Dtjytyk

The Veblen hierarchy is usually defined with $varphi_0(x) = omega^x$. As a result, we can define the Feferman–Schütte ordinal as the first fixed point of the function $varphi_alpha(0) = alpha$.

I am wondering if it is possible to exceed this by looking at fixed points for not just the Veblen function index, but simultaneously for the exponentiation base that the hierarchy is built on, as well as the argument. Let's define the following function:

$Phi_{alpha,beta}(x) = varphi_{beta}(x)$ such that $varphi_0(x) = alpha^x$

Or, in other words, the function $Phi_{alpha,beta}(x)$ is just the $beta$'th function of the Veblen hierarchy built on the base $varphi_0(x) = alpha^x$.

Given that, here are the questions:

Question 1: does a fixed point for the function $Phi_{x,x}(0) = x$ exist? If so, what is the first one, and is it larger than $Gamma_0$?

Question 2: if we let the argument change as well, no fixed points exist for $Phi_{x,x}(x) = x$ (since none exist for $Phi_{x,0}(x) = x^x neq x$). However, is there any value $x < Gamma_0$ for which $Phi_{x,x}(x) geq Gamma_0$? If so, what is the first such $x$?

Question 1: If we include finite ordinals, then the only solution is $Phi_{1,1}(0) = 1$. (and $Phi_{0,0}(0) = 0$ if one chooses to set $0^0 = 0$) For $alpha > 1$, the fixed points of $Phi_{alpha,0}(x) = alpha^x$ cannot be $alpha^0$ or $alpha^1$, so they must be at least $alpha^2$. In particular, they will be larger than $alpha$, and so $Phi_{alpha,alpha}(0)$, being a fixed point of $alpha^x$, will be larger than $alpha$ as well.

Question 2: Suppose $alpha > 1$, and $beta$ is a fixed point of the function $f(x) = alpha^x$. As above, we see that $beta > alpha$. Also, $beta = alpha^beta ge 2^beta ge beta$, so $beta = 2^beta$. Clearly $beta$ cannot be finite, and if $beta ge omega+1$ then $beta ge sup {omega+1, 2^{omega+1}, 2^{2^{omega+1}}, 2^{2^{2^{omega+1}}},cdots} = sup {omega+1, omega cdot 2, omega^2, omega^omega, omega^{omega^omega},cdots} = varepsilon_0$. Similarly, if $beta ge varepsilon_{gamma}+1$, then $beta ge varepsilon_{gamma+1}$. So $beta$ must belong to the class ${omega} bigcup {varepsilon_beta}_{beta in text{Ord}}$. Going the other direction, if $beta$ is a member of ${omega} bigcup {varepsilon_beta}_{beta in text{Ord}}$, and is larger than $alpha$, then $alpha beta = beta$, so $beta le alpha^beta le (2^alpha)^beta = 2^{(alpha beta)} = 2^beta = beta$, so $alpha^beta = beta$. Hence the members of the class ${Phi_{alpha,1}(x) | x in text{Ord}}$ are the members of the class ${omega} bigcup {varepsilon_beta}_{beta in text{Ord}}$ that are larger than $alpha$. So for $alpha$ infinite, $Phi_{alpha,1}(x) = varphi_1 (gamma + x)$ for some ordinal $gamma$ with $gamma le alpha+1$. (We need $alpha+1$, since, for instance, $Phi_{varphi_2(0),1}(0) = varphi_1(varphi_2(0) + 1)$.)

We can now use induction: Given that $Phi_{alpha,beta}(x) = varphi_beta(gamma+x)$ for some $gamma le alpha+1$, we see that for any $zeta$, $Phi_{alpha,beta+1}(zeta)$ satisfies $x = varphi_beta(gamma+x)$, and hence is larger than $gamma$. Also, $x = varphi_beta(gamma+x) ge varphi_beta(x) ge x$, so $x$ is a fixed point of $varphi_beta$. Going the other direction, if $x$ is larger than $gamma$ and is a fixed point of $varphi_beta$, then $x$, being an $varepsilon$-number, satisfies $gamma+x = x$, so $x = varphi_beta(x) = varphi_beta(gamma+x)$. So the fixed points of $varphi_beta(gamma+x)$ are precisely the fixed points of $varphi_beta(x)$ that are greater than $gamma$, and so $Phi_{alpha,beta+1}(x) = varphi_{beta+1}(gamma' + x)$, for some $gamma' le gamma+1$, with strict inequality when $gamma = alpha+1$; hence $gamma' le alpha+1$.

Furthermore, suppose $beta$ is a limit ordinal, and suppose, for all $delta < beta$, we have $Phi_{alpha,delta}(x) = varphi_delta(gamma_delta + x)$ for some ordinals $gamma_delta le alpha+1$. We see that $Phi_{alpha,beta}(x)$ must be greater than $alpha$, and also, since it is a fixed point of $varphi_delta$ for all $delta < beta$, must be of the form $varphi_beta(chi)$. Going the other direction, if $x$ is greater than $alpha$ and of the form $varphi_beta(chi)$, then $x$, being an $varepsilon$-number, satisfies $gamma_delta + x = x$ for all $delta < beta$. So for any $delta < beta$, $varphi_delta(gamma_delta + x) = varphi_delta(x) = x$, and so $x$ is of the form $Phi_{alpha,beta}(chi')$. So $Phi_{alpha,beta}(x) = varphi_beta(gamma_beta + x)$ for some $gamma_beta le alpha+1$.

So, by transfinite induction, there exists ordinals $gamma_{alpha,beta} le alpha+1$ such that, for all $alpha, beta, x,$ with $alpha$ infinite, we have $Phi_{alpha,beta}(x) = varphi_beta (gamma_{alpha,beta}+x)$. If $alpha, beta, x < Gamma_0$, then $gamma_{alpha,beta} + x < Gamma_0$, so $varphi_beta(gamma_beta+x) < Gamma_0$.