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This page uses Kripke's definition of stability[1] with a notational modification, and if f is an ordinal function where f(α0)=β, then "α0 is β-stb." can be written as "α0 is f-stb.". Note that "α is β-stb." is also sometimes used here. Lowercase Greek letters stand for ordinals, and also a function λβ.f(β) with domain Ord can be written as f(_)

The notational modification is that the symbol ⪯Σ1 is used in place of Richter and Aczel's definition[2] of ≺Σ1, the Σ1 elementary substructure relation, because a⪯Σ1b also holds if a=b. Also a≺Σ1b is used for the condition "a⪯Σ1b and a != b". Note that a≺Σ1b and a≺Σ1c doesn't necessarily imply b≺Σ1c.

Another important thing to note is that (for n>0) a≺Σnb implies a≺Πnb, because each Σn formula ϕ is equivalent (by quantifier exchange) to a negation of a Πn-formula ϕ', and since a |= ϕ ↔ b |= ϕ, and both sides are false, then (a |= ϕ')&(b |= ϕ').

Lemma(d,e,f) means lemma 5.3.2 applied to "d⪯Σ1(e u f)", for a class A, P(A) denotes the class of subsets of A, and a formula is Σn(A) (Πn(A) resp.) if it's Σnn resp.) with parameters from A[citation needed].

Stability

Definitions

For a function f:Ord→Ord, define the "sum-predecessor of f" g:Ord→Ord:

  • Given some α∈Ord, if f(α) is of the form α+(n+1) for some n∈N, then g(α):=α+n
  • Else, then g(α):=f(α)

For β∈Ord and f:Ord→Ord, define an ordinal α as being "β-ply-f-stable" if ∃((αη)η≤β∈Ordβ+1)(α0=α &
∃ν(αβ=f -(ν)) & ∀(γ,δ≤β)(γ∈δ → LαγΣ1Lαδ)), where f - is f if β is a successor ordinal, or the sum-predecessor of f if β is a limit ordinal. (we write "α0=α" to avoid ill-definedness by multiple quantification)

This definition is based off of the observation that "α0 is ω-ply-(+1)-stb." is also equivalent to Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω, where in this case αω=sup({αi:i∈ω}). The reason for this is the infinitary conjunction &{Lemma(Lα0,Lαi,Lαi+1):i∈ω} and the definition of limit stages of the constructible hierarchy.

Increasing list

This is a list of some stable ordinals (ordered by increasing size of first ordinal satisfying the condition)

Ordinals in A Zoo of Ordinals and some other important ordinals are bolded

This is a list of some stable ordinals in increasing order (ordered either using the remark from here[3], theorem/corollary 1.1, or guesses, and some small ordinals' sizes are given in A Zoo of Ordinals):

(+1)-stb

  1. α0 is (+1)-stb. ↔ Lα0Σ1Lα0+1[4] ↔ α0 is Π01-rfl.[1]
  2. α0 is (+2)-stb. ↔ Lα0Σ1Lα0+2[1]
  3. α0 is (+3)-stb. ↔ Lα0Σ1Lα0+3
  4. α0 is (+ω)-stb. ↔ Lα0Σ1Lα0 (this is equivalent to "∀(i∈ω)(α0 is (+i)-stb.)" due to the infinitary conjunction &{Lemma(Lα0,Lα0+i,Lα0+i+1):i∈ω} and the definition of limit stages of the constructible hierarchy)
  5. α0 is (+ε0)-stb. ↔ Lα0Σ1Lα00
  6. α0 is (*2)-stb. ↔ Lα0Σ1Lα0*2 ↔ α0 is (+α0)-stb. ↔ Lα0Σ1Lα00 ({α∈ω1:α is (*2)-stb.} is a subset or equal to {α∈ω1:α is (+ε0)-stb.} by corollary 1.1[5])
  7. α0 is (*2+1)-stb. ↔ Lα0Σ1Lα0*2+1 ↔ α0 is (+(α0+1))-stb. ↔ Lα0Σ1Lα0+(α0+1)
  8. α0 is (*3)-stb. ↔ Lα0Σ1Lα0*3 ↔ α0 is (+(α0*2))-stb. ↔ Lα0Σ1Lα0+(α0*2)
  9. α0 is (*ω)-stb. ↔ Lα0Σ1Lα0 ↔ α0 is (ω_+1)-stb. ↔ Lα0Σ1Lωα0+1 (this function is λβ.ωβ+1)
  10. α0 is (^2)-stb. ↔ Lα0Σ1Lα0^2 ↔ α0 is (ω_*2)-stb. ↔ Lα0Σ1Lωα0*2
  11. α0 is (^ω)-stb. ↔ Lα0Σ1Lα0 ↔ α0 is (ωω_+1)-stb. ↔ Lα0Σ1Lωωα0+1
  12. α0 is (ε_+1)-stb. ↔ Lα0Σ1Lεα0+1
  13. α0 is (ζ_+1)-stb. ↔ Lα0Σ1Lζα0+1
  14. α0 is (_+)-stb. ↔ Lα0Σ1Lα0+[4]+ is next admissible after β[1][4])
  15. α0 is (_++1)-stb. ↔ Lα0Σ1Lα0++1
  16. α0 is (_++_)-stb. ↔ Lα0Σ1Lα0+0 (different than (_+)*2)
  17. α0 is ((_+)*2)-stb. ↔ Lα0Σ1L0+)*2
  18. α0 is (ε_++1)-stb. ↔ Lα0Σ1Lεα0++1
  19. α0 is (_++)-stb. ↔ Lα0Σ1Lα0++[6]++ is 2nd next admissible after β[4], and it can also be read as (β+)+ )
  20. α0 is (_+++1)-stb. ↔ Lα0Σ1Lα0+++1
  21. α0 is (_+++_)-stb. ↔ Lα0Σ1Lα0++0
  22. α0 is (_+++_+)-stb. ↔ Lα0Σ1Lα0++0+
  23. α0 is ((_++)*2)-stb. ↔ Lα0Σ1L0++)*2
  24. α0 is (_+++)-stb. ↔ Lα0Σ1Lα0+++
  25. α0 is (_++++)-stb. ↔ Lα0Σ1Lα0++++
  26. α0 is (_(+ω))-stb. ↔ Lα0Σ1Lα0(+ω) (where β(+ω) is ωth next admissible after β)
  27. α0 is (_(+ω)+1)-stb. ↔ Lα0Σ1Lα0(+ω)+1
  28. α0 is (_(+ω)+_)-stb. ↔ Lα0Σ1Lα0(+ω)0
  29. α0 is (_(+ω)+_+)-stb. ↔ Lα0Σ1Lα0(+ω)0+
  30. α0 is (_(+ω)+_++)-stb. ↔ Lα0Σ1Lα0(+ω)0++
  31. α0 is (_(+ω)*2)-stb. ↔ Lα0Σ1Lα0(+ω)*2
  32. α0 is (ε_(+ω)+1)-stb. ↔ Lα0Σ1Lεα0(+ω)+1
  33. α0 is (_(+ω+1))-stb. ↔ Lα0Σ1Lα0(+ω+1) (where β(+ω+1) is ω+1th next admissible after β)
  34. α0 is (_(+ε0))-stb. ↔ Lα0Σ1Lα0(+ε0)
  35. α0 is (_(+(_)))-stb. ↔ Lα0Σ1Lα0(+α0)
  36. α0 is (_(+(_+1)))-stb. ↔ Lα0Σ1Lα0(+0+1))
  37. α0 is (_(+(_+)))-stb. ↔ Lα0Σ1Lα0(+0+))
  38. α0 is (least fixed point of λθ.(_(+θ)) after _)-stb. ↔ Lα0Σ1Lα1 (where α1 is the least fixed point of λθ.(_(+θ)) after α0)
  39. α0 is (next recursively inaccessible after _)-stb. - Lα0Σ1Lα1[6] where α1 is next recursively inaccessible after α0 (recursively inaccessible means "admissible and limit of admissibles"[7])
  40. α0 is (next admissible after next recursively inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is (next recursively inaccessible after α0)+
  41. α0 is (next recursively 1-inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is next recursively 1-inaccessible after α0 (recursively 1-inaccessible means "recursively inaccessible and limit of recursively inaccessibles", this is called "recursively hyper-inaccessible"[7] by Madore)
  42. α0 is (next rec. 2-inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is next rec. 2-inaccessible (rec. 1-inacessible and limit of rec. 1-inaccessibles) after α0
  43. α0 is (next rec. ω-inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is next rec. ω-inaccessible (rec. β-inacc. for all β∈ω) after α0
  44. α0 is (next rec. ω+1-inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is next rec. ω+1-inaccessible (rec. ω-inaccessible and limit of rec. ω-inaccessibles) after α0
  45. α0 is (next rec. hyper-inaccessible after _)-stb. - Lα0Σ1Lα1 where α1 is next rec. hyper-inaccessible (rec. β-inaccessible for all β<α1) after α0 (this term is different than Madore's definition of "rec. hyper-inaccessible")
  46. α0 is (next rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next rec. Mahlo after α0
  47. α0 is (2nd rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is 2nd rec. Mahlo after α0
  48. α0 is (ωth rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is ωth rec. Mahlo after α0
  49. α0 is (ε0th rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is ε0th rec. Mahlo after α0
  50. α0 is (_th rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is α0th rec. Mahlo after α0
  51. α0 is (α11th rec. Mahlo after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is α1th rec. Mahlo after α0
  52. α0 is (next "rec. Mahlo and limit of rec. Mahlo" after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "rec. Mahlo and limit of rec. Mahlo" after α0
  53. α0 is (next "rec. Mahlo and limit of "rec. Mahlo and limit of rec. Mahlo"" after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "rec. Mahlo and limit of "rec. Mahlo and limit of rec. Mahlo"" after α0
  54. α0 is (next "rec. Mahlo and limit of"ω after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "rec. Mahlo and limit of "rec. Mahlo and limit of"ω after α0 ("rec. Mahlo and limit of"ω is formalizable by creating predicates RML0(β):="β is rec. Mahlo", RMLδ+1(β):="RML0 & sup({γ:RMLδ(γ)} n β)=β", for limit δ RMLδ(β):=∀(δ'<δ)(RMLδ'(β)), then β has the ""rec. Mahlo and limit of"ω condition" if RMLω(β))
  55. α0 is (next "rec. Mahlo and limit of"ω1CK after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "rec. Mahlo and limit of "rec. Mahlo and limit of"ω1CK after α0 ("rec. Mahlo and limit of"ω1CK is formalizable as β having the ""rec. Mahlo and limit of"ω1CK condition" if RMLω1CK(β))
  56. α0 is (next "rec. Mahlo and limit of"α0 after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "rec. Mahlo and limit of "rec. Mahlo and limit of"α0 after α0 ("rec. Mahlo and limit of"α0 is formalizable as β having the ""rec. Mahlo and limit of"α0 condition" if RMLα0(β))
  57. α0 is (next "Π2-rfl. on Π2-rfl." after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next "Π2-rfl. on Π2-rfl." after α0 (this can be thought of as "rec. 1-Mahlo")
  58. α0 is (next "Π2-rfl. on Π2-rfl. on Π2-rfl." after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next "Π2-rfl. on Π2-rfl. on Π2-rfl." after α0 (this can be thought of as "rec. 2-Mahlo")
  59. α0 is (next "Π2-rfl. on"ω after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next "Π2-rfl. on"ω after α0 (this can be thought of as "rec. ω-Mahlo", "Π2-rfl. on"ω is formalizable by creating predicates RM0(β):=True, RMi+1(β):=β is Π2-rfl. on {γ:RMi(γ)}, then β has the " "Π2-rfl. on"ω condition" if ∀(i∈N)(RMi(β)))
  60. α0 is (next "Π2-rfl. on"ω after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next "Π2-rfl. on"α0 after α0 (this can be thought of as "rec. α0-Mahlo", "Π2-rfl. on"α0 is formalizable by creating predicates RM0(β):=True, RMδ+1(β):=β is Π2-rfl. on {γ:RMδ(γ)}, for limit δ RMδ(β):=∀(δ'<δ)(RMδ'(β)), then β has the " "Π2-rfl. on"α0 condition" if RMα0(β)))
  61. α0 is (next Π3-rfl. after _)-stb. &leftrightarrow; Lα0Σ1Lα1[6] where α1 is next Π3-rfl. after α0
  62. α0 is (next Π4-rfl. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next Π4-rfl. after α0
  63. α0 is (next "limit of Πn-rfl." after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "limit of Πn-rfl." after α0
  64. α0 is doubly-(+1)-stb.[6] &leftrightarrow; Lα0Σ1Lα1Σ1Lα1+1[6] &leftrightarrow; α0 is (next (+1)-stb. after _)-stb. - Lα0Σ1Lα1 where α1 is next (+1)-stb. after α0 &leftrightarrow; α0 is ((next (+1)-stb. after _)+1)-stb. (because of the transitivity of ⪯Σ1[citation needed]; using ⪯Σ1 is compatible with this definition but not the definitions of stability used later, in which case ≺Σ1 should be used instead)
  65. α0 is ((next (+1)-stb. after _)+2)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is (next (+1)-stb. after α0)+2
  66. α0 is (ε(next (+1)-stb. after _)+1)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is ε(next (+1)-stb. after α0)+1
  67. α0 is ((next (+1)-stb. after _)+)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is (next (+1)-stb. after α0)+
  68. α0 is (next rec. Mahlo after (next (+1)-stb. after _))-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next rec. Mahlo after (next (+1)-stb. after α0)
  69. α0 is (next Π3-rfl. after (next (+1)-stb. after _))-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next Π3-rfl. after (next (+1)-stb. after α0)
  70. α0 is (next Π4-rfl. after (next (+1)-stb. after _))-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next Π4-rfl. after (next (+1)-stb. after α0)
  71. α0 is (next "limit of Πn-rfl." after (next (+1)-stb. after _))-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "limit of Πn-rfl." after (next (+1)-stb. after α0)
  72. α0 is (2nd (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is 2nd (+1)-stb. after α0
  73. α0 is (3rd (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is 3rd (+1)-stb. after α0
  74. α0 is (ωth (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is ωth (+1)-stb. after α0
  75. α0 is (ε0th (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is ε0th (+1)-stb. after α0
  76. α0 is (_th (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is α0th (+1)-stb. after α0
  77. α0 is (_+1th (+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is α0+1th (+1)-stb. after α0
  78. Ignore this line
  79. α0 is (next "(+1)-stb. and limit of (+1)-stb." after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "(+1)-stb. and limit of (+1)-stb." after α0 &leftrightarrow; α1 is α1th (+1)-stb. after α0
  80. α0 is (next "Π2-rfl. on (+1)-stb." after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "Π2-rfl. on (+1)-stb." after α0
  81. α0 is (next "Π2-rfl. on Π2-rfl. on (+1)-stb." after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "Π2-rfl. on Π2-rfl. on (+1)-stb." after α0
  82. α0 is (next "Π3-rfl. on (+1)-stb." after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next "Π3-rfl. on (+1)-stb." after α0
  83. α0 is doubly-(+2)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα1+2 &leftrightarrow; α0 is (next (+2)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (+2)-stb. after α0
  84. α0 is doubly-(+ω)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα1 &leftrightarrow; α0 is (next (+ω)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (+ω)-stb. after α0
  85. α0 is doubly-(+ε0)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα10 &leftrightarrow; α0 is (next (+ε0)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (+ε0)-stb. after α0
  86. α0 is doubly-(+α0)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα10 (note that this diverges from doubly-(*2)-stb.) &leftrightarrow; α0 is (next (+α0)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (+α0)-stb. after α0
  87. α0 is doubly-(*2)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα12 &leftrightarrow; α0 is (next (+α1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (+α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1 where α1 is next (*2)-stb. after α0
  88. α0 is (next (εα1+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (εα1+1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1 where α1 is next (ε_+1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lεα1+1
  89. α0 is (next (α1+)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (α1+)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1 where α1 is next (_+)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα1+
  90. α0 is (next (next Π3-rfl. after α1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next Π3-rfl. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2 where α2 is next Π3-rfl. after α1

