User blog:Scorcher007/SLCON and almost 3000 comparisons, largest comparison table with DAN

SLCON is a partially defined way of briefly expression of large countable ordinals. A month ago, I presented my notation of expression of Large countable ordinals. Now I have compared this notation with DAN. It turned out a table containing almost 3000 comparisons.

How to get finite numbers:

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

H y (n) where y - countable ordinal; H - Hardy hierarchy;

y = PTO KP+ x [we need well-ordered ordinal notation !!!]

Some notes:

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

We can also use Veblen&#39;s Hierarchy: (nundefined0undefined0undefined0undefined...0) - m -times = (&upsih;m&times;nundefined0)

k - meas property of ordinal

0 - (&sigma;) - order

1 - (&sigma;&#39;1) - inaccessibility

2 - (&sigma;&#39;2) - mahloness

3 - (&sigma;&#39;3) - П3-reflecting

e.t.c

thing like 1/2&#123;a&#125;, 1/2&#123;a&#125;/3&#123;b&#125;&#123;c&#123;d&#125;&#125;&#123;&#123;e&#125;&#125; means combination of this property.

Example:

S[&sigma;](a)

a-th admissible (or n-th limit of admissible if we use n-th limit ordinal in a)

S[&sigma;&#39;1](bundefineda)

a-th b-inaccessible

S[&sigma;&#39;2](fundefineddundefinedbundefineda)

=

S[&sigma;&#39;2](fundefineddundefinedbundefineda)

a-th b-inaccessible limit of e-Mahlo d-limit of f-Mahlo

S[&sigma;&#39;3](nundefinedlundefinedkundefinedfundefined&#123;h &#123;&#123;i&#125;&#125;&#125;&#123;&#123;j&#125;&#125;| eundefinedbundefineda)

=

S[&sigma;&#39;3](nundefinedlundefinedkundefinedfundefined&#125;undefined&#123;j&#125;undefined| eundefinedbundefineda)

a-th b-inaccessible limit of e-Mahlo d-limit of g-Mahlo in which (i-П3-reflecting in which j-П3-reflecting are h-stationary) are stationary f-limit of k-Mahlo in which (m-П3-reflecting in which n-П3-reflecting are l-stationary) are stationary

S[&sigma;&#39;4](&rho;undefined&sigmaf;undefined&theta;undefined&beta;undefined&#123;&delta; &#123;&#123;&lambda;&#125;&#125;&#125;&#123;&#123;&mu;&#125;&#125;| yundefinedzundefinedwundefined&#125;&#123;&#123;n&#125;&#125;4&#123;&#123;&#123;o&#125;&#125;p&#123;&#123;q &#123;&#123;&#123;r&#125;&#125;&#125;&#125;&#125;&#123;&#123;&#123;s&#125;&#125;&#125; &#125;&#123;&#123;t&#123;&#123;&#123;u&#125;&#125;&#125;&#125;&#125;&#123;&#123;&#123;v&#125;&#125;&#125;| fundefined&#123;h &#123;&#123;i&#125;&#125;&#125;&#123;&#123;j&#125;&#125;| eundefinedbundefineda)

=

S[&sigma;&#39;4](&rho;undefined&sigmaf;undefined&theta;undefined&beta;undefined&#125;undefined&#123;&mu;&#125;undefined| yundefinedzundefinedwundefinedundefined&#125;undefined&#123;n&#125;undefined&#123;p&#123;o&#125;undefined&#123;q &#123;r&#125;undefined&#125;undefined&#123;s&#125;undefined &#125;undefined&#123;t&#123;u&#125;undefined&#125;undefined&#123;v&#125;undefined| fundefined&#125;undefined&#123;j&#125;undefined| eundefinedbundefineda)

a-th b-inaccessible limit of e-Mahlo d-limit of g-Mahlo in which (i-П3-reflecting in which j-П3-reflecting are h-stationary) are stationary f-limit of k-Mahlo in which (l-П3-reflecting in which (o-П3-reflecting onto r-П4-reflecting that is q-П3-reflecting onto s-П4-reflecting in which n-П3-reflecting onto v-П4-reflecting that is t-П3-reflecting onto u-П4-reflecting are p-stationary) are m-stationary) are stationary w-limit of z-Mahlo in which (x-П3-reflecting in which (&gamma;-П3-reflecting onto &lambda;-П4-reflecting that is &delta;-П3-reflecting onto &mu;-П4-reflecting in which &theta;-П3-reflecting onto &xi;-П4-reflecting that is &sigmaf;-П3-reflecting onto &rho;-П4-reflecting are &beta;-stationary) are y-stationary) are stationary

e.t.c.

Table and details here http://lihachevss.ru/SLCON.html