T3 - Small Set Theory System vs Utter Oblivion

Further to Blog T2, I conceived third thought experiment.

Let's assume there exists a Set Theory System (STS) that is made of ω layers as described below, where y is a positive integer while a, b, c and z are non-negative integers:-

1. Base layer, STS(ω) (where STS(0) = ZFC): C_a(n) = C_0,a(n) = C_0,0,a(n) = ... (this was defined in Blog T2)
2. First layer, STS(ω,ω): C_b,a(n) = C_0,b,a(n) = C_0,0,b,a(n) = ...
3. Second layer, STS(ω,ω,ω): C_c,b,a(n) = C_0,c,b,a(n) = C_0,0,c,b,a(n) = ...
4. (y-1)^th layer, STS(ω◄ω): C_z◄y(n) = C_z,z,...,z,z(n), with y of z's.

In short, all lower layers are subsets of immediate upper layer; in other words, any layer is much more powerful and more robust than all layers below it:

1. {STS(ω)} = {STS(0), STS(1), STS(2), ...}
2. {STS(ω,ω)} =
   1. {STS(0,0), STS(0,1), STS(0,2), ...} = {STS(ω)}; plus
   2. {STS(1,0), STS(1,1), STS(1,2), ...}; plus
   3. {STS(2,0), STS(2,1), STS(2,2), ...}; plus
   4. {STS(3,ω)}, {STS(4,ω)} and so on.
3. This shall continue for STS for 3 and more ω's.
4. STS(0) is the ω-order set theory system that defines Rayo's number (which is at first order); I assumed it is ZFC plus extensions of ZFC.
5. STS(1) is next level ω-order set theoty system that includes STS(0).
6. This definition shall continue for the whole STS(ω◄ω) universe.

To put these into perspective: (≍ denotes 'is similar to')

1. STS(0) to STS(1) ≍ STS(1) to STS(2) ≍ ... ≍ STS(1,0) to STS(1,1) ≍ STS(1,1) to STS(1,2) ≍ ... ≍ PA to ZFC;
2. STS(ω) to STS(1,0) ≍ STS(1,ω) to STS(2,0) ≍ ... ≍ STS(ω,ω) to STS(1,0,0) ≍ ... ≍ STS(ω,ω.ω) to STS(1,0,0,0) ≍ ... ≍ Robinson Arithmatic to ZFC.

Growth rate comparison:

C₀(n) << C₁(n) << ... << C_n(n) <<< C_1,0(n) << C_1,1(n) << ... << C_n,n(n) <<< C_1,0,0(n) << C_1,0,1(n) ...

Since this is in uncomputable set theory realm, it's inappropriate to compare it with recursive computable realm, but still it resembles {φ(1,0), φ(1,0,0), φ(1,0,0,0), ...} of SVO, e.g. a sequence of functions can be defined as {C_1,0(n), C_1,0,0(n), C_1,0,0,0(n), ...}. Hence I call it SSTS (Small Set Theory System).

Assuming both are well defined, I am confident that C_n◄n(n) is comparable to Bowers' n-System (that defines Utter Oblivion) as descibed by Ytosk.