Triply-(+1)-stb

  1. α0 is triply-(+1)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα2+1 (citation needed?) &leftrightarrow; α0 is (next (next (+1)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα2 where α2 is next (+1)-stb. after α1 &leftrightarrow; α0 is (next doubly-(+1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next doubly-(+1)-stb. after α0

ZFC can prove the existence of a triply-(+1)-stb. ordinal given for all natural n, there are ω1-many[8] "ordinals that are doubly-(least Πn-rfl. after _)-stb.". Take the infinitary intersection of each of these sets (possible using restricted separation, e.g. restricted in ω1), by this[3] remark each will be a non-strict subset of the next, so their intersection will be non-empty (note that an ordinal is (+1)-stb. iff it's Πn-rfl. for all natural n[7] due to its Π01-reflectingness[1])

  1. α0 is ((next (next (+1)-stb. after α1)-stb. after α0)+1)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is (next (next (+1)-stb. after α1)-stb. after α0)+1
  2. α0 is (next ((next (+1)-stb. after α1)+1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next ((next (+1)-stb. after α1)+1)-stb. after α0
  3. α0 is (next (next (+2)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (+2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα2+2
  4. α0 is (next (next (+ω)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (+ω)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα2+2
  5. α0 is (next (next (+α0)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (+α0)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα20
  6. α0 is (next (next (+α1)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (+α1)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα21
  7. α0 is (next (next (+α2)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (+α2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα22
  8. α0 is (next (next (_+)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (α2+)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα2+
  9. α0 is (next (next (next rec. Mahlo after _)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (next rec. Mahlo after α2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3 where α3 is next rec. Mahlo after α2
  10. α0 is (next (next (next Π3-rfl. after _)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (next Π3-rfl. after α2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3 where α3 is next Π3-rfl. after α2
  11. α0 is quadruply-(+1)-stb. &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1Lα3+1 &leftrightarrow; α0 is (next (next (next (+1)-stb. after α2)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (next (+1)-stb. afer α2)-stb. after α1)-stb. after α0 &leftrightarrow; α0 is (next triply-(+1)-stb. after _)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next triply-(+1)-stb. after α0
  12. α0 is quintuply-(+1)-stb. &leftrightarrow; α0 is (next (next (next (next (+1)-stb. after α3)-stb. after α2)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (next (next (+1)-stb. after α3)-stb. afer α2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1 where α1 is next quadruply-(+1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1Lα4Σ1Lα4+1
  13. α0 is sextuply-(+1)-stb. &leftrightarrow; α0 is (next (next (next (next (next (+1)-stb. after α4)-stb. after α3)-stb. after α2)-stb. after α1)-stb. after α0)-stb. &leftrightarrow; Lα0Σ1Lα1 where α1 is next (next (next (next (next (+1)-stb. after α4)-stb. after α3)-stb. after α2)-stb. after α1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1 where α1 is next quintuply-(+1)-stb. after α0 &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1Lα4Σ1Lα5Σ1Lα5+1
  14. α is ω-ply-(+1)-stb. (Hyp_cos's definition) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1Lα4Σ1... &leftrightarrow; ∃((αi)i<ω∈Ordω)(α0=α & ∀(γ,δ∈ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) (citation needed?) (note that n-ply-(+1)-stb. is also equivalent to ∃((αi)i<1∈Ordn+1)(α0=α & ∀(γ,δ∈n+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)), for example doubly-(+1)-stb. will cause the sequence of α's to be something like (α011+1) )

ω-ply-(+1)-stb

After ω-ply-(+1)-stb., stability becomes more difficult. From here I use original definitions and comparisons are based on guesses.

