User blog:Denis Maksudov/Slowly growing ordinal function and FS up to BHO.

In last post I tried to assign fundamental sequences (FS) for ordinals of Feferman's \(\theta\)-function up to Large Veblen ordinal and I noticed it required too much rules. That is why I decided to introduce some simplier ordinal notation to work only with  normal form based on exponentiation of \(\omega's\) and \(\Omega's\) and to assign FS up to Bachmann-Howard ordinal (BHO). To not confused with usual \(\psi\)-function, I denoted it as \(\psi'\)-function. This is rather slow-growing ordinal function, since \(\psi'(\Omega)\) is just \(\varepsilon_0\). Defenition

\(C_0(\alpha)=\{0\}\),

\(C_{n+1}(\alpha)=\{\gamma+\delta, \psi'(\eta)|\gamma,\delta\in C_n(\alpha), \eta<\alpha\}\),

\(C(\alpha)=\cup_{n=0}^\omega C_n(\alpha)\),

\(\psi'(\alpha)=\text{min}\{\xi|\xi\notin C(\alpha)\}\).

\(\psi'(\bullet+\Omega)=\text{min}\{\alpha|\alpha=\psi'(\bullet+\alpha)\}\),

\(\psi'(\bullet\times\Omega)=\text{min}\{\alpha|\alpha=\psi'(\bullet\times\alpha)\}\),

\(\psi'(\bullet\beta^\Omega)=\text{min}\{\alpha|\alpha=\psi'(\bullet\beta^\alpha)\}\),

where \(\bullet\) denotes rest part of argument - any construction with \(\Omega's\), written in normal form, before the diagonalizer.

Normal form (NF):

For uncountable ordinals NF \(\Omega^{\alpha_k}\eta_k+\cdots+\Omega^{\alpha_1}\eta_1+\eta_{0}\),

where \(\varepsilon_{\Omega+1}>\alpha_k>\cdots>\alpha_1\geq 1\) - countable or uncountable ordinals,

\(\eta_k,...,\eta_1\) - countable ordinals.

For countable ordinals NF \(\psi'(\beta_n)+\cdots+\psi'(\beta_1)\), where

\(\Omega>\psi'(\beta_n)\geq \cdots\geq \psi'(\beta_1)\geq 1\) - countable ordinals.

\(\beta_n,...,\beta_1\) - countable or uncountable ordinals.

Explanation and analysis

if \(\alpha=0\) then \(C_n(0)\) includes only zeros for all \(n\) and \(\psi'(0)=1\).

if \(\alpha=1\) then \(C_1(1)\) includes all posibble sums of zeros and \(\psi'(0)=1\). Then  \(C(\alpha)\) includes all natural numbers and \(\psi'(1)=\omega\).

if \(\alpha=2\) then \(C_1(2)\) includes all posibble sums of zeros and \(\psi'(0)=1\), \(\psi'(1)=\omega\). Then  \(C(2)\) includes all ordinals less than \(\omega^2.\) and and \(\psi'(2)=\omega^2\).

And so on: \(\psi'(\alpha)=\omega^\alpha\).

Then \(\psi'(\psi'(\psi'(0)))=\psi'(\psi'(1))=\psi'(\omega)=\omega^\omega\) since \(C(\omega)\) includes all ordinals less than \(\omega^\omega\)

\(\psi'(\psi'(\psi'(\psi'(0))))=\omega^{\omega^\omega}\)

\(\psi'^n(0)=\omega \uparrow\uparrow (n-1)\)

\(\psi'(\Omega)=\text{min}\{\alpha|\alpha=\psi'(\alpha)\}=\psi'(\psi'(\psi'(...\psi'(\psi'(0))...)))=\varepsilon_0\),

\(C(\Omega+1)\) includes all possible sums of \(\psi'(\Omega)'s\) and

\(\psi'(\Omega+\psi'(0))=\psi'(\Omega+1)=\psi'(\Omega)\omega=\varepsilon_0\omega=\omega^{\varepsilon_0+1}\),

\(\psi'(\Omega+\alpha)=\omega^{\varepsilon_0+\alpha}\),

\(\psi'(\Omega+\psi'(\Omega))=\omega^{\varepsilon_0+\varepsilon_0}=\omega^{\varepsilon_0 2}\)

\(\psi'(\Omega+\psi'(\Omega+\psi'(0)))=\omega^{\varepsilon_0+\varepsilon_0 \omega}=\omega^{\varepsilon_0 (1+\omega)}=\omega^{\varepsilon_0 (\omega)}=\omega^{\omega^{\varepsilon_0+1}}\)

\(\psi'(\Omega 2)=\psi'(\Omega+\Omega)=\psi'(\Omega+\psi'(\Omega+\psi'(...\psi'(\Omega+\psi'(0))...)))=\varepsilon_1\)

