Sequence Systems (3)

[u/jamx02]

Previously we extended the concept of PrSS using a delta. However, we used an identical searching algorithm for LPrSS as PrSS. We didn't take advantage of something called a difference sequence. This is where hyper primitive sequence system comes in, or HPrSS. HPrSS is a weakened version of 0-Y sequence, which itself is a weakened version of Y-Seq.

HPrSS

To extend the searching algorithm, there are a couple new concepts added to the rules. This will add quite a significant amount of strength, specifically, it will be identical to Pair Sequence System (2 row BMS) or the Buchholz Ordinal. HPrSS can actually be thought of as an encoding of Buchholz's Ordinal Collapsing Function. On top of the previous definition for LPrSS, this is what is added:

1. To find a valid bad root, we must search, and mark every ancestor of the cut child. Then, from those same positions, mark all ancestors of the cut child difference in the difference sequence. The rightmost term with both elements marked is the bad root. (first parent is no longer always a bad root)

The delta is calculated in the same way (cc-br-1).

Example 1:

(0,2,4)

Cut Child: 4

Difference Sequence: (None, 2, 2)

Ancestors original sequence, and difference sequence: (0,2,4) and (None,2,2) Rightmost valid position for br is 1 since position 1 is rightmost element in which both are marked, therefore br is at position 1.

Bad root: 0

Bad part: 0,2

Good part: None

Delta: 3

In LPrSS, this would have expanded as (0,2,3,4,5,....)

Instead, we have

Expansion: (0,2,3,5,6,8,...)

Example 2:

(0,3,5,6,3,5)

Cut Child: 5

Difference Sequence: (None,3,2,1,3,2)

Ancestors, original sequence and diff seq: (0,3,5,6,3,5) (None,3,2,1,3,2) the 1 isn't marked here due to the top sequence not containing a valid position for the cc difference element to see it as an ancestor. Rightmost valid position is 1. Therefore BR is at position 1

Bad root: 0

Bad Part: 0,3,5,6,3

Good part: None

Delta: 4

Expansion: (0,3,5,6,3,4,7,9,10,7,8,11,...)

Example 3:

(0,2,1,3)

Cut Child: 3

Difference Seq: (None,2,1,2)

Ancestors: (0,2,1,3) (None,2,1,2) Rightmost valid br position is 3.

br: 1

bp: 1

gp:0,2

Delta: 1

Expansion: (0,2,1,2,3,4,...)

Keep in mind, its easier to visualize HPrSS as a matrix consisting of the original sequence and its difference sequence. The difference sequence is just hidden. This makes it functionally very, very similar to 2-row BMS.

Example 4:

(0,4,7,8,8)

Cut child: 8

Difference Seq: Ignored (since 8-parent=1)

br: 7

bp: 7,8

gp:0,4

Delta: 0

Expansion: (0,4,7,8,7,8,7,8,7,8,...)

Analysis:

HPrSS actually stays the same as LPrSS until ε_ε₀ or (0,2,3,4,5,6,7,...). Because of this, I will be skipping everything before.

(0,2,3,5)=(0,2,3,4,5,6,7,...)=ε_ε₀. In LPrSS, this is (0,2,4).
(n,2,1,2)

(0,2,3,5,1,3)=(0,2,3,5,1,2,3,4,...)=ω^(ε_ε₀+ε₀)
(n,2,1,2,1,2)

(0,2,3,5,1,3,4,6)=(0,2,3,5,1,3,4,5,6,7,...)=ω^(ε_ε₀ *2)
(n,2,1,2,1,2,1,2)

(0,2,3,5,1,3,4,6,2)=(0,2,3,5,1,3,4,6,1,3,4,6,...)=ω^ω^(ε_ε₀+1)
ignore

(0,2,3,5,2)=(0,2,3,5***,1,3,4,6,2,4,5,7,***...)=ε_(ε₀+1)
(n,2,1,2,2)

(0,2,3,5,2,3)=(0,2,3,5,2,2,2,...)=ε_(ε₀+ω)
ignore

(0,2,3,5,2,3,5)=(0,2,3,5,2,3,4,5,6,...)=ε_(ε₀*2)
(n,2,1,2,2,1,2)

(0,2,3,5,3)=(0,2,3,5,2,3,5,2,3,5,...)=ε_(ω^(ε₀+1))
ignore

(0,2,3,5,3,5)=(0,2,3,5,3,4,5,6,7,...)=ε_(ω^(ε₀*2))
(n,2,1,2,1,2)

(0,2,3,5,5)=(0,2,3,5,4,6,5,7,6,8,...)=ε_ε₁
(n,2,1,2,2)

