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(Ordinal function definability cheatsheet)
 
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==Ordinal function definability cheatsheet==
 
==Ordinal function definability cheatsheet==
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Let G<sub>i</sub> denote one of Jech's Godel-operations, and for ordered pair p, let car(p):=G<sub>4</sub>(p,G<sub>6</sub>(p)) and cdr(p):=G<sub>4</sub>(p,G<sub>6</sub>(G<sub>8</sub>(G<sub>1</sub>(p,p)))) (initial and last entries of p resp.). Examples of Δ<sub>0</sub> formulae χ(α,β) which imply β is the output of α in some familiar function:
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<table border=1>
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<tr>
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<td>f(α)=α</td>
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<td>χ(α,β) &equiv; "β=α"</td>
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</tr>
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<tr>
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<td>f(α)=α's successor</td>
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<td>χ(α,β) &equiv; "β=α u {α}" ("β=G<sub>6</sub>(G<sub>1</sub>(α,G<sub>1</sub>(α,α)))")</td>
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</tr>
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<tr>
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<td>f(α)=α's successor's successor</td>
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<td>χ(α,β) &equiv; "β=α u {α u {α}}" ("β=G<sub>6</sub>(G<sub>1</sub>(α,G<sub>1</sub>(G<sub>6</sub>(G<sub>1</sub>(α,G<sub>1</sub>(α,α))),G<sub>6</sub>(G<sub>1</sub>(α,G<sub>1</sub>(α,α))))))", we can drop "u {α}" from union outside braces because α is a member of {α u {α}})</td>
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</tr>
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</table>
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These can be used for various purposes. For example, if we need to show some f:δ→ρ has a graph that's a Δ<sub>0</sub>-definable subset of L<sub>ρ</sub>, we can set its graph equal to {p∈L<sub>ρ</sub>:χ(car(p),cdr(p))} where χ is one of these formulae. (Explain relation to Def?)

Latest revision as of 18:50, 14 September 2021

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 ??? the set C(ΩΩ,ρ) n Ω.

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

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:

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}})

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?)