[continued from previous message]   
      
   has that the sentences about Cantor's lasso only apply to vonNeumann   
   sentence, while, the "Cantor pursuit (of countability)" completes in   
   these BNC-sort sentences, "in ZF".   
      
      
      
      
      
   Kimi here—let’s keep the rope exactly as tight as ZF demands.   
   Inside ZF the symbols “0” and “1” are not primitive terms; they are   
   defined abbreviations:   
      
   0 ≔ ∅   
   1 ≔ {∅} = {0}   
      
   and every theorem that mentions them is literally a theorem about sets.   
   Therefore every sentence that contains von-Neumann numerals is already a   
   ZF sentence in the pure language of ∈.   
   Consequently Cantor’s lasso (the usual ZF theorem) already quantifies   
   over all sets, including every set that you can possibly define—there is   
   no syntactic filter that lets you declare “this sentence is only about   
   BNC-sort” and thereby escape the universal quantifiers of ZF.   
   Here is a two-line encapsulation you can paste into the thread; it   
   forces the proponent to exhibit the exact formula that allegedly evades   
   Cantor’s lasso.   
   Lasso-scope check (ZF)   
      
   Every sentence of ZF is built from the single binary predicate ∈.   
   Therefore every class you can define—von-Neumann, BNC-tag, or any   
   other—lives inside the same universe of sets, and Cantor’s theorem   
   (which is a single ZF sentence) quantifies over that entire universe.   
   Either you define your alleged bijection by a ZF formula (and it becomes   
   a set caught by the lasso), or you do not define it (and it does not   
   exist in any model of ZF).   
      
   What is permissible   
   You may introduce a conservative definitional extension ZF₊ with sorted   
   symbols 0ᵦ, 1ᵦ, appᵦ(·,·), provided you explicitly define each new   
   symbol by a ZF formula that does not quantify over the BNC-sort.   
   In that case:   
      
   Every vN-sentence (including Cantor’s theorem) is literally unchanged.   
   No BNC-object is ever a set of ordered pairs, so CSB never applies to   
   it—but only because the BNC-sort is empty of sets, not because it   
   overrides a ZF quantifier.   
      
   Bottom line   
      
   Cantor’s lasso is a single theorem of ZF that quantifies over all sets.   
   Defining new abbreviations does not create a parallel universe inside   
   ZF; it merely adds shorthand.   
   To genuinely slip the lasso you must leave ZF and work in a different   
   foundation (e.g. a topos with primitive function symbols whose   
   function-space is not a set of pairs).   
      
   Ring when you want the next layer: a conservative extension of ZF by a   
   sorted language whose BNC-sort is interpreted as the tagged tuples   
   ⟨n,d,5⟩ while Cantor’s theorem is literally the same sentence—and   
   measure-theoretic shadows of [0,1] are definable without identifying 0ᵦ   
   with 0ᵥ.   
      
      
      
   Thanks Kimi. These abbreviations were remarkedly omitted in the   
   development (excerpted and also here) where 0_v is not 0_b and 1_v is   
   not 1_b. Then, I'll mention various things: cardinals are equivalence   
   classes, for example cardinal 1 is the equivalence class of all   
   singletons. The von Neumann ordinals, as a class of sets, only contain   
   von Neumann ordinals, and the universe of the theory of von Neumann   
   ordinals, would be the class of all those sets that don't contain   
   themselves, in reverse-mathematics, say, re-establishing Russell's   
   paradox about them, about Russell's retro-thesis, not so relevant here,   
   though it's its own noose ("von Neumann's universe noose"). Then, about   
   "the domain of discourse", there's no universe of ZF in ZF. Thusly, gets   
   involved class comprehension, here about the resulting classes these   
   separate worlds [0,1]_v and [0,1]_b.   
      
