a role of the constant term \(\Omega_3\), \(垓\) plays a role of the constant term \(I\) (the least weakly inaccessible cardinal).

As is shown above, \(緋_{億}\) works in a way completely similar to \(\psi_{\Omega_1}\) if the input is below the first omega fixed point \(\Phi_1(0)\). The last line in the table shows the first difference between \(想_{垓}\) and \(\psi_{\chi_1(0)}\), because \(想_{垓}(非)\) is designed as an analogue of \(\Phi_1(0)\) while \(\psi_{\chi_1}(0)\) is the least fixed point \(\Phi_{\cdot_{\cdot_{\cdot_{\Phi_0(0)}\cdot}\cdot}\cdot}(0)\) of the function \(\alpha \mapsto \Phi_{\alpha}(0)\).

Up to ψ_{Ω_1}(χ_1(0))
In order to shorten expressions, I abbreviate \(想(想_{垓}(垓)一)\) to \(亜\), and \(想_{垓}(b)\) to \(妄(b)\). Since the comparison is complicated, the result might contain many mistakes.

It is very difficult for me to analyse this realm, and hence I quit the analysis here. Instead, I exihibit expansions of rest expressions in the next section.

Up to the limit
In order shorten expression, I abbreviate \(想(一,a,b)\) to \(妄_a(b)\), \(想(一一,a,b)\) to \(幻_a(b)\), and \(想(一一一,a,b)\) to \(夢_a(b)\).

The limit of 非想非非想 notation is \(緋_{億}(想(一一\cdots,非,非))\), i.e. \(緋(想(非,非,非),想(剣(非,非)剣(非,非)\cdots,非,非))\).