Recently, we've been having some discussion of Bowers' pentational arrays. Bowers talks about pentational arrays briefly, but specific details are lacking. Here is my attempt at a rigorous definition of pentational array notation.
Definition of pentational structures[]
We define the set \(PE\) of pentational structure expressions as follows.
\(PE\) is the smallest set such that
0, 1, X are in \(PE\). If f and g are in \(PE\), then f + g, f * g, \(X^f\), and \(X \uparrow \uparrow f\) are in \(PE\).
Note that different expressions for pentational structures can express the same pentational structure.
We define the equivalence relation ~ as follows. Given that f and g are in \(PE\):
f + g ~ g + f
f + 0 ~ f
f * g ~ g * f
f * 0 ~ 0
f * 1 ~ f
\(X^0 \sim 1 \)
\(X^1 \sim X \)
\(X\uparrow\uparrow 0 \sim 1 \)
\(X \uparrow\uparrow 1 \sim X\)
\(X^{f + g} \sim (X^f) * (X^g) \)
\(X^{X\uparrow \uparrow f} \sim X \uparrow\uparrow (f+1) \)
~ is the smallest equivalence relation that satisfies the above expressions.
The set \(P\) of pentational structures is defined as the set \(PE\) modulo ~. That is, \(P\) is the set of pentational structure expressions where expressions that are related by ~ are considered the same structure.
Ordering on pentational structures[]
We need to order the pentational structures. For a pentational structure f, define N(f) to be the minimum number of symbols in an expression for f. We will define the comparsion between f and g by induction on max {N(f), N(g)}.
Case 1: 0 < g is true for all g not equal to 0.
Case 2: 1 < g is true for all g not equal to 0 or 1.
Case 3: f < 0 is false for all f.
Case 4: f < 1 is false unless f = 0.
Case 5: At least one of f or g is the sum of two or more expressions.
Let \(f = f_1 + f_2 + \ldots + f_m\) where \(f_1 \ge f_2 \ge \ldots \ge f_m\). (Note that \N(f_i) < \N(f)\) so we can assume by induction that comparisons for the \(f_i\) have already been defined.)
Let \(g = g_1 + g_2 + \ldots + g_n\) where \(g_1 \ge g_2 \ge \ldots \ge g_n\).
\(f < g \Leftrightarrow \exists i (f_i < g_i \wedge \forall j < i (f_j = g_j)) \vee (\forall i \le m (f_i = g_i) \wedge n > m) \).
(In other words, we compare the summands term by term, starting from the largest, until we find a pair that are different. Whichever summand is greater will belong to the greater sum.)
Case 6: At least one of f or g can be written as the product of two or more expressions.
Let \(f = f_1 * f_2 * \ldots * f_m\) where \(f_1 \ge f_2 \ge \ldots \ge f_m\), and no \(f_i)\) is of the form \(X^{a+b}\) where a and b are nonzero.
Let \(g = g_1 * g_2 * \ldots * g_n\) where \(g_1 \ge g_2 \ge \ldots \ge g_n\), and no \(g_i)\) is of the form \(X^{a+b}\) where a and b are nonzero.
Then as before,
\(f < g \Leftrightarrow \exists i (f_i < g_i \wedge \forall j < i (f_j = g_j)) \vee (\forall i \le m (f_i = g_i) \wedge n > m) \).
Case 7: \(f = X^F, g = X^G\)
\(f < g \Leftrightarrow F < G\).
Case 8: \(f = X \uparrow\uparrow F, g = X \uparrow\uparrow G\)
\(f < g \Leftrightarrow F < G\).
Case 9: \(f = X^F, g = X \uparrow\uparrow G\)
If \(F = X \uparrow \uparrow H\), then \(f < g \Leftrightarrow H+1 < G \). Otherwise, \(f < g \Leftrightarrow F < g\).
Case 10: \(f = X \uparrow\uparrow F, g = X^G\)
If \(G = X \uparrow \uparrow H\), then \(f < g \Leftrightarrow F < H+1 \). Otherwise, \(f < g \leftrightarrow f < G\).
This completely defines the comparison relation.
Standard form for pentational structures[]
We define the standard form for a pentational array f.
