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Hey guys, Jack Black here. And I'm here to tell you about the most fantastic shape.

Two-cardinal problem

Link: [1]

May not be useful when analyzing theories such as KP+"ω₂ exists". See context on Discord: [2]

Message contents:

~~Work in a model M of KP+GCH+"ω₂ exists". If ω = γ < α = ω₁^M < ω₂^M, then can we apply Vaught's result internally in M?~~
Edit: I don't think this has a use, since by assuming the existence of α,β s.t. <α,β>, we already assumed models of KP+GCH+"ω₂ exists" has that behavior in the first place

Relationship between OCFs and Mostowski collapse

A WIP explanation

Rathjen explains here why the intersections \(C(\alpha,\rho)\cap\pi\) resemble collapsed-down, "cut up" versions of \(\pi\)

"ψ(α) is a collapse of Ω" reasoning: \(\alpha\) isn't the set being collapsed, it just indexes how many collapses have been done (by trans. induct. hypothesis) before the one currently being performed. The actual collapsing (whether thinking of it as Mostowski collapse, or informally as "the order type obtained by removing the gaps between terms"), the set actually being collapsed is \(C(\alpha,\rho)\cap\Omega\), which is effectively \(\Omega\) for this argument. All collapses are performed using transfinite induction, assuming that each preceding collapse has been done before it.

For example, the only reason (that I can think of) for why \(\Gamma_0\) could be called a collapse of \(\Omega^\Omega\) is that \(\Gamma_0<\Omega^\Omega\). For the position "Γ₀ is the ΩΩth collapse of Ω", we assume (by trans. induct. hypothesis) that for η∈ΩΩ, the ηth collapse of Ω has already been found. Then when taking the ΩΩth collapse of Ω, we now consider the set C(ΩΩ,ρ) n Ω. We have taken each collapse ψ(η) previously, and each of these impacts the set C(ΩΩ,ρ).

To clarify:

  1. Ω is the set being collapsed, under a rule for "αth collapse of it". This rule may be "look at it through the lens of C(α,ρ)", i.e. intersect it at C(α,ρ)
  2. The resulting set after the collapse is C(α,ρ) n Ω, since that's what Ω appears as after "cut up" (intersected)
  3. ψ(α) is the order type of this collapsed set.

Notes on 2-stability

  • If $L_\alpha$ is $\Sigma_m$ $L_\alpha$-uniformizable, then there is a $\Sigma_m$ function $h$ s.t. $\textrm{dom}(h)\subseteq x\times L_\alpha$ and

$$\forall(x\in L_\alpha)(x\in h^{\prime\,\prime}(u\times\{x\})\prec_{\Sigma_m}_L\alpha)$$. [3] I don't know what some of this means

We can prove ITTM Σ is, for all n∈ω, Πn-reflecting on the Σ-stable ordinals:

Let Stb(σ) denote the Π₁ formula φ₁ from Kranakis80's theorem 1.8 - φ₁(σ) is true iff σ is stable. Letting Rn and Stsf be from Levy's "A hierarchy of formulas in set theory", let Satj(g) be the formula Rn(g,n)&∃f(Stsfj(f,g)) - this plays the role of "g is Σj and |= ⌜g⌝", and Satj(M,g) be the formula Rn(g,n)&∃(f∈M[<ω])(Stsfj(f,g)) - this plays the role of "g is Σj and M |= ⌜g⌝".

Let χn be ∀(g∈ω)(Satn+1(g) → ∃β(Stb(β) & Satn+1(Lβ,g) ) ). We have that χn is true iff Ord is Σn+1-reflecting (i.e. Πn-reflecting) on the class of stable ordinals. Note that if we have χnLξ for ordinal ξ, then ξ is Πn-reflecting on the ξ-stable ordinals. Lζ satisfies ∀ξ∃σ(σ∈ξ & χnLΣ), and this is a Π2-formula, so LΣ also satisfies this.

Large ordinals past beta 0

Ordinal function definability cheatsheet

Let Gi denote one of Jech's Godel-operations, and for ordered pair p, let car(p):=G4(p,G6(p)) and cdr(p):=G4(p,G6(G8(G1(p,p)))) (initial and last entries of p resp.). Examples of Δ0 formulae χ(α,β) which imply β is the output of α in some familiar function Ord→Ord:

f(α)=α χ(α,β) ≡ "β=α"
f(α)=α's successor χ(α,β) ≡ "β=α u {α}" ("β=G6(G1(α,G1(α,α)))")
f(α)=α's successor's successor χ(α,β) ≡ "β=α u {α u {α}}" ("β=G6(G1(α,G1(G6(G1(α,G1(α,α))),G6(G1(α,G1(α,α))))))", we can drop "u {α}" from union outside braces because α is a member of {α u {α}})

f(α)=next limit ordinal after α

χ(α,β) ≡ "β is an ordinal" & α∈β & ∀(γ∈β)∃(δ∈β)(γ∈δ) This incorrect, χ(α,β) holds for β any limit ordinal >α

These can be used for various purposes. For example, if we need to show some f:δ→ρ has a graph that's a Δ0-definable subset of Lρ, we can set its graph equal to {p∈Lρ:χ(car(p),cdr(p))} where χ is one of these formulae. (Explain relation to Def?)

For Σ1-definable functions: Marek and Srebrny cited theorem 36 of Levy's book A hierarchy of formulas in set theory, which they used to show that if f:Ord→Ord is Σ1-definable in V (β=f(α) is a Σ1 predicate), then for all α f(α)<(α+)L, where + is cardinal successor. So the output of a Σ1-definable function never has higher cardinality in L than its input.

Sources about arithmetic copies

  • Stephen G. Simpson, Subsystems of Second-order Arithmetic
  • [4]
    • Sets representable in $Z_2$ are hyperarithmetic?
    • See lemma 8.9 for possible explanation of apparent connection between $\Sigma_n$ and $\Sigma_{n+1}^1$
  • [5]