\(\psi'(\Omega\alpha)=\varepsilon_\alpha\)

\(\psi'(\Omega\psi'(\Omega))=\varepsilon_{\varepsilon_0}\)

\(\psi'(\Omega^2)=\psi'(\Omega^2)=\text{min}\{\alpha|\alpha=\psi'(\Omega\cdot\alpha)\}=\psi'(\Omega\cdot\psi'(\Omega\cdot\psi'(...\psi'(\Omega\cdot\psi'(0))...)))=\zeta_0\)

\(\psi'(\Omega^2+1)=\psi'(\Omega^2)\cdot \omega=\zeta_0 \cdot\omega=\omega^{\zeta_0}\cdot\omega=\omega^{\zeta_0+1}\)

\(\psi'(\Omega^2+2)=\omega^{\zeta_0+2}\)

\(\psi'(\Omega^2+\psi'(\Omega^2+1))=\omega^{\zeta_0+\omega^{\zeta_0+1}}=\omega^{\zeta_0(1+\omega)}=\omega^{\zeta_0\omega}=\omega^{\omega^{\zeta_0+1}}\)

\(\psi'(\Omega^2+\Omega)=\omega^{...^{\omega^{\omega^{\zeta_0+1}}}}=\varepsilon_{\zeta_0+1}\)

\(\psi'(\Omega^2+\Omega+1)=\varepsilon_{\zeta_0+1}\cdot\omega=\omega^{\varepsilon_{\zeta_0+1}+1}\)

\(\psi'(\Omega^2+\Omega+2)=\omega^{\varepsilon_{\zeta_0+1}+2}\)

\(\psi'(\Omega^2+\Omega+\psi'(\Omega^2+\Omega+1))=\omega^{\varepsilon_{\zeta_0+1}+\omega^{\varepsilon_{\zeta_0+1}+1}}=\omega^{\omega^{\varepsilon_{\zeta_0+1}+1}}\)

\(\psi'(\Omega^2+\Omega2)=\psi'(\Omega^2+\Omega^2)=\omega^{...^{\omega^{\omega^{\varepsilon_{\zeta_0+1}+1}}}}=\varepsilon_{\zeta_0+2}\)

\(\psi'(\Omega^2+\Omega\cdot\psi'(\Omega^2+\Omega2))=\varepsilon_{\zeta_0+\varepsilon_{\zeta_0+1}}=\varepsilon_{\varepsilon_{\zeta_0+1}}\)

\(\psi'(\Omega^22)=\psi'(\Omega^2+\Omega^2)=\varepsilon_{..._{\varepsilon_{\varepsilon_{\zeta_0+1}}}}=\zeta_1\)

\(\psi'(\Omega^2\cdot\psi'(\Omega^2))=\zeta_{\zeta_0}\)

\(\psi'(\Omega^3)=\zeta_{..._{\zeta_{\zeta_0}}}=\eta_0=\varphi(3,0)\)

\(\psi'(\Omega^\alpha)=\eta_0=\varphi(\alpha,0)\)

\(\psi'(\Omega^\Omega)=\varphi(1,0,0)=\Gamma_0=\theta(\Omega)\)

\(\psi'(\Omega \uparrow \uparrow (k+1))=\theta(\Omega \uparrow \uparrow k)\)

\(\psi'(\varepsilon_{\Omega+1})=\theta(\varepsilon_{\Omega+1})\)

Fundamental sequences for all limit ordinals up to BHO

Remarkable that addition of one in the end of any argument means just multiplication of function by omega

\(\psi'(\cdots+1)=\psi'(\cdots)\cdot\omega\) and \(\psi'(\cdots +\gamma+1)[n]=\psi'(\cdots +\gamma)\cdot n\),

\((\psi'(\alpha_1)+\cdots+\psi'(\alpha_k))[n]=\psi'(\alpha_1)+\cdots+\psi'(\alpha_k)[n]\),

where \(\psi'(\alpha_1)\geq\cdots\geq\psi'(\alpha_k)\).

Fundamental sequences for a limit ordinals less than \(\psi'(\Omega)\) defined as follows:

1) \((\psi'(\alpha_1)+\psi'(\alpha_2)+\cdots+\psi'(\alpha_k))[n]=\psi'(\alpha_1)+\psi'(\alpha_2)+\cdots+\psi'(\alpha_k)[n]=\)

\(=\omega^{\alpha_1}+\omega^{\alpha_2}+\cdots+\omega^{\alpha_k}[n]\),

where \(\psi'(\Omega)\geq\alpha_1\geq\cdots\geq\alpha_k\geq1\),

2) \(\psi'(\alpha)[n]=\omega^\alpha[n]=\omega^{\alpha-1}n\) if \(\alpha\) is a successor ordinal,

3) \(\psi'(1)[n]=\omega[n]=n\), 4) \(\psi'(\alpha)[n]=\omega^\alpha[n]=\omega^{\alpha[n]}\) if \(\alpha\) is a limit ordinal.