From here we use the notion of β-ply-f-stability given here. Also given the existence of ω1-many countable ordinals that are (next n-ply-(+1)-stb. after _)-stb. for each 0<n<ω, ZFC can prove the existence of an ordinal that is (next ω-ply-(+1)-stb. after _)-stb. (cf. remark after #91)

This list starts with an equivalent definition of ω-ply-(+1)-stb. and continues onward (there is much more detail between  these conditions that is omitted, some trivial parts of some conditions such as "∃ν(αω=ν)" are omitted):

  1. α is ω-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is possibly a limit ordinal and is sup({αi:i≤ω}); this is also equivalent to "∀(i∈ω)(α0 is i-ply-(+1)-stb.)" by the transitivity of ≺Σ1[citation needed]; also α1 is also ω-ply-(+1)-stb.; also the least such α0 is equal to #2.14 in A Zoo of Ordinals, cf. theorem 2.2)
  2. α is ω-ply-(+2)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+1) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is a successor ordinal, i.e. at least 1 larger than an ordinal; also α1 is also ω-ply-(+2)-stb.)
  3. α is ω-ply-(+3)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+2) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is at least 2 larger than an ordinal)
  4. α is ω-ply-(+ω)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+ω) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is at least ω larger than an ordinal)
  5. α is ω-ply-(+ε0)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+ε0) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is at least ε0 larger than an ordinal)
  6. α is ω-ply-(+α0)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+α0) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is at least α0 larger than an ordinal)
  7. α is ω-ply-(+α0+1)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν+(α0+1)) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is at least α0+1 larger than an ordinal)
  8. α is ω-ply-(*2)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ν2) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is ν2 for some ν)
  9. α is ω-ply-(ε_+1)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αων+1) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is εν+1 for some ν, which is a "successor epsilon")
  10. α is ω-ply-(_+)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω+) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is ν+ for some ν, so αω isn't (limit ordinal)th admissible)
  11. α is ω-ply-(next Π3-rfl. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next Π3-rfl. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next Π3-rfl. after ν for some ν, so αω isn't (limit ordinal)th Π3-rfl.)
  12. α is ω-ply-(next Π4-rfl. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next Π4-rfl. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next Π4-rfl. after ν for some ν, so αω isn't (limit ordinal)th Π4-rfl.)
  13. α is ω+1-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...LαωΣ1Lαω+1 (here αω+1ω+1) &leftrightarrow; α is ω-ply-(next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next (+1)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+1)-stb. ordinal, note that if αω+1 were a limit of (+1)-stb. ordinals, then it would have to be more greater than αω, which is (+1)-stb., which is unlikely (however, this note is without a formal proof); also here Lα0Σ1Lsup({α:i:i∈ω}) and Lα0Σ1Lαω+1, but this doesn't necessarily imply Lsup({αi:i∈ω})Σ1Lαω+1)
  14. α is ω-ply-((next (+1)-stb. after _)+1)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=(Next (+1)-stb. after ν)+1) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is (next (+1)-stb. ordinal after ν, so αω isn't ((limit ordinal)th (+1)-stb. ordinal)+1)
  15. α is ω-ply-((next (+1)-stb. after _)+)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=(Next (+1)-stb. after ν)+) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is (next (+1)-stb. ordinal after ν, so αω isn't ((limit ordinal)th (+1)-stb. ordinal)+)
  16. α is ω-ply-(next Π3-rfl. after next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next Π3-rfl. after next (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next Π3-rfl. after next (+1)-stb. ordinal after ν, so αω isn't next Π3-rfl. after (limit ordinal)th (+1)-stb. ordinal)
  17. α is ω-ply-(next Π4-rfl. after next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next Π4-rfl. after next (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next Π4-rfl. after next (+1)-stb. ordinal after ν, so αω isn't next Π4-rfl. after (limit ordinal)th (+1)-stb. ordinal)
  18. α is ω-ply-(2nd (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=2nd (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is 2nd (+1)-stb. ordinal after ν, so αω isn't (1-greater-than-ordinal)th (+1)-stb. ordinal)
  19. α is ω-ply-(ωth (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=ωth (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is ωth (+1)-stb. ordinal after ν, so αω isn't (ω-greater-than-ordinal)th (+1)-stb. ordinal)
  20. α is ω-ply-(α0th (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω0th (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is α0th (+1)-stb. ordinal after ν, so αω isn't (α0-greater-than-ordinal)th (+1)-stb. ordinal)
  21. α is ω-ply-(εα0+1th (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αωα0+1th (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is εα0+1th (+1)-stb. ordinal after ν, so αω isn't (εα0+1-greater-than-ordinal)th (+1)-stb. ordinal)
  22. α is ω-ply-(α0+th (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω0+th (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is α0+th (+1)-stb. ordinal after ν, so αω isn't (α0+-greater-than-ordinal)th (+1)-stb. ordinal)
  23. α is ω-ply-(_th (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=νth (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is νth (+1)-stb. ordinal after ν, for example αω isn't restricted to (ordinal <αω)th (+1)-stb. ordinal only, so α is a fixed point of the enumeration of (+1)-stb. ordinals)
  24. α is ω+1-ply-(+2)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν+2) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...LαωΣ1Lαω+1 (here αω+1ω+2) &leftrightarrow; α is ω-ply-(next (+2)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+2)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1Lα3Σ1...Lαω (here αω is next (+2)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+2)-stb. ordinal)

From here the comparisons are less detailed (there are a lot of intermediate skipped conditions)

  1. α is ω+1-ply-(+ω)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν+ω) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1ω+ω) &leftrightarrow; α is ω-ply-(next (+ω)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+ω)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is next (+ω)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+ω)-stb. ordinal)
  2. α is ω+1-ply-(+ε0)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν+ε0) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1ω0) &leftrightarrow; α is ω-ply-(next (+ε0)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+ε0)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is next (+ε0)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+ε0)-stb. ordinal)
  3. α is ω+1-ply-(+α0)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν+α0) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1ω0) &leftrightarrow; α is ω-ply-(next (+α0)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+α0)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is next (+α0)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+α0)-stb. ordinal)
  4. α is ω+1-ply-(+α1)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν+α1) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1ω1) &leftrightarrow; α is ω-ply-(next (+α1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next (+α1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is next (+α1)-stb. ordinal after ν, so αω isn't (limit ordinal)th (+α1)-stb. ordinal)
  5. α is ω+1-ply-(*2)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν*2) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is a "*2 ordinal") &leftrightarrow; α is ω-ply-((*2)-stb. after _)-stb.?? &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is αω*2; a larger "*2 ordinal" will also imply this condition)
  6. α is ω+1-ply-(*3)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=ν*3) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is a "*3 ordinal") &leftrightarrow; α is ω-ply-((*3)-stb. after _)-stb.?? &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is αω*3; a larger "*3 ordinal" will also imply this condition)
  7. α is ω+1-ply-(ε_+1)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1ν+1) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is a "successor epsilon ordinal") &leftrightarrow; α is ω-ply-((λθ.εθ+1)-stb. after _)-stb.?? &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is εαω+1; a larger "successor epsilon ordinal" will also imply this condition)
  8. α is ω+1-ply-(_+)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1+) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is a "successor admissible ordinal") &leftrightarrow; α is ω-ply-((λθ.θ+)-stb. after _)-stb.?? &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is αω+; a larger "successor admissible ordinal" will also imply this condition)
  9. α is ω+1-ply-(next Π3-rfl. after _)-stb. &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω+1=next Π3-rfl. after ν) & ∀(γ,δ≤ω+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is a "successor Π3-rfl.ordinal") &leftrightarrow; α is ω-ply-((λθ.next Π3-rfl. after θ)-stb. after _)-stb.?? &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is next Π3-rfl. after αω)
  10. α is ω+2-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω+2∈Ordω+3)(α0=α & ∀(γ,δ≤ω+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2 (here αω+2ω+1+1) &leftrightarrow; α is ω+1-ply-(next (+1)-stb. after _)-stb. &leftrightarrow;∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω=Next (+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is the next (+1)-stb. ordinal after ν, so αω+1 isn't (limit ordinal)th (+1)-stb. ordinal) &leftrightarrow; α is ω-ply-(next doubly-(+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next doubly-(+1)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is the next doubly-(+1)-stb. ordinal after ν, so αω isn't (limit ordinal)th doubly-(+1)-stb. ordinal; note that if αω+2 were a limit of doubly-(+1)-stb. ordinals, then it would have to be more greater than αω, which is doubly-(+1)-stb., which is unlikely (however, this note is without a formal proof))

Skipping even more intermediate conditions from here

  1. α is ω+2-ply-(*2)-stb. &leftrightarrow; ∃((αi)i≤ω+2∈Ordω+3)(α0=α & ∃ν(αω+2=ν*2) & ∀(γ,δ≤ω+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2 (here αω+2ω+1*2) &leftrightarrow; α is ω+1-ply-(next (*2)-stb. after _)-stb.?? &leftrightarrow;∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω=Next (*2)-stb. after ν)?? & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is the next (*2)-stb. ordinal after ν, so αω+1 isn't (limit ordinal)th (*2)-stb. ordinal) &leftrightarrow; α is ω-ply-(next doubly-(*2)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next doubly-(*2)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is the next doubly-(*2)-stb. ordinal after ν, so αω isn't (limit ordinal)th doubly-(*2)-stb. ordinal)
  2. α is ω+2-ply-(_+)-stb. &leftrightarrow; ∃((αi)i≤ω+2∈Ordω+3)(α0=α & ∃ν(αω+2+) & ∀(γ,δ≤ω+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2 (here αω+2ω+1+) &leftrightarrow; α is ω+1-ply-(next (λθ.θ+)-stb. after _)-stb.?? &leftrightarrow;∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω=Next (_+)-stb. after ν)?? & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is the next (_+)-stb. ordinal after ν, so αω+1 isn't (limit ordinal)th (_+)-stb. ordinal) &leftrightarrow; α is ω-ply-(next doubly-(λθ.θ+)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next doubly-(_+)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is the next doubly-(_+)-stb. ordinal after ν, so αω isn't (limit ordinal)th doubly-(_+)-stb. ordinal)
  3. α is ω+3-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω+3∈Ordω+4)(α0=α & ∀(γ,δ≤ω+3)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1Lαω+3 (here αω+3ω+2+1)  &leftrightarrow; α is ω+2-ply-(next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω+2∈Ordω+3)(α0=α & ∃ν(αω+2=next (+1)-stb. after ν) & ∀(γ,δ≤ω+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2 (here αω+2=next (+1)-stb. after αω+1) &leftrightarrow; α is ω+1-ply-(next doubly-(+1)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω=Next doubly-(+1)-stb. after ν)?? & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is the next doubly-(+1)-stb. ordinal after ν, so αω+1 isn't (limit ordinal)th doubly-(+1)-stb. ordinal) &leftrightarrow; α is ω-ply-(next triply-(+1)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next triply-(+1)-stb. after _)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is the next triply-(+1)-stb. ordinal after ν, so αω isn't (limit ordinal)th triply-(+1)-stb. ordinal)
  4. α is ω+4-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω+4∈Ordω+5)(α0=α & ∀(γ,δ≤ω+4)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1Lαω+3Σ1Lαω+4 (here αω+4ω+3+1) &leftrightarrow; α is ω+3-ply-(next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω+3∈Ordω+4)(α0=α & ∃ν(αω+3=next (+1)-stb. after ν) & ∀(γ,δ≤ω+3)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1Lαω+3 (here αω+3=next (+1)-stb. after αω+2) &leftrightarrow; α is ω+2-ply-(next doubly-(+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω+2∈Ordω+3)(α0=α & ∃ν(αω+2=next doubly-(+1)-stb. after ν) & ∀(γ,δ≤ω+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2 (here αω+2=next doubly-(+1)-stb. after αω+1) &leftrightarrow; α is ω+1-ply-(next triply-(+1)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω+1∈Ordω+2)(α0=α & ∃ν(αω=Next triply-(+1)-stb. after ν)?? & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1 (here αω+1 is the next triply-(+1)-stb. ordinal after ν, so αω+1 isn't (limit ordinal)th triply-(+1)-stb. ordinal) &leftrightarrow; α is ω-ply-(next quadruply-(+1)-stb. after _)-stb.?? &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=Next quadruply-(+1)-stb. after _)-stb. after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here αω is the next quadruply-(+1)-stb. ordinal after ν, so αω isn't (limit ordinal)th quadruply-(+1)-stb. ordinal)

