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The Buchholz's psi-functions are a hierarchy of single-argument ordinal functions $$\psi_\nu(\alpha)$$ introduced by Wilfried Buchholz in 1986. These functions are a simplified version of the $$\theta$$-functions, but nevertheless have the same strength as those.

Definition

$$C_\nu^0(\alpha) = \Omega_\nu$$,

$$C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}$$,

$$C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)$$,

$$\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}$$,

where

$$\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \aleph_\nu\text{ if }\nu>0\\ \end{array}\right.$$

and $$P(\gamma)=\{\gamma_1,...,\gamma_k\}$$ is the set of additive principal numbers in form $$\omega^\xi$$,

$$P=\{\alpha\in On: 0<\alpha \wedge \forall \xi, \eta < \alpha (\xi+\eta < \alpha)\}=\{\omega^\xi: \xi \in On\}$$,

the sum of which gives this ordinal $$\gamma$$:

$$\gamma=\gamma_1+\cdots+\gamma_k$$ and $$\gamma_1\geq\cdots\geq\gamma_k$$.

Thus $$C_\nu(\alpha)$$ denotes the set of all ordinals which can be generated from ordinals $$<\aleph_\nu$$ by the functions + (addition) and $$\psi_{\mu\le\omega}(\xi<\alpha)$$.

Properties

Buchholz showed following properties of this functions:

$$\psi_\nu(0)=\Omega_\nu$$,

$$\psi_\nu(\alpha)\in P$$,

$$\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi_\nu(\alpha)<\gamma\}$$,

$$\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1}$$,

$$\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0$$,

$$\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1}$$,

$$\theta(\varepsilon_{\Omega_\nu+1},0)=\psi(\varepsilon_{\Omega_\nu+1})$$ for $$0<\nu\le\omega$$.

Extension

Let me rewrite Buchholz's definition as follows:

$$C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}$$,

$$C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}$$,

$$C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)$$,

$$\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}$$,

where

$$\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \text{smallest ordinal with cardinality}\aleph_\nu \text{ if }\nu>0\\ \end{array}\right.$$

There is only one little detail difference with original Buchholz definition: ordinal $$\mu$$ is not limited by $$\omega$$, now ordinal $$\mu$$ belong to previous set $$C_n$$. For example if $$C_0^0(1)=\{0\}$$ then $$C_0^1(1)=\{0,\psi_0(0)=1\}$$ and $$C_0^2(1)=\{0,...,\psi_1(0)=\Omega\}$$ and $$C_0^3(1)=\{0,...,\psi_\Omega(0)=\Omega_\Omega\}$$ and so on.

Limit of this notation is omega fixed point $$\psi(\Omega_{\Omega_{\Omega_{...}}})$$.

Explanation

$$C_0^0(\alpha)=\{0\} =\{\beta:\beta<1\}$$,

$$C_0(0)=\{0\}$$ (since no functions $$\psi(\eta<0)$$ and 0+0=0).

Then $$\psi_0(0)=1$$.

$$C_0(1)$$ includes $$\psi_0(0)=1$$ and all possible sums of natural numbers:

$$C_0(1)=\{0,1,2,...,\text{googol}, ...,\text{TREE(googol)},...\}$$.

Then $$\psi_0(1)=\omega$$ - first transfinite ordinal, which is greater than all natural numbers by its definition.

$$C_0(2)$$ includes $$\psi_0(0)=1, \psi_0(0)=\omega$$ and all possible sums of them.

Then $$\psi_0(2)=\omega^2$$.

For $$C_0(\omega)$$ we have set $$C_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...\}$$.

Then $$\psi_0(\omega)=\omega^\omega$$.

For $$C_0(\Omega)$$ we have set $$C_0(\Omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(\omega)=\omega^\omega,...,\psi(\omega^\omega)=\omega^{\omega^\omega},...\}$$.

Then $$\psi_0(\Omega)=\varepsilon_0$$.

For $$C_0(\Omega+1)$$ we have set $$C_0(\Omega)=\{0,1,...,\psi_0(\Omega)=\varepsilon_0,...,\varepsilon_0+\varepsilon_0,...\psi_1(0)=\Omega,...\}$$.

