Plus notation is an ordinal notation which I developed. It is relatively powerful, as its countable limit is definitely greater than the Countable limit of Extended Buchholz's function, yet I have not yet fully formalized it. In this post, I will explain it. Despite me creating this, I don't understand formalizing an ordinal notation that well (I don't understand formal strings, cofinality, etc that much), so if anyone who is experienced in ordinal notations such as kanrokoti can help me in the comments, it will be much appreciated and I will put it in a new blog post.
Overview[]
This notation is an ordinal notation which I created a few weeks ago, inspired by Buchholz hydras.
Rules[]
A valid expression in plus notation must be non-empty, start with a plus, and only consist of the following symbols:
- Zero \(0\)
- Plus \(+\)
- Brackets \((\) and \()\)
- Star \(*\)
- Caret \(\textrm{^}\), note that this does not represent exponentiation
- Angle Brackets \(\langle\) and \(\rangle\)
Zero represents, well, zero, of course. Plus represents the beginning of an expression, so if there are multiple pluses in an expression, it denotes that one expression is nested in another. Brackets are included to remove potential ambiguity, among other purposes as you will see. Stars are special function symbols that are required to represent certain ordinals. Caret means that Extended Buchholz's function is used instead of regular Buchholz's function. Lastly, angle brackets mean nesting expressions. Its complicated, but the explanation is in the following list.
So, here is a list of how \(v(E)\) is calculated, where \(E\) denotes a valid expression and \(v(E)\) denotes the value of the expression (i.e. the ordinal represented by the expression):
- \(E = + \rightarrow v(E) := 0\)
- \(E = +(\underbrace{000...0}_x) \rightarrow v(E) := x\) for non-negative x
- \(E = +\underbrace{000...0}_x \rightarrow v(E) := \omega \uparrow \uparrow (x-1)\) for non-zero x
- \(E = +\underbrace{000...0}_x(\underbrace{000...0}_y) \rightarrow v(E) := \omega \uparrow \uparrow (x-1) + x-1\) for non-negative x and y
- \(E = +\underbrace{000...0}_x(0*F) \rightarrow v(E) := (\omega \uparrow \uparrow (x-1))^{v(F)}\) for non-negative x and a subexpression of E, F (added the comma to avoid ambiguity or confusion)
- \(E = +(F)(G) \rightarrow v(E) := \psi_{v(F)}(\psi_{v(G)}(0))\), where \(\psi\) represents Buchholz's function and F and G are subexpressions of E.
- \(E = \textrm{^}+(F)(G) \rightarrow v(E) := \psi_{v(F)}(\psi_{v(G)}(0))\), where \(\psi\) represents Extended Buchholz's function and F and G are subexpressions of E.
- \(E = +F\textrm{&}\langle G \rangle H \rightarrow v(E) := v(+F\underbrace{\textrm{&&&...&}}_{v(G)}H)\), where F, G and H are a subexpressions of E and \(\textrm{&}\) is a certain symbol such as \(+\) or \(0\).
Examples[]
- \(+ = 0\)
- \(+0 = 1\)
- \(+(00) = 2\)
- \(+00 = \omega\)
- \(+0(00) = \omega + 1\)
- \(+0(00)(0) = \omega 2\)
\(\omega^2\) cannot be expressed like this, and therefore we need to use the special function symbol \(*\). With this, \(\omega^2 = +0(0*+(00))\). Next we have:
- \(+000 = \omega^\omega\)
- \(+0+0 = \varepsilon_0\)
- \(+0(+0+0) = \varepsilon_1\)
- \(+0(+0)0 = \varepsilon_\omega\)
- \(+0+0+0 = \zeta_0\)
- \(+0+0+0+0 = \Gamma_0\)
- \(+0+0+0(+0)0 = \mathsf{SVO}\)
- \(+0+0+0+0+0 = \mathsf{LVO}\)
- \(+0+(00) = \mathsf{BHO}\)
- \(+0(+00) = \mathsf{BO}\)
- \(+0+0(00) = \mathsf{TFBO}\)
- \(+0(+000) = \mathsf{PTO(ID_{<\omega\textrm{^}\omega})}\)
- \(+0(+0+0) = \mathsf{PTO(ID_{<\varepsilon_0})}\)
- \(\textrm{^}+0++0 = \mathsf{PTO(Aut(ID))}\)
- \(\textrm{^}+0+\langle +00 \rangle 0 = \mathsf{EBO}\)
- \(\textrm{^}+0+((\langle +00 \rangle 0)0) = \mathsf{PTO(KPI)}\)?
- \(\textrm{^}+0+(\langle +00 \rangle 0)0 = \mathsf{PTO(ML_1W)}\)?
Classification[]
I develop classes based on plus notation to help categorize infinite ordinals. The descriptions might be a bit confusing, but I couldn't think of anything better.
- Lowest: Any number which is defined in plus notation as a string of 0s within brackets, i.e. finite numbers. The supremum of this class is \(\omega\).
- Class 1: Any number which is defined in plus notation as a string of 0s without brackets, and all numbers in between i.e. numbers which don't require uncountable arguments in Buchholz's psi function. The supremum of this class is \(\varepsilon_0\).
- Class 2: Any number which is defined in plus notation with a maximum of two pluses (excluding BHO and higher numbers), i.e. numbers which can be defined using the binary Veblen function. The supremum of this class is \(\Gamma_0\).
- Class 3: Any number which is defined in plus notation with any finite amount of pluses (excluding BHO and higher numbers), i.e. numbers which can be defined with finite tetration of uncountable arguments in Buchholz's psi function. The supremum of this class is the Bachmann-Howard ordinal \(+0+(00)\).
- Class 4: Any number which is defined in plus notation with a maximum of two pluses, where \(v(G)\) is finite. The supremum of this class is the Buchholz ordinal \(= +0+00\).
- Class 5: Any number which is defined in plus notation with a maximum of two pluses, where \(v(G)\) is countable. The supremum of this class is \(\textrm{^}+0++0\)
- Class 6: Any number which is defined in plus notation with a maximum of two pluses, where \(v(G)\) is not inaccessible. The supremum of this class is the Extended Buchholz ordinal \(\textrm{^}+0+\langle +00 \rangle 0\).
- Class 7: Any number which is defined in plus notation with a countable quantity of pluses, usually angle brackets are used. The supremum of this class is \(\textrm{^}+0+(\langle ++0 \rangle 0)0\)
- Highest: Any number which is defined in plus notation which an uncountable quantity of pluses. or \(v(E)\) itself is uncountable.
I wonder where the (countable) limit of this system would lie and how it would compare to other large countable ordinals.