ω2-ply-(+1)-stb

  1. α is ω2-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω2∈Ordω2+1)(α0=α & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1...Lαω2 (here αω2 is possibly a limit ordinal and is at least sup({αγ:γ∈ω2})) &leftrightarrow; α is ω-ply-(next ω-ply-(+1)-stb. ordinal after _)-stb. &leftrightarrow; ∃((αi)i≤ω∈Ordω+1)(α0=α & ∃ν(αω=next ω-ply-(+1)-stb. ordinal after ν) & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...Lαω (here ∀(i∈ω)(αω is i-ply-(+1)-stb.), which is equivalent to "αω is ω-ply-(+1)-stb." (cf. remark on line 105))
  2. α is ω2+1-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω2+1∈Ordω2+2)(α0=α & ∀(γ,δ≤ω2+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1...Lαω2Σ1Lαω2+1 (here αω2+1 is αω2+1) &leftrightarrow; α is ω2-ply-(next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω2∈Ordω2+1)(α0=α & ∃ν(αω2=next (+1)-stb. after ν) & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; Lα0Σ1Lα1Σ1Lα2Σ1...LαωΣ1Lαω+1Σ1Lαω+2Σ1...Lαω2 (here αω2+1 is (next (+1)-stb. after αω2))
  3. α is ω2+2-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω2+2∈Ordω2+3)(α0=α & ∀(γ,δ≤ω2+2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; α is ω2+1-ply-(next (+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω2+1∈Ordω2+2)(α0=α & ∃ν(αω2+1=next (+1)-stb. after ν) & ∀(γ,δ≤ω2+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)) &leftrightarrow; α is ω2-ply-(next doubly-(+1)-stb. after _)-stb. &leftrightarrow; ∃((αi)i≤ω2∈Ordω2+1)(α0=α & ∃ν(αω2=next doubly-(+1)-stb. after ν) & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ))
  4. α is ω3-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω3∈Ordω3+1)(α0=α & ∀(γ,δ≤ω3)(γ∈δ &rightarrow; LαγΣ1Lαδ))
  5. α is ω2-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω2∈Ordω2+1)(α0=α & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) (the least such α is the least ordinal stable up to a (limit point of limit points of _)-nonprojectible (definition here), cf. theorem 2.6)
  6. α is ε0-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ε0∈Ordε0+1)(α0=α & ∀(γ,δ≤ε0)(γ∈δ &rightarrow; LαγΣ1Lαδ))
  7. α is ω1CK-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ω1CK∈Ordω1CK+1)(α0=α & ∀(γ,δ≤ω1CK)(γ∈δ &rightarrow; LαγΣ1Lαδ))
  8. α is ν-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ν∈Ordω1CK+1)(α0=α & ∀(γ,δ≤ν)(γ∈δ &rightarrow; LαγΣ1Lαδ)), where ν is the least (+1)-stb. ordinal
  9. α is ν-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ν∈Ordω1CK+1)(α0=α & ∀(γ,δ≤ν)(γ∈δ &rightarrow; LαγΣ1Lαδ)), where ν is the least (least (+1)-stb.)-ply-(+1)-stb. ordinal
  10. α is α0-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤α0∈Ordα0+1)(α0=α & ∀(γ,δ≤α0)(γ∈δ &rightarrow; LαγΣ1Lαδ))
  11. α is (α0+1)-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤α0+1)∈Ordα0+2)(α0=α & ∀(γ,δ≤α0+1)(γ∈δ &rightarrow; LαγΣ1Lαδ))

We can also define g1-ply-g0-stability, where g1 and g0 are ordinal functions. However, this page continues using the previous concept from here.

  1. α is ν-ply-(+1)-stb. &leftrightarrow; ∃((αi)i≤ν∈Ordν+1)(α0=α & ∀(γ,δ≤α0+1)(γ∈δ &rightarrow; LαγΣ1Lαδ)), where ν is the least (next (α+1)-ply-(+1)-stb. after α)-ply-(+1)-stb. after α
  2. α is α1-ply-(+1)-stb. &leftrightarrow; α is (next α1 that is α1-ply-(+1)-stb. after α0)-ply-(+1)-stb.????1 can be thought of as having "α0's condition", e.g. α0 is α1-ply-(+1)-stb. and α1 is also α1-ply-(+1)-stb.; in fact, at least for the least such α0, here α1 is the least α1-ply-(+1)-stb. ordinal above α0[citation needed]) Is this correct?
  3. α is (α1+1)-ply-(+1)-stb.
  4. α is α1+-ply-(+1)-stb.
  5. Here

Σ_2-stability

There is a vastly large gap between the end of the last section and this section, including ordinals such as "least α0 that is α1-stb., where α1 is the least (Π2-reflecting on _)-nonprojectible (definition here)".

  1. α is Σ2-???-stb.
  2. WIP

Theorems

1.1

Lemma: Let g0 and g1 be functions mapping Ord to Ord such that ∀(α∈Ord)(g0(α)≤g1(α)). Then for an ordinal α, "α is g1-stb." implies "α is g0-stb."

Proof: This follows easily from [3], noting that because g0(α)≤g1(α), then Lg0(α) c= Lg1(α).

Sometimes, the condition "∀(α∈Ord)(g0(α)≤g1(α))" only holds for most α, but not all. So we need a weaker theorem

Theorem: Let g0 and g1 be functions mapping Ord to Ord such that ∃(γ∈(least g0-stb.))∀α(α>γ &rightarrow; g0(α)≤g1(α)). Then for an ordinal α, "α is g1-stb." implies "α is g0-stb."

Proof: The proof's condition is equivalent to ∀(α∈Ord-(least g0-stb.))(g0(α)≤g1(α)). The rest also follows from [3], noting that because g0(α)≤g1(α), then Lg0(α) c= Lg1(α), and because of the lower bound on α, we can ignore cases where g0(δ) isn't necessarily lessequal g1(δ) (and so Lg0(δ) isn't necessarily subsetequal Lg1(δ)).

Corollary: {α∈ω1:α is g1-stb.} subseteq {α∈ω1:α is g0-stb.}

1.2

Theorem: If α is (+2)-stb., then it's Πn-rfl. on (+1)-stb. ordinals for all n∈ω

Let ϕ be a formula that is Πn, such that Lα |= ϕ. Then Lα+2 |= ∃ξ(LξΣ1Lξ+1 & Lξ |= ϕ) with witness x=Lα, so by the Σ1-elementary-substructure, Lα |= ∃ξ(LξΣ1Lξ+1 & Lξ |= ϕ). So (some (+1)-stb. ordinal below α)'s rank satisfies the formula, which is Πn-reflection on (+1)-stb. ordinals below.

Remark: At first it might appear that the larger universe of Lα+2 allows for "more" witnesses of "∃x", and since the formulae are Σ1, then Lα+1 |= ϕ &rightarrow; Lα+2 |= ϕ, implying that all (+1)-stb. ordinals are also (+2)-stb. However, this is incorrect, because formulae involving Lα+2, and Σ1-expressible reflection principles (such as onto (+1)-stb ordinals) relating to it, can be satisfied at the latter universe.

This also extends to higher stability. Someone asked on Discord if for any ordinal ξ, there's an ordinal that's (+ξ)-stb. but not (+ξ+1)-stb. Assuming that a (+ξ)-stb. exists, this can be shown by proving that (+ξ+1)-stb. ordinals are, for example, Σ1-reflecting on {σ∈(least (+ξ+1)-stb.) : σ is (+ξ)-stb.}. By the Σ1-reflection, {σ∈(least (+ξ+1)-stb.) : σ is (+ξ)-stb.} is nonempty, so if we choose an ordinal from the set, this is an example of what we want.

1.3

Theorem: Let α be the least ordinal that is a limit of (+1)-stb. ordinals. Then α isn't admissible

Proof: This proof uses Richter and Aczel's notion of "Σ1-collection", note that this is different than the notion of Σn-collection given by most other sources including the page KP set theory. For contradiction, assume that α is admissible. Let ϕ(n,γ) be a Σ1-formula that is true iff γ is the nth (+1)-stb. ordinal. Then we can apply (Richter and Aczel's) Σ1-collection to "Lα |= ∀(n∈ω)∃ν(ϕ(n,ν))" to obtain "for some b∈Lα, then Lα |= ∀(n∈ω)∃ν(ϕb(n,ν))", i.e. a set within Lα contains ω-many (+1)-stb. ordinals. However, this contradicts the minimality of α.

Nonprojectibles

Intuitively, the reason why S-nonprojectibles β often satisfy interesting model-theoretic results is because Lβ thinks that the stable ordinals satisfy certain properties (characterized by S, but Lβ doesn't know this). Also note that nonprojectibility implies recursive Mahloness[9]

Definitions

An ordinal β is nonprojectible if β=sup({α∈β:LαΣ1Lβ})[6], i.e. β is a limit of β-stb. ordinals.

Definitions past here are original

For a "thinning operator" S:P(Ord)&rightarrow;P(Ord), define "β is S-nonprojectible" iff β∈S({α∈β:LαΣ1Lβ}) (i.e. β satisfies a certain thinning operation over the set of ordinals below that are stable up to β). Note that "limit point of _"-nonprojectible coincides with the original definition of nonprojectibility.

Although not directly related to nonprojectibility, a useful notion here is what I call A-admissibility:

  • For A∈P(V) (i.e. a predicate on V), let RC(A) denote A u {γ:Lγ∈A}. α is RC(A)-admissible iff Lα |= T, where T denotes KP set theory with the Σ0-collection axiom schema replaced with the "A-Σ0-collection axiom schema", which is defined:
    • For all Σ0-formulae ϕ, the following is an instance of the A-Σ0-collection schema: ∀a(∀(x∈a)(∃yϕ(x,y) &rightarrow; ∃(b∈A)(∀(x∈a)(∃(y∈b)(ϕ(x,y)))))

Theorems

2.1

Theorem: Let α be the least ω-ply-(+1)-stb., and let the ordinals αγ follow from the definition. Then αω is nonprojectible.