Then $$\psi_0(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}$$.

$$\psi_0(\Omega2)=\varepsilon_1$$,

$$\psi_0(\Omega^2)=\zeta_0$$,

$$\varphi(\alpha,1+\beta)=\psi_0(\Omega^\alpha\beta)$$,

$$\psi_0(\Omega^\Omega)=\Gamma_0=\theta(\Omega,0)$$, using Feferman theta-function,

$$\psi_0(\Omega^{\Omega^\Omega})$$ is large Veblen ordinal,

$$\psi_0(\Omega\uparrow\uparrow\omega)=\psi_0(\varepsilon_{\Omega+1})=\theta(\varepsilon_{\Omega+1},0)$$.

Okay, now let's research how $$\psi_1$$ works:

$$C_1^0(\alpha)=\{\beta:\beta<\Omega_1\}=\{0,\psi(0)=1,2,...\text{googol},...,\psi_0(1)=\omega,...,\psi_0(\Omega)=\varepsilon_0,...$$

$$...,\psi_0(\Omega^\Omega)=\Gamma_0,...,\psi(\Omega^{\Omega^\Omega+\Omega^2}),...\}$$ i.e. includes all countable ordinals.

$$C_1(\alpha)$$ includes all possible sums of all countable ordinals. Then

$$\psi_1(0)=\Omega_1$$ first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality $$\aleph_1$$.

$$C_1(1)=\{0,...,\psi_0(0)=\omega,...,\psi_1(0)=\Omega,...,\Omega+\omega,...,\Omega+\Omega,...\}$$

Then $$\psi_1(1)=\Omega\omega=\omega^{\Omega+1}$$.

Then $$\psi_1(2)=\Omega\omega^2=\omega^{\Omega+2}$$,

$$\psi_1(\psi_0(\Omega))=\Omega\varepsilon_0=\omega^{\Omega+\varepsilon_0}$$,

$$\psi_1(\psi_0(\Omega^\Omega))=\Omega\Gamma_0=\omega^{\Omega+\Gamma_0}$$,

$$\psi_1(\psi_1(0))=\psi_1(\Omega)=\Omega^2=\omega^{\Omega+\Omega}$$,

$$\psi_1(\psi_1(\psi_1(0)))=\omega^{\Omega+\omega^{\Omega+\Omega}}=\omega^{\Omega\cdot\Omega}=(\omega^{\Omega})^\Omega=\Omega^\Omega$$,

$$\psi_1^5(0)=\Omega^{\Omega^\Omega}$$,

$$\psi_1(\Omega_2)=\psi_1^\omega(0)=\Omega\uparrow\uparrow\omega=\varepsilon_{\Omega+1}$$.

For case $$\psi(\Omega_2)$$ the set $$C_0(\Omega_2)$$ includes functions $$\psi_0$$ with all arguments less than $$\Omega_2$$ i.e. such arguments as $$0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),..., \psi_1^\omega(0)$$

and then $$\psi_0(\Omega_2)=\psi_0(\psi_1(\Omega_2))=\psi_0(\varepsilon_{\Omega+1})$$.

In general case: $$\psi_0(\Omega_{\nu+1})=\psi_0(\psi_\nu(\Omega_{\nu+1}))=\psi_0(\varepsilon_{\Omega_\nu+1})=\theta(\varepsilon_{\Omega_\nu+1},0)$$.

We also can write:

$$\theta(\Omega_\nu\uparrow\uparrow(k),0)=\psi_0(\Omega_\nu\uparrow\uparrow(k+1))$$ (at least for $$1\le k<\omega; 1\le\nu<\omega$$ it must be true).

Normal form

Previously we wrote in section "Definition" $$\Omega_\nu=1$$ if $$\nu=0$$, but now, not changing the definition of $$\psi$$-function, let's define for sections "Normal form" and "Fundamental sequences" $$\Omega_0=\omega$$ for unification of NF and FS-rules for all cases $$\nu\geq 0$$. Then $$\text{cof}(\Omega_\nu)= \Omega_\nu$$ (and also let's note that $$\text{cof}(s)= 1$$, where $$s$$ is a successor ordinal, $$\text{cof}(0)= 0$$ and $$\Omega_\nu^0=1$$).

