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I encountered this mathematical structure but need help decoding the notation:

[ T = \prod_{i=1}^{\infty} \mathbb{N} ]

with hierarchy levels:

  • $!1, !2, !3\ldots$
  • $T1G, T1H, T1B\ldots$
  • $A1, A2, A3\ldots$
  • $1\text{-}1, 1\text{-}2, 2\text{-}1\ldots$ You raise excellent points, and I apologize for the lack of clarity. Let me provide the missing context:

The Notation:

  • !1, !2, !3... represent parallel hierarchies/universes
  • T1G, T1H... are element generators within each hierarchy
  • A1, A2... are atomic units forming the base elements
  • 1-1, 1-2... represent coordinate paths through the structure

The Observed Pattern: I'm studying recursive mathematical structures where each element contains the complete infinite hierarchy within itself. The pattern is self-similarity at all scales.

The Core Idea: The structure T = ∏∞i=1N describes an infinite recursive system where navigating from any element eventually reveals the entire structure.

You're right that I should have explained this initially. I'm seeking help with formalizing this self-containing property in proper set theory.

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  • $\begingroup$ Where did you encounter it? $\endgroup$ Commented 2 days ago
  • $\begingroup$ Hello Jukka Kohonen, I developed this notation in my own research on recursive mathematical structures. It emerged from studying how infinite systems can exhibit self-containing properties, where each component encodes the entire structure. I'm seeking help with the proper set-theoretic formalization of these concepts. $\endgroup$ Commented 2 days ago
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    $\begingroup$ You need our help to decode a notation you developed? $\endgroup$ Commented 2 days ago
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    $\begingroup$ Do I understand right? All these "!1, T1G, A1, 1-1" etc are your private notation, that you probably intend to mean something, but you offer us no explanation. And then you would like someone to help formalize these unexplained things? Also, what observed patterns are you talking about? Your question text explains nothing about observed patterns. $\endgroup$ Commented 2 days ago
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    $\begingroup$ There is no definition for what your notation means to mathematically handle your notations. The current question cannot be answered unless you provide a solid, formal definition for what you mean, not just vague intentions for each symbol. $\endgroup$ Commented 2 days ago

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