If f is a nontrivial sum, order the summands from largest to smallest, and express each summand in standard form.
If f is of the form \(X^{g+h}\) express it in the form \((X^g) * (X^h) \), and express g and h in standard form.
If f is a nontrivial product, order the factors from largest to smallest, and express each factor in standard form.
If f is of the form \(X^{X\uparrow\uparrow f}\), express it in the form \(X \uparrow \uparrow (f+1) \), with f expressed in standard form.
Fundamental sequences for pentational structures[]
Given f expressed in standard form, define f[n], the nth element of the fundamental sequence for f, as follows:
If \(f = 0, f[n] = 0\).
If \(f = g+1, f[n] = g\).
If \(f = f_1 + f_2 + \ldots + f_m\), then \(f[n] = f_1 + f_2 + \ldots + f_m[n]\).
If \(f = f_1 * f_2 * \ldots * f_m\), then \(f[n] = f_1 * f_2 * \ldots * f_m[n]\).
If \(f = X^1\), \(f[n] = n \).
If \(f = X^g\), \(f[n] = X^{g[n]} \).
If \(f = X\uparrow\uparrow {g + 1}, f[n] = (X\uparrow\uparrow g)*(X\uparrow\uparrow g)* \ldots *(X\uparrow\uparrow g)\), where there are n terms in the product.
If \(f = X\uparrow\uparrow g\) and g is not of the form h+1, then \(f[n] = X\uparrow\uparrow (g[n])\).
Prime blocks for pentational structures[]
We will use the same definition for the prime blocks as used by FB100Z in his Ordinal BEAF notation.
Call f a successor pentational structure if f is of the form g+1 for some g.
Call f a limit pentational structure if f is neither a successor pentational structure nor 0.
Define \(P_p (\alpha) \), the prime block of \(\alpha\), as follows:
\(P_p(0) = \lbrace \rbrace\).
\(P_p(f + 1) = \lbrace f \rbrace \cup P_p (f) \).
If f is a limit pentational structure, \(P_p (f) = P_p (f[p])\).
Pentational Arrays[]
Having defined prime blocks for pentational structures, we are now ready to define pentational arrays. A pentational array is a map from pentational structures to the natural numbers; we can notate this as
\(\lbrace (\alpha_1, n_1), (\alpha_2, n_2), \ldots, (\alpha_m, n_m) \rbrace\), where the \(\alpha_i\) are increasing. The 0 structure will map to the base number, and the 1 structure will map to the prime number. The pilot P will be the smallest structure greater than 1 mapping to a positive number, and the copilot CP will be the structure such that P = CP+1, if such a structure exists. The definition of BEAF is the usual one.
Hopefully there are no major mistakes, and we have successfully defined pentational arrays.
Relationship between structures and ordinals[]
In the following, A will stand for an arbitrary structure with corresponding ordinal \(\alpha\).
| Structure | Ordinal | |
|---|---|---|
| \(X \uparrow\uparrow X\) | \(\varepsilon_0\) | |
| \(X \uparrow\uparrow X + 1\) | \(\varepsilon_0 + 1\) | |
| \(X \uparrow\uparrow X + A\) | \(\varepsilon_0 + \alpha\) | |
| \(X \uparrow\uparrow X * 2\) | \(\varepsilon_0 * 2\) | |
| \(X \uparrow\uparrow X * A\) | \(\varepsilon_0 * \alpha\) | |
| \((X \uparrow\uparrow X) ^ 2\) | \(\varepsilon_0 ^ 2\) | |
| \((X \uparrow\uparrow X) ^ n\) | \(\varepsilon_0 ^ n\) | |
| \(X \uparrow\uparrow (X + 1) \) | \(\varepsilon_0 ^ {\omega}\) | |
| \(X ^ {X \uparrow\uparrow X} \) | \(\omega ^ {\varepsilon_0 * \omega}\) | |
| \(X ^ {X \uparrow\uparrow X} + A \) | \(\omega ^ {\varepsilon_0 * \omega} + \alpha\) | |