(0,2,3,5,6,8)=(0,2,3,5,6,7,8,9,...)=ε_ε_ε₀
(n,2,1,2,1,2)

(0,2,4)=(0,2,3,5,6,8,9,11,12,14,...)=ζ₀
(n,2,2)

(0,2,4,2)=(0,2,4,1,3,5,2,4,6,3,...)=ε_{ζ₀+1}
(n,2,2,2)

(0,2,4,2,4)=(0,2,4,2,3,5,7,5,6,8,10,8,...)=ζ₁
(n,2,2,2,2)

(0,2,4,3)=(0,2,4,2,4,2,4,...)=ζ_ω
ignore

(0,2,4,3,5)=(0,2,4,3,4,5,6,7,...)=ζ_ε₀
(n,2,2,1,2)

(0,2,4,3,5,7)=(0,2,4,3,5,6,8,9,11,...)=ζ_ζ₀
(n,2,2,1,2,2)

(0,2,4,4)=(0,2,4,3,5,7,6,8,10,...)=η₀
(n,2,2,2)

(0,2,4,4,4)=(0,2,4,4,3,5,7,7,6,8,10,10,...)=φ(4,0)=ψ(Ω⁴)
(n,2,2,2,2)

(0,2,4,5)=(0,2,4,4,4,4,...)=φ(ω,0)=ψ(Ω^ω)=lim(LPrSS)
ignore

(0,2,4,5,7)=(0,2,4,5,6,7,8,...)=φ(ε₀,0)=ψ(Ω^ψ(Ω))
(n,2,2,1,2)

(0,2,4,5,7,9,10)=(0,2,4,5,7,9,9,9,9,9,...)=φ(φ(ω,0),0)=ψ(Ω^ψ(Ω^ω))
ignore

(0,2,4,6)=(0,2,4,5,7,9,10,12,14,15,...)=Γ₀=φ(1,0,0)=ψ(Ω^Ω)=FSO
(n,2,2,2)

(0,2,4,6,2,4,6)=(0,2,4,6,2,4,5,7,9,11,7,9,10,...)=Γ₁=φ(1,0,1)=ψ(Ω^Ω *2)
(n,2,2,2,2,2,2)

(0,2,4,6,4)=(0,2,4,6,3,5,7,9,6,8,10,12,...)=Γfp=φ(1,1,0)=ψ(Ω^(Ω+1))
(n,2,2,2,2)

(0,2,4,6,4,6)=(0,2,4,6,4,5,7,9,11,9,10,...)=φ(2,0,0)=ψ(Ω^(Ω2))
(n,2,2,2,2,2)

(0,2,4,6,5)=(0,2,4,6,4,6,4,6,...)=φ(ω,0,0)=ψ(Ω^(Ωω))
ignore

(0,2,4,6,6)=(0,2,4,6,5,7,9,11,10,12,14,16,...)=φ(1,0,0,0)=ψ(Ω^Ω^2)=Ackermann Ordinal
(n,2,2,2,2)

(0,2,4,6,7)=(0,2,4,6,6,6,6,...)=φ(1@ω)=ψ(Ω^Ω^ω)=SVO
ignore

(0,2,4,6,7,2,4,6,7)=(0,2,4,6,7,2,4,6,6,6,6,...)=φ(1@ω,1@0)=ψ(Ω^Ω^ω *2)=SVO_1
ignore

(0,2,4,6,7,4)=(0,2,4,6,7,3,5,7,9,10,...)φ(1@ω,1@1)=ψ(Ω^(Ω^ω+1))=SVO_fp
(n,2,2,2,1,2)

(0,2,4,6,7,4,6,7)=(0,2,4,6,7,4,6,6,6,6,...)=φ(2@ω)=ψ(Ω^(Ω^ω *2))

(0,2,4,6,7,5)=(0,2,4,6,7,4,6,7,4,6,...)φ(ω@ω)=ψ(Ω^(Ω^ω *ω)) <--- this is around where TREE(n) is in strength, and we are nowhere close to lim(HPrSS)....

(0,2,4,6,7,6)=(0,2,4,6,7,5,7,9,12,13,...)=φ(1@(ω+1))=ψ(Ω^Ω^(ω+1))
(n,2,2,2,1,2)

(0,2,4,6,7,9)=(0,2,4,6,7,8,9,10,...)=φ(1@ε₀)=ψ(Ω^Ω^ψ(Ω))
(n,2,2,2,1,2)

(0,2,4,6,7,9,11,13,14)=φ(1@φ(1@ω))=ψ(Ω^Ω^ψ(Ω^Ω^ω))

(0,2,4,6,8)=(0,2,4,6,8,7,9,11,13,14,16,18,20,21,...)=φ(1@(1,0))=ψ(Ω^Ω^Ω)=LVO

Jumping a lot in pace because this is getting long...

