Epsilon naught, $\epsilon_0$

• Epsilon numbers

The ordinal $\epsilon_0$, commonly given the British pronunciation “epsilon naught,” is the smallest ordinal $\alpha$ for which $\alpha=\omega^\alpha$ and can be equivalently characterized as the supremum

\[\\epsilon\_0=\\sup\\{\\omega,\\omega^\\omega,\\omega^{\\omega^\\omega},\\ldots\\}\]

The ordinals below $\epsilon_0$ exhibit an attractive finitistic normal form of representation, arising from an iterated Cantor normal form involving only finite numbers and the expression $\omega$ in finitely iterated exponentials, products and sums.

The ordinal $\epsilon_0$ arises in diverse proof-theoretic contexts. For example, it is the proof-theoretic ordinal of the first-order Peano axioms.

Epsilon numbers

The ordinal $\epsilon_0$ is the first ordinal in the hierarchy of $\epsilon$-numbers, where $\epsilon_\alpha$ is the $\alpha^{\rm th}$ fixed point of the exponential function $\beta\mapsto\omega^\beta$. These can also be defined inductively, as $\epsilon_{\alpha+1}=\sup\{\epsilon_\alpha+1,\omega^{\epsilon_\alpha+1},\omega^{\omega^{\epsilon_\alpha+1}},\ldots\}$, and $\epsilon_\lambda=\sup_{\alpha\lt\lambda}\epsilon_\alpha$ for limit ordinals $\lambda$. The epsilon numbers therefore form an increasing continuous sequence of ordinals. Every uncountable infinite cardinal $\kappa$ is an epsilon number fixed point $\kappa=\epsilon_\kappa$.