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Not to be confused with ε function.

$$\varepsilon_0$$ (pronounced "epsilon-zero", "epsilon-null" or "epsilon-nought") is a small countable ordinal, defined as the first fixed point of the function $$\alpha \mapsto \omega^\alpha$$[1].

## Properties

The ordinal $$\varepsilon_0$$ has several properties:

• Smallest ordinal not expressible in Cantor normal form using strictly smaller exponents. Equivalently, this is the least nonzero ordinal that is closed under addition and $$\lambda\alpha.\omega^\alpha$$.
• In fact, $$\varepsilon_0$$ is also the least nonzero ordinal closed under only $$\lambda\alpha.\omega^\alpha$$.
• The proof-theoretic ordinal of Peano arithmetic and ACA0 (arithmetical comprehension, a subsystem of second-order arithmetic).[2]
• Informal visualizations: $$\omega^{\omega^{\omega^\cdots}}$$ or $$\omega \uparrow\uparrow \omega$$
• The second fixed point of $$x\mapsto2^x$$.
• $$\varphi(1,0)$$ using the Veblen function
• $$\psi_0(\Omega)$$ using Buchholz's function
• $$\psi(0)$$ using Madore's function

## Appearance in googology

Using the Wainer hierarchy:

$$f_{\varepsilon_0}(n)$$ with respect to the Wainer hierarchy is comparable to the Goodstein function, Beklevmishev's worm function $$\textrm{Worm}(n)$$, and its variants.

## Higher epsilon numbers and the Veblen hierarchy

The function $$\alpha \mapsto \varepsilon_\alpha$$ enumerates the fixed points of the exponential map $$\alpha \mapsto \omega^\alpha$$. Thus $$\varepsilon_1$$ is the next fixed point of the exponential map after $$\varepsilon_0$$. Formally:

• $$\varepsilon_0=\text{min}\{\alpha|\alpha=\omega^\alpha\}=\text{sup}\{0,1,\omega, \omega^\omega, \omega^{\omega^\omega},...\}$$
• $$\varepsilon_{\alpha+1}=\text{min}\{\beta|\beta=\omega^\beta\wedge\beta>\varepsilon_\alpha\}=\text{sup}\{\varepsilon_\alpha+1,\omega^{\varepsilon_\alpha+1}, \omega^{\omega^{\varepsilon_\alpha+1}},...\}$$
• $$\varepsilon_{\alpha}=\text{sup}\{\varepsilon_{\beta}|\beta<\alpha\}$$ if $$\alpha$$ is a limit ordinal.

This definition gives the following fundamental sequences for non-zero limit ordinals smaller than $$\zeta_0$$ explained later:

• If $$\alpha=\beta_1+\cdots+\beta_{k-1}+\beta_k$$, where $$k$$ is a natural number greater than $$1$$ and $$\beta_1,\ldots,\beta_{k-1},\beta_k$$ are additive principal ordinals greater than $$1$$ satisfying $$\beta_1 \geq \cdots \geq \beta_{k-1} \geq \beta_k$$, then $$\alpha[n]=\beta_1+\cdots+\beta_{k-1}+(\beta_k[n])$$
• If $$\alpha=\omega^{\beta+1}$$, then $$\alpha[n]=\omega^{\beta} \times n$$
• If $$\alpha=\omega^{\beta}$$ and $$\beta$$ is a non-zero limit ordinal which is not an epsilon number, then $$\alpha[n]=\omega^{\beta[n]}$$
• if $$\alpha=\varepsilon_0$$, then $$\alpha[0]=0$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$
• if $$\alpha=\varepsilon_{\beta+1}$$, then $$\alpha[0]=\varepsilon_\beta+1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$
• if $$\alpha=\varepsilon_{\beta}$$ and $$\beta$$ is a non-zero limit ordinal, then $$\alpha[n]=\varepsilon_{\beta[n]}$$

The first fixed point of $$\alpha \mapsto \varepsilon_\alpha$$ is called $$\zeta_0$$ (zeta-zero) or Cantor's ordinal, and $$\zeta_\alpha$$ enumerates the fixed points of $$\alpha \mapsto \varepsilon_\alpha$$.

Since we do not have an infinite number of Greek letters, we generalize this using a series of functions that form the Veblen hierarchy. Each function enumerates the fixed points of the previous one. Formally:

• $$\varphi_0(\alpha) = \omega^\alpha$$
• $$\varphi_\beta(\alpha)$$ is the $$(1+\alpha)$$th fixed point of $$\varphi_\gamma$$ for all $$\gamma < \beta$$

The first ordinal inaccessible through this two-argument Veblen hierarchy is the Feferman–Schütte ordinal.

## Sources

1. D. Madore, A Zoo of Ordinals (p.1). Accessed 2021-05-30.
2. W. Pohlers, Proof theory: The first step into impredicativity, Springer, 2009.