User blog:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm.

This notation not well-ordered, but well-formed on KP+ x,

where x - axiom of existence admissible ordinal or limit of admissible in this well-formed notation

SLCON is a partially defined way of briefly expression of large countable ordinals.

Base terms example (comparison with the terms used in OCF):

&omega; = S[&sigma;](0)

&Omega; = S[&sigma;](1)

&Omega;2 = S[&sigma;](2)

I = S[&sigma;&#39;1](0)

I2 = S[&sigma;&#39;1](1)

e.t.c.

Some notes:

n&isin;O - means &forall;n&isin;O&exist;(n terms or f(n) terms)

n&lt;O - means limit of n&lt;O term

S[...](n) - n means order of ordinal type

S[...](undefinedn) - k means property of ordinal

&upsih; - means pseudo-ordinal term used as diagonalizer for thing like &quot;hyper- x &quot; (Bachmann&#39;s Hierarchy based on CNF)

We can also use Veblen&#39;s Hierarchy: (aundefinedbundefinedcundefineddundefinede) = (&upsih;4&times;a+&upsih;3&times;b+&upsih;2&times;c+&upsih;&times;dundefinede)

k - properties:

, - undefined - undefined - order, limits and fixed-point of limits

undefined - undefined - inaccessibility

e.t.c.

Example: S[&sigma;](a) - a-th admissible; S[&sigma;&#39;1](bundefineda) - a-th b-inaccessible

How to get finite numbers:

Let&#39;s define &Theta;( x,n ) - means:

&Theta;( x,n ) is computable function which assigns to each n&isin;N the least natural number greater than or equal to the halting times of Turing machines with input 0 whose terminations admit formal proofs of length &le; n under the axiom of therory KP+ x, where x - axiom of large countable ordinal.

The following property can also be derived. It is possible to create such a system of fundamental sequences based on theory KP+ x n, that it will be valid limn&rarr;&omega;(f(PTO KP+ x 1) (n)) &gt; limn&rarr;&omega;(&Theta;( x 1,n)), but always limn&rarr;&omega;(f(PTO KP+ x 1) (n)) &lt; limn&rarr;&omega;(&Theta;( x 2,n)), where x n - axiom of large countable ordinal from list below.

Пn-reflecting:

Definition of large ordinals can be redefined in terms Пn-reflecting ordinals or Пn-reflecting on some class ordinals.

Ordinals is Пn-reflecting iff L&sigma;⊧&phi;&rarr;&exist;&beta;&lt;&sigma;(L&beta;⊧&phi;); &phi; is Пn-formula.

Ordinals is Пn-reflecting onto class B iff L&sigma;⊧&phi;&rarr;&exist;&beta;&isin;B⋂&sigma;(L&beta;⊧&phi;); &phi; is Пn-formula.

To begin, I will show how can create similar notation for small countable ordinals. This notation should be well-ordered and contain no recursive gaps, unlike SLCON. I remind that SLCON can&#39;t be considered an ordinal notation, because SLCON each time jumps from &epsilon;&alpha;+1 to admissible on &alpha;, where &alpha; is some term in the designation of the ordinal. It can only be used to definition the axioms of a large countable ordinal.

