## FANDOM

10,843 Pages

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!