Fundamental sequences for uncountable ordinals in argument of \(\psi'\) function:

Let in expression \(\psi'(\bullet+\Omega^\alpha \beta + \gamma)\) symbol \(\bullet\) denotes rest part of argument of \(\psi'\)-function and \(\Omega^\alpha \beta + \gamma\) are last terms in NF of argument and \(\alpha, \beta, \gamma\) are countable ordinals. If last term in NF of argument is \(\Omega\) with uncountable exponent, then  \(\Omega^\alpha \beta + \gamma\) are last terms in NF of exponent of last term in NF of argument and symbol \(\bullet\) denotes rest part of argument. If last term in NF for exponent again is \(\Omega\) with uncountable exponent then \(\Omega^\alpha \beta + \gamma\) are last terms in NF for exponent of last term in NF for exponent of last term in NF for argument. And so on. But in each level of tower  of exponents \(\Omega^\alpha \beta + \gamma\) are always last terms in NF with countable \(\alpha, \beta, \gamma\) and \(\bullet\) always is rest part of argument.

For example, \(\psi'(\Omega^{\Omega^{\Omega}}+\Omega^{\Omega^{\omega^2}\omega+\Omega^35+\Omega^2\omega+3})\) can be denoted as \(\psi'(\bullet+\Omega^2\omega+3)\)

5) \(\psi'(\bullet+\Omega^\alpha \beta + \gamma)[n]=\psi'(\bullet+\Omega^\alpha \beta + \gamma[n])\) if \(\gamma\) is a limit ordinal,

6) \(\psi'(\bullet+\Omega^\alpha \beta)[n]=\psi'(\bullet+\Omega^\alpha (\beta-1) + \Omega^{\alpha-1}\cdot \Omega)=\)

\(=\psi'(\bullet+\Omega^\alpha (\beta-1) + \Omega^{\alpha-1}\cdot (\psi'(\bullet+\Omega^\alpha \beta)[n-1]))\)

if \(\alpha, \beta\) are a successor ordinals,

7) \(\psi'(\bullet+\Omega^\alpha \beta)[n]=\psi'(\bullet+\Omega^\alpha \cdot(\beta[n]))\) if \(\beta\) is a limit ordinal,

8) \(\psi'(\bullet+\Omega^\alpha \beta)[n]=\psi'(\bullet+\Omega^\alpha \cdot(\beta-1)+\Omega^{\alpha[n]} \cdot \Omega)=\)

\(=\psi'(\bullet+\Omega^\alpha \cdot(\beta-1)+\Omega^{\alpha[n]} \cdot (\psi'(\bullet+\Omega^\alpha \beta)[n-1]))\)

if \(\alpha\) is a limit ordinal and \(\beta\) is a successor ordinal,

9) \(\psi'(\bullet+\Omega)[n+1]=\psi'(\bullet+(\psi'(\bullet+\Omega)[n]))\),

10) \(\psi'(\bullet+\beta^{\Omega})[n+1]=\psi'(\bullet+\beta^{\psi'(\bullet+\beta^{\Omega})[n]})\) where \(\bullet\) denotes rest part of argument of \(\psi'\)-function and \(\beta\) is any tower of \(\Omega's\) (one or more \(\Omega's\)) and \(\beta^{\Omega}\) is last term in NF for argument or last term in NF for exponent of last term in NF and so on.

11) \(\psi'(\varepsilon_{\Omega+1})[n]=\psi'(\varepsilon_{\Omega+1}[n])\) and \(\varepsilon_{\Omega+1}[n+1]=\Omega^{\varepsilon_{\Omega+1}[n]}\).

The searching of \(\Omega\)-diagonalizer among other \(\Omega's\) has same algorithm as searching of final \(\omega\) according rules of Wainer hierarchy for ordinals, written in Cantor normal form. In general we can say, it works with transfinity ordinals almost same way as fast-growing hierarchy works with natural numbers.

Compare:

\(f^{n+1}(\omega^{\omega^2+\omega\cdot\omega},a)=f(\omega^{\omega^2+\omega\cdot f^{n}(\omega^{\omega^2+\omega\cdot\omega},a)},...)\)

and

\(\psi'(\Omega^{\Omega^2+\Omega\cdot\Omega})[n+1]=\psi'(\Omega^{\Omega^2+\Omega\cdot\psi'(\Omega^{\Omega^2+\Omega\cdot\Omega})[n]})\).

This ordinal notation easy can be extended by applying of \(\Omega_2\), \(\Omega_3\) and so on but presently it need not since the aim is assignation FS for limit ordinals up to Bachmann-Howard ordinal. Any way rules of FS for \(\Omega_k\) will be similar.