The ordinal $$\alpha$$ is an additive principal number ($$\alpha\in P$$) if $$\beta+\gamma<\alpha$$ for all $$\beta,\gamma<\alpha$$

1) If $$\alpha\notin P$$ (i.e. $$\alpha$$ is not additive principal number) then normal form for $$\alpha:$$

$$\alpha=\alpha_1+\cdots+\alpha_k$$ where $$\alpha_1,...,\alpha_k\in P$$ and $$\alpha>\alpha_1\geq\cdots\geq\alpha_k$$ and each $$\alpha_i$$ also is written in normal form, $$i \in \{1,...,k\}$$.

2) If $$\alpha\in P$$ and $$\alpha=\Omega_\nu^\beta \gamma$$ then normal form for $$\alpha:$$

$$\alpha=\Omega_\nu^\beta \gamma$$ where $$\beta<\Omega_\nu^\beta\wedge\gamma<\Omega_\nu\wedge\gamma\in P$$ and $$\nu, \beta, \gamma$$ also are written in normal form.

3) If $$\alpha\in P$$ and $$\alpha=\psi_\nu(\beta)$$ then normal form for $$\alpha:$$

$$\alpha=\psi_\nu(\beta)$$ and $$\nu, \beta$$ also are written in normal form.

If $$\alpha\notin P$$ is an ordinal, such that $$\Omega_\nu<\alpha<\varepsilon_{\Omega_\nu+1}$$ (between $$\omega$$ and $$\varepsilon_0$$ if $$\nu=0$$) then normal form for $$\alpha$$:

$$\Omega_\nu^{\beta_1}\gamma_1+\Omega_\nu^{\beta_2}\gamma_2+\cdots+\Omega_\nu^{\beta_k}\gamma_k$$, where

• $$\alpha>\Omega_\nu^{\beta_1}\gamma_1\geq \cdots \geq \Omega_\nu^{\beta_k}\gamma_k$$,
• $$\text{cof}(\beta_m)\le\Omega_\nu\wedge\beta_m\geq 0\wedge\text{cof}(\gamma_m)<\Omega_\nu\wedge\gamma_m<\Omega_\nu\wedge\gamma_m\in P$$ for $$1\le m \le k$$,

and Cantor normal form is partial case when $$\nu=0$$:

$$\omega^{\beta_1}+\cdots+\omega^{\beta_k}$$, where $$\beta_1\geq\cdots\geq\beta_k$$ and each $$\beta_m$$ can be successor or limit ordinal with cofinality 1 or $$\omega$$.

Fundamental sequences

The fundamental sequence for an ordinal with cofinality $$\Omega_\nu$$ is a distinguished strictly increasing sequence with length $$\Omega_\nu$$, which has the ordinal as its limit. Let $$\alpha\in s$$ denotes $$\alpha$$ is a successor ordinal and $$\alpha\in L_\nu$$ denotes $$\alpha$$ is a limit ordinal with cofinality $$\Omega_\nu$$ (with length of fundamental sequence $$\Omega_\nu$$). For example, $$\alpha\in L_0$$ denotes a limit countable ordinal with cofnality $$\Omega_0=\omega$$.

For ordinals, written in mentioned normal form, fundamental sequences are defined as follows:

1) If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_k$$, where $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k$$ then $$\text{cof}(\alpha)=\text{cof}(\alpha_k)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_k[\eta])$$,

2) if $$\alpha=\psi_\nu(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[n]=\psi_\nu(\beta)\cdot n$$, and note that $$\psi_0(0)=1$$ and $$\psi_\nu(0)=\Omega_\nu$$ for $$\nu>0$$,

3) if $$\alpha=\psi_\nu(\beta)$$ and $$\beta\in L_{\mu\le\nu}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_\nu(\beta[\eta])$$,