Proof (credit to Hyp_cos): Let α0 be the least ω-ply-(+1)-stb. ordinal. Then by definition there exists a chain Lα0Σ1Lα1Σ1Lα2Σ1...Lαω, and in this case αω=sup({αi:i∈ω}) (c.f. #106). Also each αi is stable up to αω (by transitivity of ⪯Σ1[citation needed]), and αω is their supremum (c.f. #106), so αω is nonprojectible.

2.2

Lemma: Let α be the least ω-ply-(+1)-stb. α is less than the least (ω+1)-ply-(+1)-stb.

Proof: Let ϕ formalize "there exists an ω-ply-(_+)-stb. ordinal". Then Lleast (ω+1)-ply-(+1)-stb. |= ∃γ(ϕLγ), so . On the other hand, ... doesn't hold. I think that Lleast (ω+1)-ply-(+1)-stb. is sufficiently accurate on statements about ω-ply-(_+)-stb. ordinals[citation needed], so the least ω-ply-(_+)-stb. ordinal is a member of Lleast (ω+1)-ply-(+1)-stb. but not Lα.

Theorem: Let νn be the nth (0-indexed) ω-ply-(+1)-stb. ordinal, for ξ∈ω. Then {ν : LαΣ1Lν}={νn : n∈ω\0} u {Least npr.}.

Proof: Let A denote {ν : LαΣ1Lν}, and for notational convenience, let λξ.δξ enumerate A (under ∈). From lemma 2.2, α can't be ω+1-ply-(+1)-stb., so the order type of A under ∈ must be at most ω+1, and A's order type can't be less than ω+1 by the ω-(+1)-stability(Is this correct?). So A has order type ω+1. We now consider cases for δξ:

  • ξ∈ω. Apply proposition 7.4.(ii) from [Barwise75, Admissible Sets and Structures (p.179)] on members of A to obtain that LiΣ1Lj for any i,j∈ω, and therefore all members of A n δω are ω-ply-(+1)-stb. In fact, for any ω-ply-(+1)-stb. ordinals θ,η less than the least nonprojectible (cf. theorem 2.1), we have "θ∈η &rightarrow LθΣ1Lη" by Barwise's proposition 7.4.(iii), so because A n δω is unbounded in the least nonprojectible, θ,η∈A n δω.
  • ξ=ω. WIP

2.3

Theorem: Let β be nonprojectible. Then any α that is β-stb. is also ω-ply-(+1)-stb.

Proof (credit to Hyp_cos): There are ℵ0-many ordinals αi∈β such that LαiΣ1Lβ by the definition of the nonprojectibility, and the set of them has order type ω under ∈. Let αi∈αi+1. Let αj be the jth such ordinal. Because each αii+1<β, and LαiΣ1Lβ, then LαiΣ1Lαi+1 (cf. Richter and Aczel's remark[3]). So due to the transitivity of ⪯Σ1[citation needed], given a natural n, then the transfinitary conjunction of the previous logic for all natural i≥n implies LαnΣ1Lαn+1Σ1Lαn+2Σ1..., and by the transitivity of ⪯Σ1, ∀(j,k∈ω)(n≤j<k &rightarrow; LαjΣ1Lαk). We can also show that αω exists by the infinitary conjunction &{Lemma(Lα0,Lαi,Lαi+1):i∈ω}. So αn is ω-ply-(+1)-stb. (cf #105) (to use #106's definition of ω-ply-(+1)-stability, we need to prove αω exists), and because n is arbitrary, then any ordinal that is β-stb. is also ω-ply-(+1)-stb.

Corollary: "β is nonprojectible" implies the existence of ℵ0-many ω-ply-(+1)-stb. ordinals below β, and they have order type ≥ω under ∈. Also the least ω-ply-(+1)-stb. isn't nonprojectible.

Remark: Also note that informally some "important" ordinals (e.g. the least ω-ply-(+1)-stb.) can be described in terms of the least ordinal stable up to the least S-nonprojectible ordinal for a thinning operator S.

2.4

Theorem: Let β and β' be nonprojectible ordinals such that β'∈β. Then there exists a β-stb. ordinal α such that α≥β'.

Proof: For contradiction, assume that there doesn't exist such α, i.e. the β-stable ordinals that are <β are upper-bounded by β'. However, because β is nonprojectible, the β-stable ordinals must be unbounded in β (e.g. not bounded by β'), which is a contradiction.

2.5

Theorem: Let νγ denote the γth ω-ply-(+1)-stb. ordinal. Then for n∈ω, the nth (0-indexed) nonprojectible ordinal is equal to sup({νi:i∈ω(n+1)}).

Proof: Proceed by induction on n:

  • For n=0, this is true by theorem 2.1.
  • Assume that this has been proven for all k∈n. Then by theorem 2.4 using β'=(nth nonprojectible) and β=((n+1)th nonprojectible), then there is some α that is β-stb. such that α>β', and by theorem 2.2, α is ω-ply-(+1)-stb. So ∃((αi)i≤ω∈Ordω+1)(α0=α & ∀(γ,δ≤ω)(γ∈δ &rightarrow; LαγΣ1Lαδ)), and each αi for i∈ω is ω-ply-(+1)-stb. and >β', guaranteeing the existence of ω-many ω-ply-(+1)-stb. ordinals between β and β'. Also β=αω because of the nonprojectibility of β, and αω=sup({αi:i∈ω})[citation needed].

Remark: Let ν(m,n) denote the mth ordinal that is (nth nonprojectible)-stb. For m∈ω, the sequence (ν(m,i))i∈ω doesn't prove sup({ith nonprojectible:i∈ω})'s nonprojectibility (also it likely isn't nonprojectible), e.g. Lν(m,i)Σ1Lmth nonprojectible doesn't necessarily imply Lν(m,i)Σ1Lν(m,i+1)

2.6

Theorem: Let α be the least ω2-ply-(+1)-stb. ordinal, and let the ordinals αγ follow from the definition. Then αω2 is a limit of nonprojectible ordinals.

Proof: Construct a sequence of nonprojectible ordinals (αγ')γ∈Lim n ω2 where Lim denotes the class of limit ordinals:

  • For γ∈Lim n ω2 (i.e. γ∈{ω(i+1):i∈ω}), define αγ'=sup({αδ:δ<γ})

Let δ<γ. First, prove that LαδΣ1Lαγ':

  • Take the infinitary conjunction &{Lemma(Lαδ,Lαδ+i,Lαδ+i+1):i∈ω}, which implies LαδΣ1Lαγ' by definition of αγ' and the definition of limit stages of the constructible hierarchy.

Now, prove that each αγ' is nonprojectible as follows:

  • Let γ' denote the least ordinal such that γ'+ω=γ. For each i∈ω, γ'+i<γ, and Lαγ'+iΣ1Lαγ' by the above proof. Also αγ'=sup({αγ+i:i∈ω}) by definition, so αγ' is nonprojectible.

Finally, prove that αω2≥sup({αωi':i∈ω}) (which is a limit of nonprojectibles). One way to show this is to show that sup({αω(i+1)':i∈ω}) by taking the infinitary conjunction &{Lemma(Lαω',Lαω(i+1)',Lαω(i+2)'):i∈ω}. So sup({αω(i+1)':i∈ω})=αω2 (cf. remark on line #106) is a limit of nonprojectible ordinals.

Remark: Note that this theorem doesn't directly imply that αω2 is "limit point of limit points of _"-nonprojectible.

2.7

Lemma 2.7.1: Given an ordinal β that is "limit point of limit points of _"-nonprojectible, there exists a β-stb. ordinal that is ω2-ply-(+1)-stb.

Proof: We construct a sequence (αi)i∈ω2 of β-stb. ordinals as follows:

  • Let ξ∈ω2. Define αξ := min({η:LηΣ1Lβ & ∀(ξ'∈ξ)(η>αξ')})

This also follows from the "limit point of limit points of _"-nonprojectibility, e.g. for all j∈ω, the suprema sup({αi':i'∈ω(j+1)}) are limits of β-stb. ordinals, and the supremum of those suprema is β (citation needed). By Richter and Aczel's remark[3], because each αi is β-stb., and αi+1∈β, then each αi is αi+1-stb. So ∃((αi)i≤ω2∈Ordω2+1)(αω2=β & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ)), and trivially, ∃((αi)i≤ω2∈Ordω2+1)(αω2=β & ∃ν(αω2=ν) & ∀(γ,δ≤ω2)(γ∈δ &rightarrow; LαγΣ1Lαδ)) (note the small change). For example, α0 is ω2-ply-(+1)-stb., and by the transitivity of ≺Σ1, α0 is also β-stb. (in fact, this is true for any αi with i∈ω2, due to how all such αi are also ω2-ply-(+1)-stb.)

Lemma 2.7.2: Let α denote the least ω2-ply-(+1)-stb. ordinal. There are no "limit of limit points of _"-nonprojectibles below α.

Proof: For contradiction, assume that there is an ordinal β∈α that is "limit of limit points of _"-nonprojectible. By the minimality of α, there is no β-stb. ordinal that is ω2-ply-(+1)-stb. Then, the contradiction follows from lemma 2.7.1.

Theorem: Let β be "limit point of limit points of _"-nonprojectible. Then any α that is β-stb. is also ω2-ply-(+1)-stb.

Proof: There are ℵ0-many ordinals αi∈β such that LαiΣ1Lβ by the definition of the nonprojectibility, and the set of them has order type ω2 under ∈. For ξ∈ω2, let αξ be the ξth such ordinal (cf. lemma 2.7.1's ordinals αξ for a formal definition; they're equivalent as well), and let αξ∈αξ+1. Because each αξξ+1<β, and LαξΣ1Lβ, then LαξΣ1Lαξ+1 (cf. Richter and Aczel's remark[3]). Also limit cases can be shown by the infinitary conjunctions &{Lemma(Lαωj,Lαωj+i,Lαωj+(i+1)):i∈ω} for each j∈ω-{0}. So due to the transitivity of ⪯Σ1, given an ordinal θ∈ω2, then ∀(γ,δ∈ω2)(θ≤γ<δ &rightarrow; LαγΣ1Lαδ). We can also show that αω2 exists by the infinitary conjunction &{Lemma(Lα0,Lαω(i+1),Lαω(i+2)):i∈ω}. So αθ is ω2-ply-(+1)-stb. (cf. #148), and because θ is arbitrary, then any ordinal that is β-stb. is also ω2-ply-(+1)-stb.

Remark: Theorem 2.4. is also useful for comparing "limit point of limit points of _"-nonprojectible ordinals. Also note that for any j∈ω-{0}, the supremum sup({αi':i'∈ωj}) isn't necessarily β-stb.

