## FANDOM

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Note that for all of this blog post $\lambda$ ranges over the limit ordinals.

I don't provide a method for exponentiation nor tetration for succesor bases. in other words $(\lambda+1)^\beta$ is not something i here provide a sequence for. (Nor do i think it is as trivial to do so at this moment, but i might be wrong!)

I begin with a reminder of how to reach $\lambda^\beta$

A.- for succesor exponents $\lambda^{\beta+1}$ the sequence is $\lambda^\beta*\lambda[1]$, $\lambda^\beta*\lambda[2]$, $\lambda^\beta*\lambda[3]$ ...where $\lambda[n]$ is the nth member in the fundamental sequence of $\lambda$

B.- for limit exponents $\lambda^\beta$ the sequence is $\lambda^{\beta[1]}$,$\lambda^{\beta[2]}$, $\lambda^{\beta[3]}$... where $\beta[n]$ is the nth member in the fundamental sequence of $\beta$

Note that A works even If $\beta$ is a fixed point such that $\lambda^\beta$ -> $\beta$

ε0 is related to tetration by the sequence $^1\omega$,$^2\omega$, $^3\omega$... and can be represented as the uglier but still intuitive $^\omega\omega$

i give instructions on how to express tetration as exponentiation for natural numbers here:

The above Works for natural numbers only, for ordinals substraction is not defined, so the instructions must be changed a bit, since for example $^3\omega = {(\omega^\omega)}^{(\omega^{\omega-1})}$

leads to an undefined result, because it involves ω-1

So the way $^\beta\lambda$ works is this

A.- for ordinals with succesor "heights"

$^{\beta+1}\lambda= {^2(^\beta\lambda)}$

B.- for ordinals with limit "heights"

$^{\beta}\lambda= LIM ( ^{\beta[1]}\lambda, ^{\beta[2]}\lambda, ^{\beta[3]}\lambda, ...)$ where $\beta[n]$ is the nth member in the fundamental sequence of $\beta$

A few Examples

$^3\omega = {^2\omega}^{^2\omega} = {(\omega^\omega)}^{(\omega^\omega)} = \omega^{(\omega^1 * \omega^\omega)} = \omega^{\omega^{1+\omega}} = \omega^{\omega^\omega}$

$^4\omega = {^3\omega}^{^3\omega} = {(\omega^{\omega^\omega})}^{(\omega^{\omega^\omega})} = \omega^{(\omega^\omega * \omega^{\omega^\omega})} = \omega^{\omega^{(\omega+\omega^\omega)}}= \omega^{\omega^{\omega^\omega}}$

$^{n+1}\omega = {^n\omega}^{^n\omega}$

$^{(\omega+1)}\omega = {^\omega\omega}^{^\omega\omega} = {\varepsilon_0}^{\varepsilon_0}$

$^{(\omega+2)}\omega = {^{(\omega+1)}\omega}^{^{(\omega+1)}\omega} = ({\varepsilon_0}^{\varepsilon_0})^{({\varepsilon_0}^{\varepsilon_0})} = {^3\varepsilon_0}$

$^{(\omega2)}\omega = Lim(^{(\omega+1)}\omega , ^{(\omega+2)}\omega , ^{(\omega+3)}\omega , ...) = {\varepsilon_1}$