Monotonic FS systems
I don't know how you understand "a monotonic FS system", but in this blog post, I'll use "monotonic" for some special FS systems.
Definition 1: In a fundamental sequence system \(S:\mu\cap\text{Lim}\times\mathbb N\rightarrow\mu\), let \(\lambda\alpha\ge0\).
- If \(\beta\) is a successor ordinal, then let \(\beta=\gamma+1=\lambda[n_1]_S[n_2]_S\cdots[n_k]_S[]_S\) where \(n_k>0\) (\(k\ge0\)). \(\alpha
Are these functions well-defined?
Here are two extensions of subcubic graph function, but I don't know whether they are well-defined.
Graph A is called a graph minor of graph B if A can be obtained from B by these 3 operations:
- Delete an edge;
- Delete an isolated vertex;
- Edge contraction - for an edge ab, delete this edge, and replace vertex a and b by one vertex connecting to the edges which a or b formerly connect to. (If a and b are linked by another edge, it will become a self-loop after edge contraction.)
Graph A is called a topological minor of graph B if A can be obtained from B by these 3 operations:
- Delete an edge;
- Delete an isolated vertex;
- Smoothing - for a (degree=2) vertex a which links to b and c, delete vertex a, and replace edge ab and ac by one edge bc.
For (unlabeled) subcubic graphs, graph m…
General fundamental sequences for OCFs
In definitions of some ordinal collapsing functions (OCFs), there is always a series of \(C_n(\text{some arguments})\) sets. \(C_0(\text{something})\) may contains some argument in the "something", and may contains a "large" ordinal for collapsing; \(C_{n+1}(\text{something})\) is usually obtained from \(C_n(\text{something})\) by applying some operations with some limitations (avoiding "loop" definition); and \(C(\text{something})\) is the union of all \(C_n(\text{something})\). By this structure, we can define general fundamental sequences for OCFs.
Here this "general" definition works on the notation using a weakly compact cardinal. The definition of the ordinal notation is:
- Let \(K\) denote the weakly compact cardinal, \(\Omega_0=0\) and…
Advanced approximation project
Current approximations can be further improved, and this will take large amount of edits, so I start this discussion.
BEAF is not well-defined beyond tetrational arrays. More accurately, \(X\uparrow\uparrow A\&n\) where \(A\) is not a single number (e.g. \(X+1\), \(2X\), \(X^2\), \(X^X\), etc.) and beyond are not well-defined. But \(X\uparrow\uparrow m\&n\) and \(X\uparrow\uparrow X\&n\) are still considered as well-defined.
My suggestion about that is:
- Approximations of numbers defined in ill-defined BEAF should be removed.
- Approximations in ill-defined BEAF of any numbers should be removed.
Growth Rate ε(0)−1
What's the growth rate of Goodstein function? \(\varepsilon_0\)?
No. Unlike Hydra function and Worm function, which are comparable to \(f_{\varepsilon_0}(n)\), Goodstein function has some "cost" - \(G(2\uparrow\uparrow n)\approx f_{\varepsilon_0}(n)\). By the definition of "functional approximation":
- \(f\ge^*g\) if there exists k such that for all n > 0, \(f(n+k)\ge g(n)\)
and
- \(f\approx g\) if \(f\ge^*g\) and \(g\ge^*f\)
Goodstein function is not comparable to \(f_{\varepsilon_0}(n)\). Currently we don't have a compact notation to express the growth rate of it. However, in this blog post I won't consider this thing. Instead, I'll consider what it "costs" and extend it. Goodstein has a 3-level cost under \(\varepsilon_0\), since \(G(f_3(n))\app…