The External Axiomatic Requirement Hypothesis
Takuya Aoki
Abstract
The Structure–Observer Interpretation Hypothesis (SOIH) describes the relationship between a structure S and the observer intelligence O that interprets it as I = f(S, O). The Observer-Relative Knowability Hypothesis (ORKH), in turn, treats knowability K as relative to an observer intelligence O and its epistemic capability level L: K = K(O, L). This paper extends these two formulations to their theoretical totalities. Let S* denote the totality of all internal structures, whether known or unknown: S* = Sknown ∪ Sunknown. Let O* denote not the set of currently existing individual intelligences, but the theoretical totality encompassing possible observer-intelligence configurations and their possible epistemic capability levels L: O* = ⋃i,L Oi(L). Here, i does not enumerate individual observers; it distinguishes different configurations of observer intelligence. Likewise, L denotes theoretically possible epistemic capability levels without presupposing either a fixed maximum or a restriction to discrete stages. In ordinary terms, S* corresponds to the “world,” while O* corresponds to the “totality of observer intelligence.” An internal system employing S* and O* may describe relations within them, but it cannot, using only that same internal system, provide a non-circular grounding for the existence of S* and O* that it already presupposes. This problem is not confined to the totalities themselves. The presently observed world Spresent and presently existing observer intelligences, including ourselves, Opresent, are also internal to S* and O*, respectively. The fact that they can be observed and described therefore does not imply that the grounds for their own existence can be supplied from within the same system. At this point, two hypotheses may be considered: one in which an external axiomatic position A exists and one in which it does not. This paper does not claim to observe or prove the existence of A. Without A, however, the existence of the internal system—including the present world and ourselves—remains, ultimately, an ungrounded premise. Accordingly, under the condition that one requires grounding and continues the deductive construction rather than stopping at such brute premises, this paper posits an external axiomatic position A from necessity: A ∉ M(S*, O*). A does not denote an unknown structure within S*, nor does it denote God, a creator, a higher universe, or any other specific entity. Its content remains unspecified: A = X. Furthermore, if some content X′ currently occupying the A-side later becomes connected to the internal descriptive space through a newly available orthogonal basis and thereby becomes observable and describable as internal structure, X′ is no longer A. Once such a descriptive connection is established, X′ becomes internally mappable and loses its external status as A. The hypothesis therefore does not establish the concrete existence or content of A. It advances a conditional requirement: if deductive construction is to continue in search of the grounds for the existence of an internal system that includes the present world and ourselves, an external axiomatic position A must be posited.