2.8

Lemma: Let A∈P(V), and let α be RC(A)-admissible. Then the set of sets whose existence is asserted by the A-Σ0-collection schema is unbounded in Lα (i.e. Lα thinks they're unbounded).

Proof: By contradiction, assume that Lα doesn't think they're unbounded, i.e. it thinks there's a set b' that contains all such sets. Let ϕ be some Σ0 map from Lα &rightarrow; (Lα)-(transitive closure of b'). Then by the assumption of the boundedness of such b, ϕ doesn't satisfy A-Σ0-collection, which is a contradiction. (Note that this lemma also implies that α is a limit point of RC(A) n Ord)

Theorem: For A∈P(V), α is RC(A)-admissible iff α is Π2-rfl. on RC(A).

Proof (credit to W. Richter, P. Aczel (1973) Inductive Definitions and Reflecting Properties of Admissible ordinals (p.19); note that the original proof used an equivalently strong schema referred to as "Σ1-collection" instead):

  • Let α be Π2-rfl on RC(A). If α'<α, then Lα |= !(α'∈α'). By the Π2-reflection of α, there is a β<α such that Lβ |= !(α'∈α'), i.e. α'<β<α. So α is a limit ordinal. So Lα |= ∀x∃y(x∈y), which implies that there is a β<α such that Lβ |= ∀x∃y(x∈y). Hence α is a limit ordinal >ω. Using lemma 6 of K. Devlin, (1974) An introduction to the fine structure of the constructible hierarchy, Richter and Aczel have remarked that it is not hard to show that Lα satisfies rudimentary set theory for any limit ordinal α. Hence it remains only to show that Lα satisfies RC(A)-Σ0-collection. Let Lα |= ∀(x∈a)ϕ where ϕ is a Σ0 formula. Then by Π2-reflection on RC(A) (Π1 is a subset of Π2) there is a β∈RC(A) n α such that Lβ |= ∀(x∈a)ϕ. Let b=Lβ∈Lα, then Lα |= ∀(x∈a)ϕβ as required, also β∈RC(A), so by the definition of RC, also Lβ∈RC(A). (Note that Richter and Aczel's paper uses a different but equivalently strong formulation of collection).
  • Conversely, let α>ω be RC(A)-admissible, and let ϕ be a Π2 sentence such that Lα |= ϕ. We may assume that ϕ is of the form ∀x1,...,xm∃y1,...,yn(χ), where χ is Σ0. Let a be an arbitrary set in Lα. Also ϕ implies ∀(x1∈a,...,xm∈a)∃(y1,...,yn)(χ). Then by RC(A)-Σ0-collection, there exists some set b∈RC(A) such that Lα |= ∀(x1∈a,...,xm∈a)∃(y1∈b,...,yn∈b)(χ). However, in order for the Π2-reflection on RC(A) to be satisfied, we need to prove that some Lβ also satisfies this, where β∈RC(A).
    Such sets satisfying b's property (note that ∃(y1∈b)... implies ∃(y1∈c)... for b c= c) are unbounded in Lα (cf. above lemma), so we can choose some β<α such that b c= Lβ c= Lα and Lβ∈RC(A) (cf. note after the above lemma). So since such β∈RC(A) exists, α satisfies the definition of Π2-reflection on RC(A).

2.9

Theorem: Let α be the least α-ply-(+1)-stb. ordinal, i.e. the least ordinal such that ∃((αi)i≤α∈Ordα+1)(α0=α & ∀(γ,δ≤α0)(γ∈δ &rightarrow; LαγΣ1Lαδ)). Then αα0 is not (Π2-reflecting on _)-nonprojectible.

Proof sketch:

  • Where n encodes a Σ1-formula and S is a free variable, let χ(n,S) be a Σ0-formula that is a satisfaction predicate for the formula encoded by n at S. This can be used to formalize the map λθ.Lθ, however this map is Σ1. The Σ1-truth-predicate χ can be used to define ≺Σ1, and from these and the definition of θ-ply-(+1)-stability, the predicate "ϕ(ν,θ) &leftrightarrow; ν is θ-ply-(+1)-stb." is Σ1.
  • Consider the Π2-formula ∀(γ∈α0)∃ν(ϕ(ν,γ)) (note that this is logically equivalent to ∀(γ)(γ∈α0 &rightarrow; ∃(ν)(ϕ(ν,γ))), which can be shown to be Π2). Since we're working in Lαα0, this is satisfied. So by Π2-reflection there must exist β∈α0 such that Lβ satisfies this. Let η be the least ordinal such that β∈αη (note that η<α0 because α0 is a limit ordinal), however then the formula isn't satisfied for ordinals such as γ=η+1, so β and also Lβ can't be closed under the map "λθ.αθ with domain α0", which is a contradiction.

2.10

Theorem: Let α be the least ordinal such that ∃((αi)i≤αα...∈Ordαα...+1)(α0=α & ∀(γ,δ≤αα...)(γ∈δ &rightarrow; LαγΣ1Lαδ)), where αα... denotes sup({(λθ.αθ)i(0):i∈ω}), where the superscript denotes function iteration. Then αα... is not (Π2-reflecting on _)-nonprojectible.

Proof sketch:

  • Where n encodes a Σ1-formula and S is a free variable, let χ(n,S) be a Σ0-formula that is a satisfaction predicate for the formula encoded by n at S. This can be used to formalize the map λθ.Lθ, however this map is Σ1. The Σ1-truth-predicate χ can be used to define ≺Σ1, and from these and the definition of θ-ply-(+1)-stability, the predicate "ϕ(α,θ) &leftrightarrow; α is θ-ply-(+1)-stb." is Σ1.
  • Consider the Π2-formula ∀γ∃ν(ϕ(ν,γ)) (note that since ϕ(ν,γ) is Σ1 when in prenex normal form, then "∃ν(ϕ(ν,γ))" is also Σ1). Since we're working in Lαα..., this is satisfied. So by Π2-reflection there must exist β∈αα... such that Lβ satisfies this. Let η be the least ordinal such that β∈αη (note that η<αα... because αα... is a limit ordinal), however then the formula isn't satisfied for ordinals such as γ=η+1, so β and also Lβ can't be closed under the map λθ.αθ, which is a contradiction.

2.11

Theorem about α that is α1-ply-stb.?

Let f:Ord&rightarrow;Ord enumerate {α∈ω1 : ∃(ξ>α)(α is ξ-ply-stb.)}, and let f' be its derivative. Then the least α0 that is α1-ply-stb. is f'(1) is this correct??

Remark (WIP): If we instead let f enumerate ordinals α that are min{ξ:ξ is ξ-ply-stb.}-ply-stable, we get f'(1) being the second ordinal in the chain. But f'(1) isn't f'(1)-ply-stb.

Relation between stability and nonprojectible-like conditions

Some stability conditions are related to nonprojectibility conditions. For example, let S:P(Ord)&rightarrow;P(Ord) be the thinning operation defined as S(A):={β∈Ord:∃(γ∈A)(γ∈β)}. Then an ordinal α is (+1)-stb. iff it's β-stb. for some S-nonprojectible ordinal β (proof needed). Also α is (+1)-stb. iff Lα satisfies "there exists a stable ordinal"(proof needed).

Stronger substructures

Other extensions of stability given by various authors such as Rathjen[10] and ??? include stronger substructures, e.g. Σ2-elementary-substructures and Σn-elementary-substructures for n∈ω. This page uses the terminology "β-Σn-stability" to refer to Rathjen's notion of "n-β-stability". As for n=1, the reason[11] why even "α that is (α+1)-Σ1-stb." ("normal" (+1)-stability) is a strong condition seems to be that if α satisfies a certain formula ϕ (with certain restrictions), then Lα+1 |= ∃x(isTrans(x)&ϕx), and by the Σ1-elementary-substructure, then also Lα |= ∃x(isTrans(x)&ϕx), so smaller such ordinals satisfying ϕ exist. (Note that formulae such as ∃x∃(y∈x)(isTrans(y)&ϕy) are false in both structures). Also note that we lose some properties like "larger substructure implies smaller substructure" (Barwise justified this as "consequences of the persistence of Σ1-formulae", however these are Σ2 or higher)

An important result by Rathjen is here[12]

Definitions

We also define a variant of relativization (⌜χ⌝)x for some of these proofs. In order to avoid reference to χ which is an object in the meta-theory, we define the following as kx where k is a Godel-number of a formula and x is a class (note that in case x is an ordinal, kx doesn't necessarily correspond to ordinal exponentiation):

  • If k encodes an atomic formula a0=a1, then kx :&leftrightarrow; a0=a1 & a0,a1∈x
  • If k encodes an atomic formula a0∈a1, then kx :&leftrightarrow; a0∈a1 & a0,a1∈x
  • WIP

Theorems

3.1

Theorem: "∃α∃(β>α)(LαΣ2Lβ&β is a successor ordinal)" is false

Proof (credit to Wojowu): For contradiction, assume that ∃α∃(β>α)(LαΣ2Lβ&β is a successor ordinal). Let ϕ be a Σ2-formula formalizing "there exists a largest ordinal", i.e. "the ordinals are not unbounded" (e.g. ∃(γ∈Ord)(∀(δ∈Ord)(δ≤γ))). Then because α is a limit ordinal, Lα doesn't satisfy ϕ, but Lβ |= ϕ is true, which contradicts the definition of ≺Σ2. (Note that there isn't a way to translate this formula to Σ1)

Corollary: For an ordinal α, ∀β(LαΣ2Lβ&rightarrow;(β is a limit ordinal))

3.2

P進大好きbot has shown this proof is incorrect

Theorem: Let α be any ordinal, and χ be a setence such that the set {η:χη} is unbounded in α. Then "∃(β>α)(LαΣ2Lβ&({η:χη} is bounded in β))" is false.

Proof: For contradiction, let ϕ be the Σ2 formula ∃γ∀δ(χδ&rightarrow;δ∈γ) (note that χδ is Σ0), i.e. "the set of all ranks whose relativization of χ is true are bounded". For contradiction, assume that ∃α∃(β>α)(LαΣ2Lβ&({η:χη} is bounded in β)). Then, Lα doesn't satisfy ϕ (which is given), but Lβ |= ϕ is true (because {η:χη} is bounded in β), which contradicts the definition of ≺Σ2.

Remark: This is an extension of theorem 3.1. For example, let χ formalize "the universe is an admissible ordinal", and such formula is proven to exist in [13] (and in this source such formula is denoted by σ1). Also, each ordinal α that is β-Σ2-stb. for some β is also β'-Σ1-stb. for some β', and β'-Σ1-stb. ordinals are limits of admissibles (cf. [4] and theorem 5.1). Applying theorem 3.2 yields that β must be a limit of admissibles as well.

