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fix: proofread Set Theory II part - #338

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fix: proofread Set Theory II part#338
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@vEnhance vEnhance commented Jul 9, 2026

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Proofreads the Set Theory II: Model Theory and Forcing part (issue #315), covering tex/set-theory/models.tex, tex/set-theory/forcing.tex, and tex/set-theory/CH.tex.

All findings are minor typos / notational slips; no mathematical arguments were rewritten.

models.tex — Reflection problem hint

  • The hint quantified if there exists $a \in M$, but M is never defined in this problem (it's about $V_\kappa$). Changed to a \in V_\kappa, matching the solution.
  • "use the fact that $\kappa$ is cofinal" → "$\kappa$ is regular". Cofinality isn't a standalone property of a cardinal here, and the solution itself invokes regularity.

forcing.tex

  • Non-existence-of-generic example: $G \cap M = \varnothing$$G \cap D = \varnothing$. The point is that $G$ misses the dense set $D = \Po\setminus G$; $G \cap M$ is just $G$.
  • Motivation for the forcing relation: $\left<\tau_r\right>$$\left<\tau, r\right>$ (missing comma in the pair).
  • Definition of the forcing relation: "we write $\tau \Vdash \varphi$" → "$p \Vdash \varphi$" ($p \in \Po$ was just introduced; the body uses $p \Vdash$ throughout).
  • Two phrasing fixes for consistency with the standard term: "dense below in $p$" → "dense below $p$", and "dense in $p$" → "dense below $p$".

CH.tex — countable chain condition

  • $\Delta$-system lemma proof: "there exists an $a \in C$ contained in uncountably many $F \in C$" → "an element $a$ contained in …" ($a$ is a point that lies in uncountably many members, not itself a member of $C$).
  • ccc proof of $\opname{Add}(\omega,\kappa)$: $B = \{p_\alpha \mid \dom(p_\alpha) \in R\}$\in \ol C (we keep the conditions whose domains lie in the $\Delta$-system $\ol C$; $R$ is the root, so $\dom(p_\alpha)\in R$ is a type error). Also clarified that $p_\alpha$ is restricted to $R$ before comparing (each $p_\alpha\restriction R \colon R \to {0,1}$), since finitely many such restrictions forces a collision.

🤖 Generated with Claude Code

https://claude.ai/code/session_014PXK5YtzH5S2FyB9XcXmE1


Generated by Claude Code

Proofread the "Set Theory II: Model Theory and Forcing" part
(tex/set-theory/models.tex, forcing.tex, CH.tex). Fixes minor
typos and notational slips:

- models.tex (Reflection hint): correct undefined "a \in M" to
  "a \in V_\kappa", and "kappa is cofinal" to "kappa is regular"
  (matching the solution).
- forcing.tex: "G \cap M" -> "G \cap D" (the dense set); missing
  comma in the pair "<\tau, r>"; "we write \tau \Vdash" -> "p \Vdash";
  and two "dense below p" phrasing fixes.
- CH.tex (ccc proof): "a \in C" -> "an element a"; the set B should
  collect conditions with domain in \ol C (not R); restrict p_alpha
  to R before comparing.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014PXK5YtzH5S2FyB9XcXmE1
@vEnhance vEnhance added the proofreading-campaign Systematic full proofreading of text label Jul 9, 2026 — with Claude
@vEnhance vEnhance mentioned this pull request Jul 9, 2026
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vEnhance merged commit d723824 into main Jul 9, 2026
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