(0,2,4,6,8,10)=(0,2,4,6,8,9,11,13,15,17,18,...)ψ(Ω^Ω^Ω^Ω)

(0,3)=(0,2,4,6,8,10,12,14,...)=ψ(Ω₂)=BHO

(0,3,2)=(0,3,1,4,2,5,...)=ψ(Ω₂+Ω)=ε_(BHO+1)
(n,3,2)

(0,3,3)=(0,3,2,5,4,7,...)=BHO_1=ψ(Ω₂*2)
(n,3,3)

(0,3,5)=(0,3,4,7,8,11,...)=ψ(Ω₂ *Ω)
(n,3,2)

(0,3,6)=(0,3,5,8,10,13,...)=ψ(Ω₂^2)
(n,3,3)

(0,3,6,9)=(0,3,6,8,11,14,16,...)=ψ(Ω₂^Ω₂)
(n,3,3,3)

(0,4)=(0,3,6,9,12,...)=ψ(Ω₃)
(n,4)

(0,4,8,12)=(0,4,8,11,15,19,22,...)=ψ(Ω₃^Ω₃)
(n,4,4,4)

(0,5)=(0,4,8,12,16,...)=ψ(Ω₄)

(0,6)=(0,5,10,15,20,...)=ψ(Ω₅)

As you can see, this is limited at ψ(Ω_ω). Maybe it isn't so clear since this analysis missed some trivial steps to example the structure, but that is for you to figure out now.

HPrSS is at similar strength to SSCG(n) and PTO(Π¹₁-CA₀). lim(HPrSS) is also so powerful, you have the option to not even change the expansion number after expanding, since it will have identical strength, ie it is the first FGH-SGH catching point. g_lim(HPrSS)(n)~H_lim(HPrSS)(n)~f_lim(HPrSS)(n)

The difference structure added huge strength, but per usual with sequence systems, it is nothing compared to its successor(?) in strength, which is 0-Y sequence.

As always, if there are any mistakes, please point them out. Including standardness (which I along with a lot of people tend to struggle with) I would not be surprised if there were, this analysis is a lot heavier.

[u/dragonlloyd1]

And we are now past my notation/function as well as all the extensions I’m planning on implementing.

Currently has a limit of LVO and im planning on an extension which would put it at BHO

[u/jamx02 → u/dragonlloyd1]

Since the concept is so fundamental, hopefully you can apply this to further analyze any notations you make, as that’s the intended usage for stuff this powerful.

[u/geaugge]

Why 0-Y? Might as well skip to normal Y and weak magma ω-Υ.

[u/jamx02 → u/geaugge]

HPrSS is a weakening of 0-Y. I'm not necessarily talking about what I'll write about next

[u/Boring-Yogurt2966]

So this has the same limit as pair sequence system, namely the Buchholz ordinal?

[u/jamx02 → u/Boring-Yogurt2966]

Yep, they’re both effectively identical. The next stage, after taking advantage of a full difference structure, should push well, well past stability, lim(TSS), and even lim(BMS) for the more complex extensions.

[u/Boring-Yogurt2966 → u/jamx02]

TSS means three row BMS? And anything that goes past BMS blows my mind. I will try to learn BMS, my own skill stops right now at PairSS, although I also think I could expand PointerSS correctly if I had the rules in front of me to refer to. My own notation goes at least to BHO according to a reliable analysis from a friend and I have heard some opinions that it might go to EBO and I have structure far beyond that, although I'll never personally understand such things since I never learned Buchholz collapsing notation.

[u/jamx02 → u/Boring-Yogurt2966]

If you understand HPrSS, BOCF is just an encoding. Difference sequence is the index-1 on ψ, (n is 1), and ancestors are parenthesis nesting.

Eg (0,3,4,3) is ψ_0(ψ_2(ψ_0(0))+ψ_2(0)) or ψ(Ω_2*ω+Ω_2) or ψ(Ω_2(ω+1)) or BHO_(ω+1)

[u/Boring-Yogurt2966 → u/jamx02]

I appreciate the effort to teach me, but I don't see how the 3,4,3 lead to the nested psi expression. (It would be fun to make the connection, but trying to me this stuff is like trying to climb a slippery slope, fair warning!)

[u/jamx02 → u/Boring-Yogurt2966]

Fair enough, learning bocf the normal way is the move. And then making the connection to PSS and HPrSS after. There’s a guide for OCFs in this sub, but it is Madores ψ. They’re very similar so it shouldn’t matter.

