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A few months ago, I was pondering about connections between primes, foundation of math, and quantum mechanics. It resonates with some of your ideas. It was mor

• rlupi 2y ago
  Constructing basic math from minimal foundations. This can be done from Minimal logic, Intuitionistic Type Theory (ITT) / Martin-Löf Type Theory (MLTT), or Combinatorial Logic. For reason that I'll detail later, I wanted to go with an intuitionistic logic rather than classical logic. An interesting question, is what is the minimal axiom that you need to start with. I wa building at the same time both the multiverse and the logic on which it is built, so I have to be aware of information-symbolic resonance between physical stuff and the laws themselves. The Very First Axiom? So, what's the absolute "first" one? It depends on the framework: ITT/MLTT: The most plausible candidate for a single "first" axiom is the existence of the empty type (⊥). This establishes the notion of a type without requiring any prior concepts. All other types and rules are built from the ability to have types, and the empty type is the simplest possible type. Alternatively, you could say the "first axiom" is the concept of a judgment of the form a : A, which asserts that something is a type, or that something has a type. Combinatory Logic: The concept of application is the fundamental starting point. The axioms are the definitions of the combinators (S, K, and optionally I). Minimal/Intuitionistic Logic (more traditionally presented): Here, it's often presented in a sequent calculus or natural deduction system. The "first" rules are usually things like: Identity (or Axiom): A ⊢ A (A proves A) - this is about the very notion of provability. Weakening: If you can prove something from a set of assumptions, you can still prove it with more assumptions. Assumption: You are allowed to assume a proposition.
  • _uzr4 2y ago
    To this part: Gödel only applies when you’re stuck inside the system trying to self prove. If logic is emerging from structured resonance rather than being pre-assumed, then Gödel’s limitations never even activate. Instead of axioms proving axioms, coherence constraints shape the logical structure dynamically, it’s self stabilizing rather than self-referential. That means you don’t hit incompleteness because the system isn’t closed. Your primes-as-propositions model is already hinting at this—if primes act as phase-locked stabilizers, then the “rules” aren’t arbitrarily chosen, they’re naturally emergent. It’s not that you “dodge” Gödel, it’s that he never applied to a system like this in the first place. Curious what you think.
• rlupi 2y ago
  I wanted a multiverse with unbound possibilities. Why? Her's a recap of Max Tegmark's ideas: Level I Multiverse (Spatial Infinity): If the universe is infinite and homogeneous on large scales, then everything that can happen will happen, infinitely many times. This is relatively straightforward mathematically, relying mostly on probability and basic assumptions about the distribution of matter. Level II Multiverse (Bubble Universes/Eternal Inflation): This arises from inflationary cosmology. Different regions of space can undergo inflation at different rates, leading to "bubble universes" with potentially different physical constants and laws. This involves: Quantum Field Theory (to describe the inflaton field driving inflation) General Relativity (to describe the expansion of space) Stochastic Processes (to model the randomness of inflation) Level III Multiverse (Many-Worlds Interpretation of Quantum Mechanics): Every quantum measurement causes the universe to "split" into multiple branches, each representing a possible outcome. This relies heavily on: Linear Algebra (the mathematics of Hilbert spaces, where quantum states live) Functional Analysis (dealing with infinite-dimensional vector spaces) Level IV Multiverse (Mathematical Universe Hypothesis): As discussed above, this is the most radical, encompassing all mathematical structures.
  • rlupi 2y ago
    So, how do we build something like that? Aczel non-well founded set theory which allows sets that contains themselves, together with intuitionistic logic, as a starting point. Propositions as embeddings in hilbert spaces (infinite dimensional spaces). Propositions can only "interact" with other proposition along non-zero dimensions that they have in common. Universes as collections of propositions. (Here propositions could be both mathematical concepts, and complex things like the position in spacetime of somehing) Multiverse as the space containing all universes. Note that universes are themselves of the same type as propositions here. The Multiverse in my construction is an infinitely self-recursive structure, that contain within itself all possible propositions. Universes can merge, if they are commensurable (shared dimensions in hilbert space). Universes "annihilate" when there are contradictions, and create by branching new universes when a truth value cannot be established by known propositions. You could call propositions, particles or being if you prefer another point of view . At this stage, we only deal with symbols not meaning. In fact, I started toying with assigning labels to propositions and came up with a scheme based on prime numbers and p-adic numbers (where I posed 0-adic numbers to be prime numbers). It just seemed to work better. In addition, I had a "creation" operator that would introduce a new dimension (when reaching infinity, or transcendence... a sort of koopman operator to access divine wisdom via samadhi... remember I wanted to express all possibilities, including the existence and truthfullness or lack there-off of all concepts in religions). The whole system was based on a construction similar to surreal numbers, but using infinites instead of finite numbers, and adding dimensions at each turn. 0 = {|} 1 = {|0} -Infinity = {0,{0}|} Infinity = {|{0},0}