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The Feferman-Schütte ordinal, $\Gamma_0$

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.

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Veblen hierarchy

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

Proof When $f$ is increasing, $f(\alpha)\geq \alpha$ for all $\alpha$; when also continuous,

$$ f ( \cup_n f^n (\alpha + 1)) = \cup_n f^n (\alpha + 1) $$ is a fixed point greater than $\alpha$

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

(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:

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 \(0<\alpha<\Gamma_0=\min\{\beta|\varphi(\beta,0)=\beta\}\)

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

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 \(\alpha\)

The Feferman-Schütte ordinal, \(\Gamma_0\) is the least ordinal not in \(T\).

Gamma function

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


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