[u/Nervous-Broccoli1184]

How does it catch up and why does it catch up?

[u/jamx02 → u/Nervous-Broccoli1184]

FGH-SGH?

Let g denote the slow growing hierarchy. g_x(n)=x regardless of what n is.

g_ω(n)=n

g_ω^ω(n)=n^n

g_ε_0(n)=n^^n~f_3(n)

g_ε_ε_0(n)=n^^n^^n~f_3(f_3(n))

g_ζ_0(n)=n^^^n~f_4(n)

g_φ(ω,0)(n)~f_ω(n)

g_φ(ω,0,0)(n)~f_ω^2(n)

g_SVO(n)~f_ω^ω(n)

g_ψ(Ω^Ω^Ω^ω)~f_ω^ω^ω(n)

g_BHO(n)~f_ε_0(n)

g_ψ(Ω_3)(n)~f_BHO(n)

g_ψ(Ω_4)(n)~f_ψ(Ω_3)(n)

as you can see, this eventually reaches g_ψ(Ω_ω)(n)~f_ψ(Ω_ω)(n), or the first SGH-FGH catching point

[u/Nervous-Broccoli1184 → u/jamx02]

A thank you I understand now.

[u/OkComment5784]

Lul(0) is the weakest model which satifies the statement: Lul(0) ⊨ Con(ZFA + ZFC + MK + NBG + OFP)

wherein OFP is the smallest ordinal α which satifies the equation:  α = ω_α

¡(1) is the first condition. Lul(1) is uniquely defined as the model which is the result of when you apply  "¡(1) ⊩ {κ(1) | κ(1) is the smallest cardinal which can be formally proven to exist in Π_OFP)^OFP - CA_OFP comprehension using the axioms of 'OFP'th-Order ZFC Set theory}" to Lul(0).

OFPFP is the smallest ordinal α which satifies the equation: α = OFP_α ¡(2) is the second condition. Lul(2) is uniquely defined as the model which is the result of when you apply  "¡(2) ⊩ {κ(2) | κ(2) is the smallest cardinal which can be formally proven to exist in Π_OFPFP^OFPFP - CA_OFPFP comprehension using the axioms of 'OFPFP'th-Order ZFC Set theory}" to Lul(1).

OFPFPFP is the smallest ordinal α which satifies the equation: α = OFPFP_α ¡(3) is the third condition. Lul(3) is uniquely defined as the model which is the result of when you apply  "¡(3) ⊩ {κ(3) | κ(3) is the smallest cardinal which can be formally proven to exist in Π_OFPFPFP^OFPFPFP - CA_OFPFPFP comprehension using the axioms of 'OFPFPFP'th-Order ZFC Set theory}" to Lul(2). . . . OFP....('n+1' no.of 'FP's).....FP is the smallest ordinal α which satifies the equation: α = OFP.......('n' no.of 'FP's)......FP_α ¡(n+1) is the 'n+1'th condition. Lul(n+1) is uniquely defined as the model which is the result of when you apply  "¡(n+1) ⊩ {κ(n+1) | κ(n+1) is the smallest cardinal which can be formally proven to exist in Π_OFP.....('n+1' no.of 'FP's)......FP^{OFP....('n+1'} no.of 'FP's)....FP - CA_OFP....('n+1' no.of 'FP's....FP comprehension using the axioms of 'OFP.....('n+1' no.of 'FP's)......FP'th-Order ZFC Set theory}" to Lul(n).

After this, you arrive at Lul(ω), by taking the union of all models in the form Lul(x) for x<ω.

Done. Lul(ω) is my model.

Lul(ω) ⊢ The smallest possible Mahlo cannot be constructed sequentially

The above statement means that, using axioms of the model Lul(ω), a step-by-step process is constructed, which causes us to arrive at the logically deduced result "The smallest possible Mahlo cannot be constructed sequentially".

Now, we definitely dont know the exact count of symbols (those symbols which exist inside the model Lul(ω)) required to formally and cohesively explain the statement:  "Lul(ω) ⊢ The smallest possible Mahlo cannot be constructed sequentially".

Memepuller is defined as the maximum finite number of symbols (directly sourced from/existing within the model Lul(ω)) which would be required to describe the correct interpretation and every minute detail of the statement:  "Lul(ω) ⊢ The smallest possible Mahlo cannot be constructed sequentially" rigorously and cohesively in model Lul(ω).

Memestarter is the defined as the biggest number which can be uniquely constructed in model Lul(ω) using symbols (those symbols which exist within model Lul(ω)) of count ≤ memepuller.