&quot;P&quot; - П1-reflecting ordinals

&quot;+1&quot; - next ordinal

1 - 1st successor

P - &omega; - 1st P

P+1 - &omega;+1 - 1st successor after 1st P

P(1) - &omega;&times;2 - 2nd P

P(P) - &omega;2 - (1st P)-th P

P(P)+1 - &omega;2+1 - 1st successor after (1st P)-th P

P(P+1) - &omega;2+&omega; - (1st successor after 1st P)-th P

P(P(1)) - &omega;2&times;2 - (2nd P)-th P

P(P(P)) - &omega;3 - ((1st P)-th P)-th P

P(1,0) - &omega;&omega; - 1st P on P = 1st 2-P

P(P(1,0)) - &omega;&omega;+&omega; - 1st P after 2-P

P(P(P(1,0))) - &omega;&omega;+&omega;2 - (1st P)-th P after 2-P

P(1,1) - &omega;&omega;&times;2 - 2nd 2-P

P(1,P) - &omega;&omega;+1 - (1st P)-th 2-P

P(1,P(1)) - &omega;&omega;+1&times;2 - (2nd P)-th 2-P

P(1,P(P)) - &omega;&omega;+2 - ((1st P)-th P)-th 2-P

P(1,P(1,0)) - &omega;&omega;&times;2 - (2-P)-th 2-P

P(2,0) - &omega;&omega; 2 - 1st P on 2-P = 1st 3-P

P(2,1) - &omega;&omega; 2 &times;2 - 2nd 3-P

P(2,P) - &omega;&omega; 2+1 - (1st P)-th 3-P

P(2,P(1,0)) - &omega;&omega; 2+&omega; - (1st 2-P)-th 3-P

P(2,P(2,0)) - &omega;&omega; 2&times;2 - (1st 3-P)-th 3-P

P(3,0) - &omega;&omega; 3 - 1st P on 3-P = 1st 4-P

P(P,0) - &omega;&omega; &omega; - 1st (1st P)-P

P(P+1,0) - &omega;&omega; &omega;+1 - 1st (1st successor after 1st P)-P

P(P(1),0) - &omega;&omega; &omega;&times;2 - 1st (2nd P)-P

P(P(P),0) - &omega;&omega; &omega; 2 - 1st ((1st P)-th P)-P

P(P(1,0),0) - &omega;&omega; &omega; &omega; - 1st (1st 2-P)-P

P(P(P,0),0) - &omega;&omega; &omega; &omega; &omega;  - 1st (1st (1st P)-P)-P

P(P(P(1,0),0),0) - &omega;&omega; &omega; &omega; &omega; &omega;   - 1st (1st (1st 2-P)-P)-P

P(&upsih;,0) = P(1,0,0) - &epsilon;0 - 1st hyper-P

P(&upsih;,1) = P(1,0,1) - &epsilon;0&times;2

P(&upsih;+1,0) = P(1,1,0) - &epsilon;1

P(&upsih;&times;2,0) = P(2,0,0) - &zeta;0

P(&upsih;2,0) = P(1,0,0,0) - Г0

P(&upsih;&omega;,0) - LVO

P(&upsih;&upsih;,0) - SVO

P(n&upsih;,0) - BHO

Redefinition of large countable ordinals through Пn-reflecting:

1st admissible &gt; &omega; = &omega;1CK = 1st П2-reflecting

2nd admissible &gt; &omega; = &omega;2CK = 2nd П2-reflecting

limit of (n&lt;&omega;)-th admissible = sup(&omega;nCK)|n&lt;&omega; = 1st П1-reflecting onto П2-reflecting;

limit of (&omega;+(n&lt;&omega;))-th admissible = sup(&omega;&omega;+nCK)|n&lt;&omega; = 2nd П1-reflecting onto П2-reflecting

1st fixed point limits of admissible = &alpha;↦&omega;&alpha;CK = 1st П1-reflecting onto П1-reflecting onto П2-reflecting = 2-П1-reflecting onto П2-reflecting

1st fixed point of fixed point limits of admissible = 3-П1-reflecting onto П2-reflecting

1st hyper-fixed point limits of admissible = 1st hyper-П1-reflecting onto П2-reflecting

1st recursively inaccessible = 1st П2-reflecting that is П1-reflecting onto П2-reflecting

1st recursively inaccessible = 2nd П2-reflecting that is П1-reflecting onto П2-reflecting

limit of (n&lt;&omega;)-th inaccessible = 1st П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting)

limit of (&omega;+(n&lt;&omega;))-th inaccessible = 2nd П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting)

1st fixed point limits of inaccessible = 1st 2-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting)

1st fixed point of fixed point limits of inaccessible = 1st 3-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting)

1st hyper-fixed point limits of inaccessible = 1st hyper-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting)

1st recursively 2-inaccessible = 1st П2-reflecting that is П1-reflecting onto П1-reflecting onto П2-reflecting =1st П2-reflecting that is 2-П1-reflecting onto П2-reflecting

1st recursively hyper-inaccessible = 1st П2-reflecting that is hyper-П1-reflecting onto П2-reflecting

1st recursively Mahlo = 1st П2-reflecting onto П2-reflecting;

Reference

zoo - means refer to http://www.madore.org/~david/math/ordinal-zoo.pdf (Madore D., Zoo of ordinals, 2017)

S - Large Countable Ordinal Notation (SLCON ). Chapter I, Up to KPm.