| \(X ^ {X \uparrow\uparrow X} * 2 \) | \(\omega ^ {\varepsilon_0 * \omega} * 2\) | |
| \(X ^ {X \uparrow\uparrow X} * A\) | \(\omega ^ {\varepsilon_0 * \omega} * \alpha\) | |
| \(X ^ {X \uparrow\uparrow X + 1} \) | \(\omega ^ {\varepsilon_0 * \omega + 1} \) | |
| \(X ^ {X \uparrow\uparrow X + A} \) | \(\omega ^ {\varepsilon_0 * \omega + \alpha} \) | |
| \(X ^ {X \uparrow\uparrow X} * X \uparrow\uparrow X\) | \(\omega ^ {\varepsilon_0 * (\omega + 1)} \) | |
| \(X ^ {X \uparrow\uparrow X} * X \uparrow\uparrow X * A\) | \(\omega ^ {\varepsilon_0 * (\omega + 1 + \alpha)} \) | |
| \(X ^ {X \uparrow\uparrow X} * (X \uparrow\uparrow X)^2\) | \(\omega ^ {\varepsilon_0 ^ 2 } \) | |
| \(X ^ {X \uparrow\uparrow X} * (X \uparrow\uparrow X)^n\) | \(\omega ^ {\varepsilon_0 ^ n } \) | |
| \(X ^ {X \uparrow\uparrow X * 2}\) | \(\omega ^ {\varepsilon_0 ^ {\omega} } \) | |
| \(X ^ {X \uparrow\uparrow X * (1 + A}\) | \(\omega ^ {\varepsilon_0 ^ {\omega * \alpha} } \) | |
| \(X ^ {(X \uparrow\uparrow X)^2}\) | \(\omega ^ {\varepsilon_0 ^ {\varepsilon_0} } \) | |
| \(X ^ {(X \uparrow\uparrow X)^n}\) | \(\omega ^ {\omega ^ {\varepsilon_0 * n} } \) | |
| \(X ^ {X ^ {X \uparrow\uparrow X}} \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega}} \) | |
| \(X ^ {X ^ {X \uparrow\uparrow X}} + A \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega}} + \alpha \) | |
| \(X ^ {X ^ {X \uparrow\uparrow X}} * A \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega}} * \alpha \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X}}) ^ n \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega} * n} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X + 1}}) \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega + 1}} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X + A}}) \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega + \alpha}} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X} * X \uparrow \uparrow X}) \) | \(\omega ^ {\omega ^{\varepsilon_0 * \omega + \varepsilon_0}} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X} * (X \uparrow \uparrow X)^n}) \) | \(\omega ^ {\omega ^{\varepsilon_0 ^ n}} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X * 2} }) \) | \(\omega ^ {\omega ^{\varepsilon_0 ^ {\omega} }} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X * ( 1 + A)} }) \) | \(\omega ^ {\omega ^{\varepsilon_0 ^ {\omega * \alpha}}} \) | |
| \((X ^ {X ^ {X \uparrow\uparrow X * A} }) \) | \(\omega ^ {\omega ^{\varepsilon_0 * (\omega * \alpha)}} \) | |
| \(X ^ {X ^ {(X \uparrow\uparrow X)^2}}\) | \(\omega ^ {\omega ^ {\varepsilon_0 ^ {\varepsilon_0} } } \) | |
| \(X ^ {X ^ {(X \uparrow\uparrow X)^n}}\) | \(\omega ^ {\omega ^ {\omega ^ {\varepsilon_0 * n} } } \) | |
| \(X ^ {X ^ {X ^ {X \uparrow\uparrow X}}} \) | \(\omega ^ {\omega ^ {\omega ^{\varepsilon_0 * \omega}}} \) | |
| \(X \uparrow \uparrow (2X) \) | \(\varepsilon_1 \) | |
| \(X \uparrow \uparrow ((A+1) X) \) | \(\varepsilon_(\alpha) \) | |
| \(X \uparrow \uparrow (X \uparrow \uparrow X) \) | \(\varepsilon_{\varepsilon_0} \) | |
| \(X \uparrow \uparrow (X \uparrow \uparrow (X \uparrow\uparrow X)) \) | \(\varepsilon_{\varepsilon_{\varepsilon_0}} \) | |
| \(X \uparrow \uparrow \uparrow X \) | \(\phi_2 (0) \) |