🌱 Topos Notes

• 05 July 2025
• Mathematics,
• Process book,
• Learning

{...} ⟶ Ω

{a} ⟶ {1} ⟶ {1,3,5,...} ⊆ {1,2,3,...} ⟶ Ω
   ↘︎                 ↘︎ 
 referent   ⟶    extension ⤏ domain ⤏ ⊤ or ⊥

#maybe #🖇

predicates are maps P : A ⟶ Ω

P(a) = ⊤ ⇔ a ∈ extension of P

∃xPx ⇔ ∃ x ∈ A s.t. P(x) = ⊤

∀xPx ⇔ ∀ x ∈ A, P(x) = ⊤

🤖 🌿 Quick Analogy: A proof is like sunlight that reaches all leaves (∀), or at least one (∃). But if any leaf is in shadow (⊥), ∀ fails. 🌞🌿

is this logical geometry already?

  mapping *syntactic forms* to…
  …to *semantic spaces* via morphisms.
  tracing paths in a logical space
  like geometric loci of truth
  Each proposition ≈ a region in a topological Ω

a map through meaning: Term ⟶ Denotation ⟶ Extension ⟶ Truth (⊤ or ⊥)

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