P進大好きbot has also mentioned that the relativization χη isn't related to reflection properties of Lη. In the example of Richter and Aczel's sentences σ0 and σ1, these are different sentences (although both Π3). THe sentences aren't the same, but σ1 emulates a property that sets satisfying σ0 have. For applying this result however, a Discord user has remarked that proving existence of χ such that χη iff η is a Π3-reflecting ordinal is a very difficult problem.

3.3

Theorem: "∃α∃(β>α)(LαΣ2Lβ& !(Lβ |= χ))" is false, where χ is any Σ2-formula such that Lα |= χ.

Proof: For contradiction, assume that "∃α∃(β>α)(LαΣ2Lβ& !(Lβ |= χ))" is true. It's given that Lα |= χ, so by the Σ2-elementary-substructure, then Lβ |= χ is true. However, it's also given that "Lβ |= χ" is false, which is a contradiction.

Remark: This is similar to theorem 3.2, and can be applied in different scenarios. Also note that adapting this theorem to Σ1-stability becomes much weaker, as no universal quantification is possible in χ.

3.4

Theorem: Let α be the least ordinal such that ∃(β>α)(LαΣ2Lβ). Then LαΣ1Lβ+1.

Proof: Let χ be a Σ1-definition of "γ is (+1)-stb." (which exists by corollary 5.4). Also α is (+1)-Σ1-stb., because LαΣ2Lβ, so LαΣ1Lβ, so by [3], then LαΣ1Lα+1. So we can apply theorem 3.3, yielding that β must be (+1)-Σ1-stb.

3.5

Theorem: There exists a Σ2 formula ϕ with one free variable such that ϕ(α) iff LαΣ1L

Proof (credit to Wojowu): Let N denote a set of Godel-numbers of Σ1(Lα) formulae. Then ϕ can be a formula formalizing "∃A∀(k∈N)(A encodes a truth value (at Lα) for k & (k is satisfied at Lα &rightarrow; A contains a witness for the formula k encodes) & (k isn't satisfied at Lα &rightarrow; !∃(x∈L)(k is satisfied at x)) )"

3.6

Theorem: Let α be a countable ordinal such that ∃(β>α)(LαΣ2Lβ). Then α is nonprojectible.

Proof: For θ∈α, Let ϕ(θ) denote a Σ2-formalization of "∃ν(ν is Σ1-stb. & θ∈ν)". Given an arbitrary θ∈α, because Lβ |= ϕ(θ) (because α is a witness of "∃ν(ν is Σ1-stb. & θ∈ν)"), then by the Σ2-elementary-substructure, then Lα must satisfy this as well, i.e. there is an Σ1-α-stb. ordinal above θ. Because θ is arbitrary, then α is nonprojectible.

Update 06-22-21: Look through this source?

3.6 for later

Theorem: If α is β-Σ2-stb., then α is Πn-reflecting on {α'∈α:α' is α-Σ1-stb.}

Proof: Let \(\chi\) be a first-order formula (in the coded theory) such that \(L_\alpha\vDash\chi\), and let \(\phi\) denote a \(\Sigma_2\) formula formalizing \(\exists x(\textrm{isTrans}(x)\land\chi^x\land x\prec_{\Sigma_1}L)\).

\(L_\beta\) satisfies this (for example \(x=L_\alpha\) can be a witness of the existential quantifier), so by the \(\Sigma_2\)-elementary-substructure, so does \(L_\alpha\). So there must be some transitive \(x\in L_\alpha\) such that \(x\prec_{\Sigma_1}L_\alpha\) and \(\chi^x\). This asserts \((L_\alpha\vDash\chi)\rightarrow\chi^x\), c.f. \(\Pi_n\)-reflection on the set of sets that are \(\Sigma_1\)-elelemtary-substructures of \(L_\alpha\).

3.7

Lemma 3.7.1: There is a Σ2 formalization of "there is a Σ1(Lβ) map of a bounded subset of β cofinally into β".

Proof: Let the bounded subset of β be called T (i.e. T c θ ∈ β), and call the map f. The property "∃θ∃T∃f(T c θ & ∀η(∃(γ∈T)(η=fγ)) & f is Σ1-definable with parameters)" is Σ2-formalizable.

Lemma 3.7.2: Let α be a countable ordinal such that ∃(β>α)(LαΣ2Lβ). Then β is nonprojectible.

Proof: By [Devlin, An introduction to the fine structure of the constructible hierarchy (p.39)], β will be nonprojectible iff there is no Σ1(Lβ) map of a bounded subset of β cofinally into β. For contradiction, assume such a map exists. By lemma 3.7 there is a Σ2 formalization of this property, which is false at Lα. So by the Σ2-elementary-substructure, then it's also false at Lβ, i.e. β is nonprojectible.

Remark: A direct formalization of "the stable ordinals are unbounded" is Π3.

Lemma 3.7.3: Let α0 be a countable ordinal such that ∃(α10)(Lα0Σ2Lα1), and let β = max{ξ:Lα0Σ1Lξ}. Then β is no greater than α1.

Proof: For contradiction, assume β>α1. Then Lβ |= ∃ξ∃(ν01∈ξ)(Lν0Σ1Lν1), and by the Σ1-elementary-substructure, then Lα0 |= ∃ξ∃(ν01∈ξ)(Lν0Σ1Lν1) (the same formula). However, since the latter is false, i.e. ∃(ν01∈α0)(Lν0Σ1Lν1) (by transitivity of ∈ on ordinals) is false, this is a contradiction.

Remark: There may be an issue with this theorem if Lα0 isn't accurate about ≺Σ2, i.e. which ordinals ν01∈α1 have Lν0≺Lν1, if ν1=OrdLα0. If ν1<OrdLα0, then see theorem 3.8 about its accuracy.

Theorem: Let α0 be a countable ordinal such that ∃(α10)(Lα0Σ2Lα1), and let β be the greatest ordinal such that Lα0Σ1Lβ. Then β is nonprojectible.

Proof: First, we show

3.8

Lemma: Let α, β, γ be ordinals where α∈β∈γ and γ is a limit ordinal, and n be a natural number. Then LαΣnLβ iff Lγ |= LαΣnLβ.

Proof sketch: The form of "LαΣnLβ" is ∀(k∈G)(Lα |= k &leftrightarrow; Lβ |= k) where G is a (Δ0-definable) set of Godel-codings for Σn formulae, and this form is itself Σn. Because γ is a limit ordinal >β, then Lγ is closed under formation of ordered pairs from Lβ[14], so Lγ can appropriately formalize variable assignments for "∀(k∈G)(Lα |= k &leftrightarrow; Lβ |= k)". Note that we can't use this formulation to formalize elementary substructures of "L" (at Lγ)

Remark: This lemma is useful for looking at set-sized elementary substructures inside of a universe. Note that there are still discrepancies for "proper-class-sized" elementary substructures, and that Lγ thinks that (the class defined by the formula defining) Lγ is a proper class. Also applying one of Arai's results [15] to Lmin{ν:∃(ξ>ν)(LνΣ2? TypoLξ)}, we can form Σ2-elementary-substructures which weren't there before (in the source's terminology between Lα and Lκ, which we call Lν here), to yield "illusory" ν-Σ2-stb. ordinals, i.e. ordinals which Lν thinks are Σ2-stb., but in reality aren't ν-Σ2-stb. (is this correct?)

Stable-up-to list

Another property of stability is the ordinals that other ordinals are stable up to. These are ordinals β where ∃(α∈β)(LαΣ1Lβ). Here are some of these ordinals:

  • (Least (+1)-stb.)+1
  • (Least (+2)-stb.)+1
  • (Least (+2)-stb.)+2

Here are some ordinals not on this list:

  • 0
  • 1
  • ω
  • Least (+1)-stb.
  • Least (*2)-stb. (i.e. there is no ordinal α where LαΣ1LLeast (*2)-stb.)

Miscellaneous results

5.1

Theorem: If α is Πn-reflecting, then the witnesses of "∃(β∈α)(Lβ |= ϕ)" (i.e. such β that it holds) are unbounded in α.

Proof (credit to Wojowu): Let β be some witness of "∃(β∈α)(Lβ |= ϕ)", and let χ(η) denote the formula "ϕ & (η=η)". Given an arbitrary ordinal γ≥β, Lα satisfies χ(γ), and χ(γ) is Π2 (with free variables), so there must exist β'∈α such that Lβ satisfies χ(γ), i.e. there exists another witness of α's Π2-reflection that is greater than γ. So such β' are unbounded in α.

5.2

Lemma: If an ordinal α is recursively Mahlo, then it's also recursively inaccessible.

Proof: See citation 6 in this article

Theorem: If an ordinal α is Π3-reflecting, then it's a limit of admissible ordinals.

Proof: If α is Π3-reflecting, then it's also recursively Mahlo[16], so it's also recursively inaccessible (cf. lemma 5.2), so it's a limit of admissible ordinals.

5.3

Thanks to Discord users Wojowu and N/U for help with the proofs of these lemmas

Lemma 5.3.1: For sets a,b,c where a is transitive, if a⪯Σ0b and a⪯Σ0c, then a⪯Σ0(b u c). Proof:

Assuming a⪯Σ0b (named "assumption 1") and a⪯Σ0c (named "assumption 2"), we will prove that for all Σ0 formulae ϕ, a |= ϕ(p) &leftrightarrow; b u c |= ϕ(p), where p c= a & |p|<ℵ0 (p is allowed to be the empty set, but this never occurs for Σ0 formulae):

Proof by strong induction on the structure of ϕ(p):

  • Base case: If ϕ(p) is of the form x=y, x∈y, !ψ, ψ&χ, or ψ|χ, then it's true because of the absoluteness of each relation (the universe is changed between different elementary substructures)
  • Inductive case 1, if ϕ(p) is of the form ∀(xi∈y n a)χ(xi,p) where y∈p (this accounts for parameters y, the set of parameters p may change from step to step of the induction). In this case y n a=y because y∈a and a is transitive. Also because xi∈y c= a, xi∈a, so it's possible to use the inductive assumption (since the set of parameters p u {xi} c= a):

Assume that the lemma holds for χ(xi,p). Then b u c |=ϕ(p) iff for all xi∈y, b u c |=χ(xi,p). y is a parameter, so y must be in p, and y∈p c= a c= b c= b u c (the transitivity of a requires xi∈a, so xi is in this universe). So y must be a member of b u c, so y∈b|y∈c. If y∈b, then ∀(xi∈y)(y∈a &rightarrow;χ(xi,p)) implies ∀(xi∈y)(y∈a u b &rightarrow;χ(xi,p)), and the left-hand side of this implication is true by assumption 1 (it follows from the definition of p, and the use of (FreeVariables)∈a follows from assumption 1). The right-hand side is equivalent to this case of the lemma.

