Difference between revisions of "User:C7X"

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==Large ordinals past beta 0==
==Large ordinals past beta 0==
==Alpha-recursion theory terminology==
==Ordinal function definability cheatsheet==
None of these are sourced yet, but this is what I think terms in α-recursion theory mean. Organized in a table for clarity
    Property of set:  Member of    Σ₁-def. subset of
L_α                  α-finite    α-recursive

Revision as of 18:54, 13 September 2021

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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