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Note: My new notation is further down the page.

Define \(H\) to be the smallest possible set such that:

\(\\\{(x\rightarrow0), (x\rightarrow1), (x\rightarrow x)\}\subseteq H \\f,g\in H\Rightarrow (x\rightarrow f(x)+g(x))\in H \\f,g\in H\Rightarrow (x\rightarrow x\uparrow^{f(x)}g(x))\in H\)

(\((x\rightarrow f(x))\) notates the function that maps \(x\) to \(f(x)\). The above functions are defined on \(\mathbb{N}_0\).)

\(H\) is the set of hypernomial functions, a set of functions of \(x\). The order type of the hypernomial functions, under eventual domination, is (probably) \(\Gamma_0\).

We can use hypernomial functions to create a simple notation, comprable to SuperJedi224's X-Sequence notation.

Hypernomials

Define Hypernomial Exponential Notation (HEX) as follows:

\(\\c\langle 0\rangle n'\#=c\langle 0\rangle n+x \\X\langle a\rangle 1\#=x \\X\langle a\rangle X\#=(X\langle a\rangle x)' \\c\langle a'\rangle b'\#=c\langle a\rangle c\langle a'\rangle b \\X\langle A\rangle b'\#=X\langle A\#\rangle (X\langle A\rangle b)' \\n\langle A\rangle b\#=n\langle A\#\rangle b\)

Where:

  • \(A\) is a limit; it is not of the form \(a+1\), for any \(a\).
  • \(a\) and \(b\) are anything.
  • \(n\) is a nonnegative integer.
  • \(c\) is a nonnegative integer, or \(X\).
  • \(x\) is the rightmost number.
  • \(\langle a\rangle\) is right-associative.
  • \(+\) is treated as right-associative.
  • \(X\) does not have a value, but could be compared to \(\omega\).
  • \(a'\) is the successor of \(a\), \(a+1\).
  • \(\#\) is a symbol with no special meaning, and is repeatedly appended to the structure. The rules above are then followed in order to reduce the structure, until it is reduced to a single number.


The limit of HEX is \(\Gamma_0\).

We can make a relation between the hypernomial functions defined at the start and the hypernomials by replacing every \(X\) with the base, \(x\), to get a hypernomial function of \(x\). All hypernomial functions are representable in this way (note that \(a\langle b\rangle c=a\uparrow^bc\)).

Analysis

By Example

As a rule of thumb, the following approximation should hold for FGH ordinal conversions: \(X\langle1+\alpha\rangle\omega\beta\approx\varphi(\alpha,\beta)\).

Here, \(K~\alpha\) is shorthand for \(a[K]a\approx. f_\alpha(a)\). We also use the notations for multiplication (\(Xa\)), exponentiation (\(X^a\)), and tetration (\(X\uparrow\uparrow a\)) as shorthand for \(X\langle0\rangle a\), \(X\langle1\rangle a\) and \(X\langle2\rangle a\) respectively, as this is exact for natural numbers.

\(\\ b\sim b+1 \\ X\sim\omega \\ X+k\sim\omega+k \\ X2\sim\omega2 \\ Xk_1+k_0\sim\omega k_1+k_0 \\ X^2\sim\omega^2 \\ \cdots X^2k_2+Xk_1+k_0\sim\cdots \omega^2k_2+\omega k_1+k_0 \\ X^X\sim\omega^\omega \\ X^{X^X+X^43+X2}+X^{X^2}+X^62+X^2+3\sim\omega^{\omega^\omega+\omega^43+\omega2}+\omega^{\omega^2}+\omega^62+\omega^2+3 \\ X\uparrow\uparrow X\sim\varepsilon_0 \\ X\uparrow\uparrow(X+k)\sim\varepsilon_0 \\ (X\uparrow\uparrow X)+1\sim\varepsilon_0+1 \\ (X\uparrow\uparrow X)2\sim\varepsilon_02 \\ X^{X\uparrow\uparrow X+1}\sim\varepsilon_0\omega \\ X^{(X\uparrow\uparrow X)+X}\sim\varepsilon_0\omega^\omega \\ X^{X^{X\uparrow\uparrow(X+1)}}\sim\varepsilon_0^\omega \\ X\uparrow\uparrow(X2)\sim\varepsilon_1 \\ X\uparrow\uparrow(X3)\sim\varepsilon_2 \\ X\uparrow\uparrow(X^2)\sim\varepsilon_\omega \\ X\uparrow\uparrow(X^X)\sim\varepsilon_{\omega^\omega} \\ X\uparrow\uparrow X\uparrow\uparrow X\sim\varepsilon_{\varepsilon_0} \\ X\langle3\rangle X\sim\zeta_0 \\ X\langle4\rangle X\sim\varphi(3,0)\)

(Edit: probably not) To be completed. Sorry!


By Generalisation

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