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# Alemagno12

## aka Nish

My favorite wikis
• I live in Laniakea Supercluster
• My occupation is GO INTO GOOGOLOGY SERVER
• I am a
• ## Some set theory questions

December 5, 2017 by 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?

• ## An attempt at formalizing Username5243's Pi Notation

December 3, 2017 by 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:

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

Then, we have the cases for sequence ordinals:

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

Then, we have the cases for of limit ordinals:

• Ï€(Ï€(a,b,Ï€(c,d)),0,x) = [WIP]
• ## How strong is this function? (2)

November 10, 2017 by 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 Ï‰. Closed.

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

• 0 âˆˆ OP
• If a,b âˆˆ OP, then (a,b) âˆˆ 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 âˆˆ S and x is not 0, then x > 0
• If (a,b) âˆˆ S, then these conditions must be true: a,b âˆˆ 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 xâ€¦

• ## Analysis of Edwin Shade's ''three-symbol'' notation for ordinals

November 5, 2017 by Alemagno12

Here's the notation.

WIP!

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

• (|) = Ï‰
• (|)| = Ï‰+1
• (|)|| = Ï‰+2
• (|)(|) = Ï‰2
• (|)(|)| = Ï‰2+1
• (|)(|)(|) = Ï‰3
• ((|)) = Ï‰2
• ((|))| = Ï‰2+1
• ((|))(|) = Ï‰2+Ï‰
• ((|))((|)) = Ï‰22
• ((|))((|))((|)) = Ï‰23
• (((|))) = Ï‰3
• (((|)))(((|))) = Ï‰32
• ((((|)))) = Ï‰4
• (((((|))))) = Ï‰5

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

• [(]|[)] = Ï‰Ï‰
• [(]|[)][(]|[)] = Ï‰Ï‰2
• ([(]|[)]) = Ï‰Ï‰+1
• (([(]|[)])) = Ï‰Ï‰+2
• [(][(]|[)][)] = Ï‰Ï‰2
• [(][(][(]|[)][)][)] = Ï‰Ï‰3
• ([(])|([)]) = Ï‰Ï‰2
• ([(])[(]|[)]([)]) = Ï‰Ï‰2+Ï‰
• ([(])([(])|([)])([)]) = Ï‰Ï‰22
• (([(]))|(([)])) = Ï‰Ï‰3
• ((([(])))|((([)]))) = Ï‰Ï‰4

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

• ## How strong is this function?

October 28, 2017 by 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.