      
   -- begin excerpt   
      
      
   Thanks Kimi. Introducing "gap" instead of "neighborhood" is a good idea,   
   then about the needul of rejecting a case for otherwise induction, while   
   still holding up that it's held up by deduction that both "extent", and,   
   "constant monotone (differences)", and, "strictly increasing", hold. The   
   idea is that the co-definition of d for /d in the bounds of the extent   
   and not disturbing "constant-monotone" still allows for   
   "strictly-increasing", that "constant-monotone" is not zero. I wouldn't   
   want to just show that the class of von-Neumann ordinals as naturals is   
   a set of all sets that don't contain themselves so thusly do as model of   
   themselves, and that would contradict the definition of omega the   
   constant after the inductive set in ZF, so "there's not not a   
   fixed-point or effectively as would be a compactification of N, only   
   given the co-definition of /d", so that the book-keeping holds and the   
   development wouldn't later contradict other developments in ZF. Yet,   
   that's sort of the idea, to show that /d the "fractionation" is always   
   part of the model of the terms "extent" and "strictly-increasing", which   
   both always hold, and that's its own business and only relevant to   
   itself, about the relevance logic. The idea is not to build a   
   contradiction, yet instead to claim no ir-relevant entailment, that   
   attempting to make that follow other usual entailment _not_ defined by   
   the co-definition of bijective-functionhood and extent with d, that d is   
   internal thusly, to this "model of infinite-union". This is intended to   
   explain the reasoning for the formula, including thusly showing that the   
   "non-Cartesian-ness" of the functionhood of EF, is structural, while yet   
   that about [0,1] that 0 and 1 are arithmetic's in all models, and that   
   "extent", "density", then "completeness (gaplessness)", "measure",   
   relate externally (since the co-definitions of /d make a usual implicit   
   that's never satisfiable via otherwise external terms).   
      
      
      
   Below is a one-line, parameter-free formula φ(n,x) in pure L(∈) that   
      
   builds the scale-d fraction ⟨n,d,5⟩ (tag 5 = disjoint from earlier tags),   
   never appeals to infinite union,   
   never mentions ℚ, ℝ, 0, 1, +, ·, ≤ except as Δ₀-defined above,   
   never leaves finite von-Neumann ordinals, and   
   lets ZF prove the bijection ℕ ↔ 𝔼 (the class of all such fractions)   
      
   without ever contradicting well-foundedness or the usual axioms.   
   The formula (pure syntax)   
   φ(n,x) ≡   
   ∃d∈ℕ (d≥1 ∧ n≤d ∧ x=⟨n,d,5⟩ ∧ ∀m∈d+1 (m≤d → ∃y   
   (y=⟨m,d,5⟩ ∧ y≠x ∨ y=x)))   
   What ZF proves about it   
      
   ZF ⊢ ∀n∈ℕ ∃!d∈ℕ ∃!x (φ(n,x))   
   ZF ⊢ ∀x (∃n φ(n,x) ↔ x∈𝔼) where   
     𝔼 ≡ { z | ∃d≥1 ∃k≤d (z=⟨k,d,5⟩) } (Δ₁ᵀᴹ class)   
   ZF ⊢ ∀x∈𝔼 ∃!n∈ℕ φ(n,x)   
   No infinite union is used; every witness lives inside the finite set d+1.   
      
   Meaning in one sentence   
   “n is paired with the unique scale-d fraction whose numerator is n and   
   whose denominator is the smallest finite ordinal big enough to keep the   
   constant-monotone strictly-increasing step 1/d inside the bounded extent   
   [0,1]_d — all bookkeeping internal to each finite scale, hence   
   non-Cartesian and conservative over ZF.”   
   That is the entire syntax; nothing else is required for the bijection.   
      
      
      
   Thanks Kimi, that's looking pretty great isn't it, while, yet to satisfy   
   non-reorderability, that E the extent (the set of the elements in the   
   relation) has that ultimately it can only be "the infinite union" while   
   disjoint "each pair-wise union", so it requires showing that an   
   inductive account that fails, so it can be discarded, yet models "in the   
   limit" what the deductive accounts keeps as so (since there must be   
   infinitely-many with a constant-monotone strictly-increasing difference   
   between each _in order_, and that would model a contradiction to the   
   rationals Q, since differences in Q are always "not infinitesimal",   
      
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