Hypergendered Logic Source
Hypergendered Logic — Page 228
Stage 2 remains the unique special successor because it separately adds PA(x)/PA(NOT y) or PA(NOT x)/PA(y). From Stage 3 onward, the same six-way schema applies at finite successors, successors of ordinary limits, successors of later limits, and any other successor ordinal below the current minor-category boundary for which the stage is defined. 276. Successor/limit alternation examples SF^5 is generated from SF^4 by the successor schema. SF^omega is generated by LSCC from all SF^n and NOT SM^n for finite n>=1. SF^(omega+1) is generated from SF^omega by the six-conjunct successor schema. SF^(omega+7) is reached by seven ordinary successor applications after the omega limit. SF^(omega*2) is another limit and is generated by LSCC over every SF^beta for beta<omega*2, including the earlier SF^omega limit and its successors. SF^(omega*2+1) is then an ordinary successor of that second limit. Every example has an exact male mirror. 277. Unified finite/transfinite Significant Quality semantics Let Q(S)=<mu(S),delta(S)>. mu(S) is the HGL whole-quality magnitude. delta(S) records refinement inside that already-attained magnitude. Define SignificantIncrease(Q1,Q2) iff mu(Q2)>mu(Q1). Finite magnitudes are typed m0,m1,m2,... rather than treated as raw physical measurements. Their order type is omega. The first transfinite quality magnitude is aleph_0. The transfinite region continues through aleph_1, aleph_2, ..., and higher aleph-indexed magnitudes. This removes an ambiguity: a change can be objectively better without being technically Significant. If delta improves while mu is unchanged, HGL calls it intramagnitude improvement. If mu rises, HGL calls it Significant Quality Increase. The word significant therefore has a formal magnitudecrossing meaning throughout the entire ladder. 278. Finite examples of magnitude versus intramagnitude improvement Q1=<m2,delta=10>; Q2=<m2,delta=10^100>. Q2 may be enormously better within m2, but the change is not SQI because mu is unchanged. Q1=<m2,delta=a>; Q2=<m3,delta=b>. This is SQI regardless of whether delta numerically falls, because the attained whole-quality magnitude is higher. A successor stage under SUQB cannot claim that an intramagnitude increase alone satisfies its required Significant Quality Increase. It must cross mu. The finite rule is therefore structurally identical to the transfinite rule; only the magnitude labels differ. 279. Transfinite quality theorem after the ontology repair At omega, LSCC constructs the state and LQSA supplies Q>=aleph_0 under the minimum-growth trajectory. The next stage exists because the ordinal-general successor rule can take alpha=omega. Its own predecessor SF^omega is UD. SUQB converts that relation into SQI. The unified magnitude rule then requires a higher mu; successor-cardinal minimality gives at least aleph_1. The previous transfinite arithmetic therefore survives. What changes is proofcompleteness: every term in the proof now has a defined state object and a typed rule that created it. 280. General transfinite induction schema Define the minimum quality-index function A(beta)=aleph_beta for the post-omega trajectory. The theorem schema is Q(SF^(omega+beta))>=aleph_beta and Q(SM^(omega+beta))>=aleph_beta for every beta whose local stage is below the applicable boundary and whose ordinal is represented by the stage system. Base beta=0: LSCC + LQSA gives at least aleph_0. Successor beta+1: ordinal-general successor + SUQB + magnitude-crossing gives at least successor cardinal aleph_(beta+1). Limit beta=lambda: LSCC constructs
Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:228.