Similar logic applies to if y∈c (and assumption 2 instead of 1)

  • Inductive case 2, if ϕ(p) is of the form ∃(xi∈y)χ(xi,p) where y∈p. In this case y\cap a=y because y∈a and a is transitive. Also because xi∈y c= a, xi∈a, so it's possible to use the inductive assumption (since the set of parameters p u {xi} c= a):

Assume that the lemma holds for χ(xi,p). Then b u c |=ϕ(p) iff there exists xi∈y such that b u c |=χ(xi,p). y is a parameter, so y must be in p, and y∈p c= a c= b c= b u c (the transitivity of a requires xi∈a, so xi is in this universe). So y must be a member of b u c, so y∈b| y∈c.

If y∈b, then ∃(xi∈y)χ(xi,p)& y∈a implies ∃(xi∈y)χ(xi,p)& y∈a u b, and the left-hand side of this implication is true by assumption 1. The right-hand side is equivalent to this case of the lemma.

Similar logic applies to if y∈c (and assumption 2 instead of 1)

Lemma 5.3.2: For sets a,b,c where a is transitive, if a⪯Σ1b and a⪯Σ1c, then a⪯Σ1(b u c). Proof:

Assuming a⪯Σ1b (named "assumption 3") and a⪯Σ1c (named "assumption 4"), we will prove that for all Σ1 formulae ϕ, a |=ϕ(p) &leftrightarrow; b u c |=ϕ(p), where p c= a & |p|<ℵ0:

Proof by strong induction on the structure of ϕ(p) (note that each Σ1 formula is a possibly empty block of free existential quantifiers followed by a Σ0 formula):

  • Base case: ϕ is also a Σ0 formula. cf. lemma 5.3.1
  • Inductive case: ϕ is of the form ∃ xi:χ(xi,p). Due to assumption 3, xi∈b, so xi∈b u c. Also due to assumption 4, xi∈c, and if xi∈c, so xi∈b u c. (This logic can also apply to b n c, but that's irrelevant to this result)

5.4

Corollary (credit to Barwise): For γ>α with γ∈Ord, there is a Σ1-formalization of "α is γ-stb."[17].

Remark: Because these are Σ1, it might at first appear that Σ1-formulae like "∃ν(ν is γ-stb.)" can be used to disprove existence of those γ-stb. ordinals (e.g. by contradiction, letting α be the least one, then reflecting its Σ1-formalization down below it, contradicting its minimality). However, it's important to be careful to whether Lγ does in fact satisfy such formulae (cf. here). In fact, in the case that Lγ doesn't satisfy it, then by the biconditional, Lα must not either, agreeing that α is the least γ-stb. ordinal.

Lemma (credit to Kranakis): For each n≥1, there exists a Πn-formula ϕn(β) with exactly one parameter such that for any α∈β, Lβ |= ϕn(α) &leftrightarrow; LαΣnLβ[18]

Remark: This is a stronger version of the corollary, and more versatile as well. It's important to carefully watch how the parameters behave in smaller universes (for example, we can't accurately apply Lα |= ϕ1(α) in order to disprove existence of (+1)-stb. ordinals). Also note that the assertion "Lβ |= ⌜LαΣnLβ⌝" is trivially false when working with set-sized arguments of ≺

5.5

Theorem 5.5.1: ωωCK is not admissible.

Proof: Let χ denote the Π3 formula given in [19], let ϕ(a) denote the Σ0 formula ⌜∃(b∈a)(χb)⌝, and apply Σ0-collection to ⌜∀(n∈ω)ϕ(n)⌝.

Remark: See Jensen's uniformization theorem?

Theorem 5.5.2: The least ordinal that is Π2-reflecting on the set of Π3-reflecting ordinals below is not Π3-reflecting.

Proof: In RichterAczel??

Theorem 5.5.3: Given an ordinal α, then "α is Π1-rfl. on ordinals γ∈α where "for all n, γ is Π1-rfl. on set of Πn-rfl. ordinals below"" iff "for all n, γ is Π1-rfl. on Π1-rfl. on set of Πn-rfl. ordinals below".

Proof: WIP

Remark: These ordinals correspond to terms with degree εΩ+1+1 in Taranovsky's notation "Degrees of Reflection".

5.6

Theorem: For \(n\in\omega\) and \(A\in\mathcal P(\textrm{Ord})\), \(\Pi_n\)-reflection on those ordinals \(\Pi_n\)-reflecting on \(A\) implies \(\Pi_n\)-reflection on \(A\).

Proof: For contradiction, assume that there exists \(\alpha\) that is \(\Pi_n\)-rfl. on those ordinals \(\Pi_n\)-rfl. on \(A\), but not \(\Pi_n\)-rfl. on \(A\). Then there must exist a \(\Pi_n\) \(\phi(\vec a)\vDash L_\alpha\) (with \(\vec a\in (L_\alpha)^{<\omega}\)) such that \(\nexists(\beta\in\alpha)(L_\beta\vDash\phi\land\beta\in A)\). However, take a witness \(\alpha'\) of \(\alpha\)'s reflection, and \(\alpha'\) must be \(\Pi_n\)-rfl. on \(A\) (note that \(\alpha'\) is a limit ordinal and \(\vec a\in (L_{\alpha'})^{<\omega}\) as well). In fact, a witness \(\alpha\) of \(\alpha'\)'s reflection will be a member of \(A\), and \(L_\alpha\vDash\phi(\vec a)\rightarrow L_{\alpha'}\vDash\phi(\vec a)\), which in turn implies \(L_{\alpha}\vDash\phi(\vec a)\). However this contradicts the assertion "\(\nexists(\beta\in\alpha)(L_\beta\vDash\phi\land\beta\in A)\)" from earlier.

Remark: The case for \(\Pi_0\) or \(\Pi_1\)-rfl. is also quote simply stated in terms of unboundedness of \(A\cap\alpha\) in \(\alpha\). To construct a subset of \(A\cap\alpha\) cofinal in \(\alpha\), choose elements of subsets of \(A\) cofinal in "ordinals that are limit of \(A\)".

5.7

Definition: A thinning operator \(S:\mathcal P(\textrm{Ord})\rightarrow\mathcal P(\textrm{Ord})\) has the property "iteration is continuous in the superscript" if for any limit ordinal \(\alpha\) and \(A\in\mathcal P(\textrm{Ord})\), \(S^\alpha(A)=\{\beta:\forall(\alpha'\in\alpha)(\beta\textrm{ limit of }S^{\alpha'}(A)\cap\beta)\}\), where \(S^\gamma(A)\) denotes transfinite composition of \(S\) (i.e. \(\bigcap\{S(S^{\gamma'}(A)):\gamma'\in\gamma\}\).

Theorem: If \(S(A)\) is a thinning operator of the form \(\textrm{Set of limit points of }(S'\cap A)\) for \(S'\in\mathcal P(\textrm{Ord})\), then \(S\)'s iteration is continuous in the superscript.

Proof: For contradiction, assume that \(S\)'s iteration is not continuous in the superscript. Then there exists \(\alpha'\in\alpha\) such that \(S^{\alpha'}(A)\neq WIP\)

5.8

Theorem: Use Rathjen's definition[20] of α-Πn-reflection. for any ordinals α and β, then "β is α+1-Π0-rfl." is equivalent to "for all n∈ω, β is α-Πn-rfl.".

Proof: Let ϕ denote a Πn formula. We can find a Π0 formula ϕ+ such that Lβ+1 |= ϕ iff Lβ |= ϕ+:

  • Let a be a (possibly 0-length) tuple of the parameters of ϕ, and let (ai)0≤i<length(a) enumerate the entries of a. Then, let a+ be a with Lβ appended as an entry, and let there be a similar enumeration (e.g. a+length(a)+1=a+length(a+)=Lβ). Noting that all entries of a are in Lβ and all entries of a+ are in Lβ+1, we let ϕ+ be ϕ with all of its quantifiers bounded to a+length(a)+1=Lβ.

We now prove that Lβ+1 |= ϕ iff Lβ |= ϕ+:

  • Let Qn be an existential quantifier from ϕ that quantifies a variable xn, and let ϕ- denote the subformula of ϕ not including any quantifiers Qm if m<n (is this correct?). WIP

Remark: A similar, but superficially weaker statement was given as a corollary by Rathjen in the same paper "The Art of Ordinal Analysis".

Sources

  1. 1.0 1.1 1.2 1.3 1.4 W. Richter and P. Aczel, Inductive Definitions and Reflection Properties of Admissible Ordinals (1973) (p.16), accessed September 2020
  2. W. Richter and P. Aczel, Inductive Definitions and Reflection Properties of Admissible Ordinals (1973) (p.15)
  3. 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 W. Richter and P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.44)
  4. 4.0 4.1 4.2 4.3 4.4 D. Madore, A Zoo of Ordinals (2017) (p.4)
  5. Using g0(δ)=δ+ε0 and g1(δ)=δ*2, and γ=ε0, noting that γ∈(least (+ε0)-stb. ordinal)
  6. 6.00 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.10 D. Madore, A Zoo of Ordinals (2017) (p.5)
  7. 7.0 7.1 7.2 D. Madore, A Zoo of Ordinals (2017) (p.3)
  8. They're a superset of the ω1-many [J. Barwise, Admissible Sets and Structures (p.178)] countable stable ordinals
  9. J. Barwise, Admissible sets and structures (p.188)
  10. M. Rathjen, The Higher Infinite in Proof Theory (p.19)
  11. W. Richter, P. Aczel, Inductive Defintions and Reflecting Properties of Admissible Ordinals (1973) (p.45)
  12. M. Rathjen, The Higher Infinite in Proof Theory (p.19)
  13. W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.25)
  14. K. Devlin, An introduction to the fine structure of the constructible hierarchy (p.2)
  15. T. Arai, A sneak preview of proof theory of ordinals (p.14)
  16. W. Richter, P. Aczel, Inductive Definitons and Reflecting Properties of Admissible Ordinals (1973) (p.13)
  17. J. Barwise, Admissible Sets and Structures (p.181)
  18. E. Kranakis, Reflection and partition properties of admissible ordinals (p.217)
  19. W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.20)
  20. M. Rathjen, The Art of Ordinal Analysis (p.19)
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