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}

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