# Difference between revisions of "Feferman-Schütte"

The Feferman-Schütte ordinal, denoted $\Gamma_0$ ("gamma naught"), is the first ordinal fixed point of the Veblen function. It figures prominently in the ordinal-analysis of the proof-theoretic strength of several mathematical theories.

## Veblen hierarchy

Every increasing continuous ordinal function $f$ has an unbounded set of fixed points;

Since the set of fixed points is an unbounded, well-ordered set, there is an ordinal function $\varphi^{[f]}$ listing these fixedpoints; it is in turn increasing and continuous. The Veblen Hierarchy is the sequence of functions $\varphi_\alpha$ defined by

• $\varphi_0 x = \omega^x$
• $\varphi_{\alpha + 1} = \varphi^{[\varphi_\alpha]}$
• for $0 \lt \beta = \cup \beta$, $\varphi_\beta(x)$ enumerates the fixedpoints common to all $\varphi_\alpha$ for $\alpha \lt \beta$

(For $\alpha \lt \beta$, the fixed point sets of $\varphi_\alpha$ are all closed sets, and so their intersection is closed; it is unbounded because $\cup_\alpha \varphi_\alpha(t+1)$ is a common fixed point greater than $t$)

In particular the function $$\varphi_1$$ enumerates epsilon numbers i.e. $$\varphi_1(\alpha)=\varepsilon_\alpha$$

The Veblen functions have the following properties:

• if $$\beta<\gamma$$ then $$\varphi_\alpha(\beta)<\varphi_\alpha(\gamma)$$
• if $$\alpha<\beta$$ then $$\varphi_\alpha(0)<\varphi_\beta(0)$$
• if $$\alpha>\gamma$$ then $$\varphi_\alpha(\beta)=\varphi_\gamma(\varphi_\alpha(\beta))$$
• if $$\alpha=\varphi_\beta(\gamma)$$ then $$\alpha$$ is an additive principal number.

An ordinal $$\alpha$$ is an additive principal number if $$\alpha>0$$ and $$\alpha>\delta+\eta$$ for all $$\delta, \eta<\alpha$$. Let $$P$$ denote the set of all additive principal numbers.

We define the normal form for ordinals $$\alpha$$ such that $$0<\alpha<\Gamma_0=\min\{\beta|\varphi(\beta,0)=\beta\}$$

• $$\alpha=_{NF}\varphi_\beta(\gamma)$$ if and only if $$\alpha=\varphi_\beta(\gamma)$$ and $$\beta,\gamma<\alpha$$
• $$\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n$$ if and only if $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ and $$\alpha>\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$ and $$\alpha_1,\alpha_2,...,\alpha_n\in P$$

Let $$T$$ denote the set of all ordinals which can be generated from the ordinal number 0 using the Veblen functions and the operation of addition

• $$0 \in T$$
• if $$\alpha=_{NF}\varphi_\beta(\gamma)$$ and $$\beta,\gamma \in T$$ then $$\alpha\in T$$
• if $$\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n$$ and $$\alpha_1,\alpha_2,...,\alpha_n\in T$$ then $$\alpha\in T$$

For each limit ordinal number $$\alpha\in T$$ we assign a fundamental sequence i.e. a strictly increasing sequence $$(\alpha[n])_{n<\omega}$$ such that the limit of the sequence is the ordinal number $$\alpha$$

• if $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_k$$ then $$\alpha[n]=\alpha_1+\alpha_2+\cdots+(\alpha_k[n])$$
• if $$\alpha=\varphi_0(\beta+1)$$ then $$\alpha[n]=\varphi_0(\beta)\times n$$
• if $$\alpha=\varphi_{\beta+1}(0)$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\varphi_\beta(\alpha[n])$$
• if $$\alpha=\varphi_{\beta+1}(\gamma+1)$$ then $$\alpha[0]=\varphi_{\beta+1}(\gamma)+1$$ and $$\alpha[n+1]=\varphi_\beta(\alpha[n])$$
• if $$\alpha=\varphi_{\beta}(\gamma)$$ and $$\gamma$$ is a limit ordinal then $$\alpha[n]=\varphi_{\beta}(\gamma[n])$$
• if $$\alpha=\varphi_{\beta}(0)$$ and $$\beta$$ is a limit ordinal then $$\alpha[n]=\varphi_{\beta[n]}(0)$$
• if $$\alpha=\varphi_{\beta}(\gamma+1)$$ and $$\beta$$ is a limit ordinal then $$\alpha[n]=\varphi_{\beta[n]}(\varphi_{\beta}(\gamma)+1)$$

The Feferman-Schütte ordinal, $$\Gamma_0$$ is the least ordinal not in $$T$$. If $$\alpha=\Gamma_0$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\varphi(\alpha[n],0)$$

## Gamma function

The Gamma function is a function enumerating ordinal numbers $$\alpha$$ such that $$\varphi(\alpha,0)=\alpha$$

• if $$\alpha=\Gamma_0$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\varphi(\alpha[n],0)$$
• if $$\alpha=\Gamma_{\beta+1}$$ then $$\alpha[0]=\Gamma_{\beta}+1$$ and $$\alpha[n+1]=\varphi(\alpha[n],0)$$
• if $$\alpha=\Gamma_{\beta}$$ and $$\beta$$ is a limit ordinal then $$\alpha[n]=\Gamma_{\beta[n]}$$

## References

Oswald Veblen. Continuous Increasing Functions of Finite and Transfinite Ordinals. Transactions of the American Mathematical Society (1908) Vol. 9, pp.280–292