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Alemagno12

aka Nish

  • I live in Laniakea Supercluster
  • My occupation is GO INTO GOOGOLOGY SERVER
  • I am a
  • Alemagno12

    What is the cardinality of the set of all well-orderings over the set of natural numbers?


    If a cardinal is not compatible with the axioms of some set theory, does it mean that it is larger than all cardinals that are compatible with those axioms?

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  • Alemagno12

    NOTE: This pi notation is slightly different from the original one to make it easier to define. But it still has the same strength.

    The original definition can be found here.

    First we have the base cases:

    • π(0,x) = x+1
    • π(x,y)[n] = π(x,y,n)

    Then, we have the cases for sequence ordinals:

    • π(π(0,z),0,0) = 0
    • π(π(0,z),0,π(0,x)) = π(z,π(π(0,z),0,x))
    • π(π(0,z),0,π(a,b))[n] = π(π(0,z),0,π(a,b,n))
    • π(π(0,z),π(0,y),0) = π(0,π(π(0,z),y))
    • π(π(0,z),π(0,y),π(0,x)) = π(z,π(π(0,z),π(0,y),x))
    • π(π(0,z),π(0,y),π(a,b))[n] = π(π(0,z),0,π(a,b,n))
    • π(π(0,z),π(a,b),x) = π(π(0,z),π(a,b,x))

    Then, we have the cases for of limit ordinals:

    • π(π(a,b,π(c,d)),0,x) = [WIP]
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  • Alemagno12

    EDIT: I found out that f(2) for n greater than 2 don't exist, since there are infinitely many possible ordinal pairs for ω. Closed.

    Define the set of all ordinal pairs OP using these inductive rules:

    • 0 ∈ OP
    • If a,b ∈ OP, then (a,b) ∈ OP

    Next, define an ordinal pair system as a well-ordering on the elements of a subset of OP (which we will call S), where:

    • If x ∈ S and x is not 0, then x > 0
    • If (a,b) ∈ S, then these conditions must be true: a,b ∈ S and a,b < (a,b).

    We can use this well-ordering to label the elements of S: 0 is labelled with, well, 0, and (a,b) is labelled with the smallest ordinal x such that x is greater than the labels of all elements of S smaller than (a,b).

    Finally, let f(n) be the largest possible value of the smallest ordinal x…

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  • Alemagno12

    Here's the notation.

    WIP!


    • [empty] = 0
    • | = 1
    • || = 2
    • ||| = 3
    • n |'s = n

    • (|) = ω
    • (|)| = ω+1
    • (|)|| = ω+2
    • (|)(|) = ω2
    • (|)(|)| = ω2+1
    • (|)(|)(|) = ω3
    • ((|)) = ω2
    • ((|))| = ω2+1
    • ((|))(|) = ω2
    • ((|))((|)) = ω22
    • ((|))((|))((|)) = ω23
    • (((|))) = ω3
    • (((|)))(((|))) = ω32
    • ((((|)))) = ω4
    • (((((|))))) = ω5

    Let [] is a bigger pair of brackets than ().

    • [(]|[)] = ωω
    • [(]|[)][(]|[)] = ωω2
    • ([(]|[)]) = ωω+1
    • (([(]|[)])) = ωω+2
    • [(][(]|[)][)] = ωω2
    • [(][(][(]|[)][)][)] = ωω3
    • ([(])|([)]) = ωω2
    • ([(])[(]|[)]([)]) = ωω2
    • ([(])([(])|([)])([)]) = ωω22
    • (([(]))|(([)])) = ωω3
    • ((([(])))|((([)]))) = ωω4

    Question: How would [(][(][)]|[(][)][)] be solved?

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  • Alemagno12

    Let f(n) be the length of the largest string that satisfies the following conditions:

    • The string can have at most n types of characters
    • No substring in the string appears inmediately before a copy of itself

    I believe f(n) has tetrational growth rate, but I'm not sure. Also, sorry for the short blog post.

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