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First I made up separators for my [X]a function, then I realized that it may be used for any functions of multiple variables.
means n separators (commas) between variables with zeros between them (zeros may be omitted). So, is n "empty" variables (zeros):
= ,0
= ,0,0
= ,0,0,0
if is not before a variable or
= ,
= ,0,
= ,0,0,
if is before a variable
For example, let f  function of 15 variables.
f(1,3,0,0,0,0,0,0,0,0,0,0,0,0,1) may be written as f(1,31)
Particularly, I used for Veblen function:
Î“_{0} = Ï†(1,0,0) = Ï†(1)
Ï†(1,0,0,0) = Ï†(1)
Ï†(1)  small Veblen ordinal
Also may be used to describe multidimensional arrays of variables:
is n "empty" variables
is n "empty" rows
is n "empty" planes
For example,
is three "empty" variables, two "empty" rowâ€¦
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Ordinal subtraction is operation opposite to ordinal addition.
Ordinal addition is noncommutative, so, there are two ordinal subtractions (as for exponentiation, which is also noncommutative and also has two opposite operations: root and logarithm):
1. Î±Î²: (Î±Î²)+Î² = Î±
2. Î²+Î±: Î²+(Î²+Î±) = Î±
(Note: in Ordinals array function blog I denoted both subtractions as Î±Î², and later I started to denote second subtraction as Î²+Î±).
As we can extend natural numbers and zero to integer numbers using subtraction, we can extend ordinals to "integer ordinals".
Subtraction of Î² we may descibe as addition of opposite "integer ordinal" Î²:
Î±Î² = Î±+(Î²)
Î²+Î± = (Î²)+Î±
Î± is opposite to Î±:
Î±+(Î±) = 0
(Î±)+Î±=0
To get opposite to Î± we need write Cantor normal form of Î± bacâ€¦
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Hi!
This summer I derived rules of ordinal arithmetic. I used Wikipedia article Ordinal arithmetic. Some of this I rediscovered independently, including Cantor normal form, addition and multiplication rules, which are contained in the same article in Cantor normal form section. Rules for my [X]a function need verification.
Let Î±, Î², Î³, Î´  ordinals. Î² (as any ordinal) is 0, or a successor ordinal, or a limit ordinal.
Let Î±+1 is the successor of Î±.
Definition of addition (Î±+Î²):
1. Î±+0 = Î± (rule for Î² = 0)
2. Î±+(Î²+1) = (Î±+Î²)+1 (rule for Î² is a successor ordinal)
3. Î±+Î² = limit of the Î±+Î´ for all Î´ 0)
(zc(Î±) and n_{i} are multiplied as natural numbers)
3. Î±Â·Ï‰^{Î±i}Â·n_{i} = Ï‰^{vs(Î±)+Î±i}Â·n_{i} (if Î±_{i} > 0, n_{i} > 0)
Examples.
Î± = Ï‰^{9}Â·2 + Ï‰^{7} + Ï‰^{4}Â·3 + 1
zp(Î±) = Ï‰^{9}Â·2
rt(Î±) = Ï‰^{7} + Ï‰^{4}Â·3 + 1 â€¦
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I interested in ordinals, and I asked myself: "Why we use \(\omega\), \(\omega2\), \(\omega^2\), \(\omega^\omega\), \(\epsilon_0\), \(\zeta_0\), \(\phi(3,0)\), \(\Gamma_0\) etc. instead of \(\omega\uparrow\uparrow\uparrow\omega\), \(\omega\uparrow\uparrow\uparrow\uparrow\omega\) etc.?"
But I realized this doesn't work beyond \(\epsilon_0\) since \(\omega\uparrow\uparrow(\omega+1) = \omega^{\omega\uparrow\uparrow\omega} = \omega^{\epsilon_0} = \epsilon_0 = \omega\uparrow\uparrow\omega\).
I experimented and created a BEAFlike system, suitable also for ordinals.
It is family of functions [X]a of ordinal a.
Let X  sequence of ordinals (or "negative ordinals", i.e. ordinals with "" sign before them) with separators.
All zeros we may omit.
All lefâ€¦
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