4) if $$\alpha=\psi_\nu(\beta)$$ and $$\beta\in L_{\mu+1>\nu}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\left\{\begin{array}{lcr} \alpha[\eta]=\psi_\nu(\beta[\gamma[\eta]])\\ \gamma[0]=\Omega_\mu \text{ if }\mu\geq 1\\ \gamma[0]=0\text{ if }\mu=0\\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.,$$

5) if $$\alpha=\Omega_\nu^\beta$$ and $$\beta\in s$$ then $$\text{cof}(\alpha)=\Omega_\nu$$ and $$\alpha[\eta]=\Omega_\nu^{\beta-1}\cdot \eta$$,

6) if $$\alpha=\Omega_\nu^\beta \gamma$$ and $$\gamma \in L_{\mu<\nu}$$ then $$\text{cof}(\alpha)=\text{cof}(\gamma)$$ and $$\alpha[\eta]=\Omega_\nu^\beta (\gamma[\eta])$$,

7) if $$\alpha=\Omega_\nu^\beta$$ and $$\beta \in L_{\mu\le\nu}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\Omega_\nu^{\beta[\eta]}$$,

8) if $$\alpha=\Omega_{\mu+1}$$ then $$\text{cof}(\alpha)=\Omega_{\mu+1}$$ and $$\alpha[\eta]=\eta$$ (as well as $$\omega[n]=n$$),

9) if $$\alpha=\Omega_{\beta}$$ and $$\beta\in L_{\mu}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\Omega_{\beta[\eta]}$$.

Rules 1-9 assign FS for each limit ordinal up to omega fixed point $$\psi(\Omega_{\Omega_{\Omega_{...}}})$$, where $$\psi$$ without lower-line index denotes $$\psi_0$$. After the definition of fundamental sequences up to the omega fixed point we can use this notation for fast-growing hierarchy.

Detailed example for illustrating working of rules for fundamental sequences

Let's define $\psi_\Iota(0)[0]=0$ and $\psi_\Iota(0)[n+1]=\Omega_{\psi_\Iota(0)[n]}$

Let us consider the following example and find $\psi(\psi_\Iota(0))[2]$ where $\psi$ denotes $\psi_0$ Since $\text{cof}(\psi_0)=\Omega_0=\omega$ as well as $\text{cof}(\psi_\Iota(0))=\omega$

consequently use rule 3 $\psi(\psi_\Iota(0))[2]=\psi(\psi_\Iota(0)[2])=\psi(\Omega_{\Omega_0})=\psi(\Omega_\omega)$

$$\Omega_\omega[\eta]=\Omega_{\omega[\eta]}$$ (rule 9)

Thus, $$\text{cof}(\Omega_\omega)=\omega$$

and, consequently, use rules 3 and 9.

$$\psi(\Omega_\omega)[2]=\psi(\Omega_\omega[2])=\psi(\Omega_{\omega[2]})=\psi(\Omega_2)$$

$$\text{cof}(\Omega_2)=\Omega_2>\text{cof}(\psi_0)=\omega$$

and, сonsequently, use rule 4 and 8, taking into account

$$\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha}$$ for $$\nu>0$$ and $$\alpha<\varepsilon_{\Omega_nu+1}$$;

$$\psi_0(\alpha)=\omega^{\alpha}$$ for $$\alpha<\varepsilon_0$$

Then, according rules 4 and 8

$$\psi(\Omega_2)[2]=\psi(\Omega_2[\gamma[2]])=\psi(\gamma[2])$$ where

$$\gamma[0]=\Omega_1=\Omega$$

$$\gamma[1]=\psi_1(\Omega_2[\Omega])=\psi_1(\Omega)=\omega^{\Omega+\Omega}=\Omega^2$$

$$\gamma[2]=\psi_1(\Omega_2[\Omega^2])=\psi_1(\Omega^2)=\omega^{\Omega+\Omega\cdot\Omega}=\omega^{\Omega\cdot\Omega}=\Omega^\Omega$$

$$\psi(\Omega_2)[2]=\psi(\Omega^\Omega)$$

Next step:

$$\Omega^\Omega[\eta]=\Omega^{\Omega[\eta]}$$ (rule 7)

$$\text{cof}(\Omega^\Omega)=\Omega>\text{cof}(\psi_0)=\omega$$

and, сonsequently, again use rule 4 and 8

$$\psi(\Omega^\Omega)[2]=\psi(\Omega^{\gamma[2]})$$

where

$$\gamma[0]=0$$

$$\gamma[1]=\psi(\Omega^\Omega[0])=\psi(\Omega^{\Omega[0]})=\psi(\Omega^0)=\psi(1)=\omega$$

$$\gamma[2]=\psi(\Omega^\Omega[\omega])=\psi(\Omega^{\Omega[\omega]})=\psi(\Omega^\omega)$$

$$\psi(\Omega^\Omega)[2]=\psi(\Omega^{\psi(\Omega^\omega)})$$

Next step:

According rules 3 and 7

$$\omega=\text{cof}(\Omega^\omega)=\text{cof}(\psi(\Omega^\omega))=\text{cof}(\Omega^{\psi(\Omega^\omega)})=\text{cof}(\psi(\Omega^{\psi(\Omega^\omega)}))$$

and, сonsequently,

$$\psi(\Omega^{\psi(\Omega^\omega)})[2]=\psi(\Omega^{\psi(\Omega^{\omega[2]})})=\psi(\Omega^{\psi(\Omega^2)})$$

Next step:

$$\Omega^2[\eta]=\Omega\cdot\eta$$ and $$\text{cof}(\Omega^2)=\Omega$$ (rule 5)

$$\text{cof}(\psi(\Omega^{\psi(\Omega^2)}))=\text{cof}(\Omega^{\psi(\Omega^2)})=\text{cof}(\psi(\Omega^2))=\omega$$ (rules 3 and 7)

and, сonsequently,

$$\psi(\Omega^{\psi(\Omega^2)})[2]=\psi(\Omega^{\psi(\Omega^2)[2]})=$$

$$=\psi(\Omega^{\psi(\Omega\cdot\gamma[2])})$$

where

$$\gamma[0]=0$$

$$\gamma[1]=\psi(\Omega\cdot 0)=\psi(0)=\omega^0=1$$

$$\gamma[2]=\psi(\Omega\cdot 1)=\psi(\Omega)$$

Thus $$\psi(\Omega^{\psi(\Omega^2)})[2]=\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))})$$

Next step:

According rules 3, 6 and 7

$$\omega=\text{cof}(\psi(\Omega))=\text{cof}(\Omega\cdot\psi(\Omega))=\text{cof}(\psi(\Omega\cdot\psi(\Omega)))=$$

$$=\text{cof}(\Omega^{\psi(\Omega\cdot\psi(\Omega))})=\text{cof}(\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))}))$$

and, сonsequently,

$$\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega))})[2]=\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega)[2])})$$

and $$\psi(\Omega)[2]=\psi(\Omega[\gamma[2]])=\psi(\Omega[\omega])=\psi(\omega)=\omega^\omega$$

where $$\gamma[0]=0$$

$$\gamma[1]=\psi(\Omega[0])=\psi(0)=1$$

$$\gamma[2]=\psi(\Omega[1])=\psi(1)=\omega$$

That is why $$\psi(\Omega^{\psi(\Omega\cdot\psi(\Omega)[2])})=\psi(\Omega^{\psi(\Omega\cdot\omega^\omega)})$$

Next step:

$$\psi(\Omega^{\psi(\Omega\cdot\omega^\omega)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega^2)})$$

$$\psi(\Omega^{\psi(\Omega\cdot\omega^2)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega\cdot2)})$$

$$\psi(\Omega^{\psi(\Omega\cdot\omega\cdot2)})[2]=\psi(\Omega^{\psi(\Omega\cdot(\omega+2))})=$$

$$=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})$$

Next step:

According rules 1 and 8

$$(\Omega\cdot\omega+\Omega+\Omega)[\eta]=\Omega\cdot\omega+\Omega+\Omega[\eta]=\Omega\cdot\omega+\Omega+\eta$$

and $$\text{cof}(\Omega\cdot\omega+\Omega+\Omega)=\Omega>\omega=\text{cof}(\psi)$$

and, сonsequently, use rule 4

$$\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})[2]=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\gamma[2])})$$

where $$\gamma[0]=0$$

$$\gamma[1]=\psi(\Omega\cdot\omega+\Omega+0)=\psi(\Omega\cdot\omega+\Omega)$$

$$\gamma[2]=\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega))$$,

That is why $$\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\Omega)})[2]=$$

$$=\psi(\Omega^{\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega+\psi(\Omega\cdot\omega+\Omega)))})$$

and so on.

Examples of application of normal form

1) $$\psi_1(\psi_0(\psi_0(1))+\psi_0(1))+\psi_0(\psi_0(1))+\psi_0(1)$$

is valid since

$$\psi_1(\psi_0(\psi_0(1))+\psi_0(1))>\psi_0(\psi_0(1))>\psi_0(1)$$

$$\psi_1(\psi_0(\psi_0(1))+\psi_0(1)), \psi_0(\psi_0(1)), \psi_0(1) \in P$$,

and all of arguments also are written in normal form. Note: $$P$$ denotes class of additive principal numbers.

2) We can write this expression as

$$\omega^{\Omega_1+\omega^\omega+\omega}+\omega^\omega+\omega$$

this is also valid, since $$\omega=\Omega_0$$ and this is

$$\Omega_0^{\Omega_1+\Omega_0^{\Omega_0}+\Omega_0}+\Omega_0^{\Omega_0}+\Omega_0=$$

$$=\Omega_0^{\alpha}\beta+\Omega_0^{\gamma}\beta+\Omega_0^{\delta}\beta$$

where

$$\Omega_0^{\alpha}\beta>\Omega_0^{\gamma}\beta>\Omega_0^{\delta}\beta$$

$$\Omega_0^{\alpha}\beta, \Omega_0^{\gamma}\beta, \Omega_0^{\delta}\beta \in P$$

$$\alpha<\Omega_0^{\alpha}, \gamma<\Omega_0^{\gamma}, \delta<\Omega_0^{\delta}$$

$$\beta=1, \beta<\Omega_0=\omega, \beta \in P$$

and all of exponents $$\alpha, \gamma, \delta$$ also are written in normal form.

3) We can write this expression as

$$\Omega_1\cdot(\omega^\omega)+\Omega_1\cdot\omega+\omega^\omega+\omega=\Omega_1\cdot(\omega^\omega)+\Omega_1\cdot\omega+\Omega_1^0\cdot(\omega^\omega)+\Omega_1^0\cdot\omega$$

and this is also valid, since

$$\Omega_1\cdot(\omega^\omega)>\Omega_1\cdot\omega>\omega^\omega>\omega$$

$$\Omega_1\cdot(\omega^\omega),\Omega_1\cdot\omega,\omega^\omega,\omega \in P$$

$$\omega^\omega<\Omega_1,\omega<\Omega_1$$

For example,

$$\omega^\Omega$$ is not valid, because the condition $$\beta<\Omega_\nu^\beta$$ is not satisfied, but $$\omega^{\Omega+1}$$ is valid, since$$\omega^{\Omega+1}=\Omega\omega>\Omega+1$$.

$$\Omega_1\cdot(\omega^\omega+\omega)$$ is not valid, since $$\gamma=\omega^\omega+\omega$$ is not additive principal number (i.e. $$\gamma \notin P$$).

Examples of additive principal numbers are $$\omega^\alpha$$ starting with $$\omega^0=1$$, since 1 can not be obtained as sum of zeros.

If $$\alpha=\Omega_\nu^\beta\cdot \gamma$$ and $$\gamma$$ is successor ordinal, then $$\gamma=1$$.

That is why the condition $$\Omega_\nu^\beta\cdot\gamma$$, where $$\gamma$$ is a successor ordinal, was excluded from rule set for FS.

Note: the post was written using materials from Deedlit's posts [1], [2] and Buchholz article [3].