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  • Rgetar

    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:

    Γ0 = φ(1,0,0) = φ(1)

    φ(1,0,0,0) = φ(1)

    φ(1) - small Veblen ordinal


    Also may be used to describe multi-dimensional arrays of variables:

    is n "empty" variables
    is n "empty" rows
    is n "empty" planes

    For example,

    is three "empty" variables, two "empty" row…

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  • Rgetar

    Ordinal subtraction is operation opposite to ordinal addition.

    Ordinal addition is non-commutative, so, there are two ordinal subtractions (as for exponentiation, which is also non-commutative and also has two opposite operations: root and logarithm):

    1. α-β: (α-β)+β = α

    2. -β+α: β+(-β+α) = α

    (Note: in Ordinals array function blog I denoted both subtractions as α-β, and later I started to denote second subtraction as -β+α).

    As we can extend natural numbers and zero to integer numbers using subtraction, we can extend ordinals to "integer ordinals".


    Subtraction of β we may descibe as addition of opposite "integer ordinal" -β:

    α-β = α+(-β)

    -β+α = (-β)+α

    -α is opposite to α:

    α+(-α) = 0

    (-α)+α=0

    To get opposite to α we need write Cantor normal form of α bac…


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  • Rgetar

    Ordinal arithmetic

    October 3, 2017 by Rgetar

    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 α, β, γ, δ - ordinals. β (as any ordinal) is 0, or a successor ordinal, or a limit ordinal.

    Let α+1 is the successor of α.

    Definition of addition (α+β):

    1. α+0 = α (rule for β = 0)

    2. α+(β+1) = (α+β)+1 (rule for β is a successor ordinal)

    3. α+β = limit of the α+δ for all δ 0)

    (zc(α) and ni are multiplied as natural numbers)

    3. α·ωαi·ni = ωvs(α)+αi·ni (if αi > 0, ni > 0)

    Examples.

    α = ω9·2 + ω7 + ω4·3 + 1

    zp(α) = ω9·2

    rt(α) = ω7 + ω4·3 + 1 …


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  • Rgetar

    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 BEAF-like 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 lef…


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