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Day fifty-four -- Constructed linguistic universe (VII)

      From Tienzen:
      The concept of "functional equal" is not new. But it is new as an operator in algebra and in set theory. For two sets, A and B which are not equal in algebra nor in traditional set theory but can be "functionally equal" with definition 11. Now, the internal dynamics of this "constructed linguistic universe" can be analyzed.

      First, let's review some definitions.
         2. Definition two: Set Vx = {syx; syx is a symbol in Lx}.
         3. Definition three: Wx is a "word" in Lx if and only if the following two conditions are met.
               1. Wx is a syx of Lx.
               2. Wx has the following attributes:
                     1. Wx has a unique topological form.
                     2. Wx carries, at least, one unique completed sound note.
                     3. Wx carries, at least, one unique meaning.
                        ...
         6. Definition six: Sx is a "sentence" in Lx if and only if the following two conditions are met.
               1. Sx must have, at least, two wx. That is, Sx = Opc (syxa, syxb, ...).
               2. Sx must be an operand of Opd. That is, Sx = Opd (Opc (syxa, syxb, ...)).
            Note: Definition 6.a -- If Sx has only one wx, Sx = Opd (wx) is a "degenerated" sentence.
         7. Definition seven: Px is a "phrase" in Lx if and only if the following two conditions are met.
               1. Px must have, at least, two wx. Px = Opc (syxa, syxb, ...)
               2. Px must "not" be an operand of Opd.
      Second, the word/sentence layer (ws - sphere) -- this sphere has three sub-layers
         1. the word sphere
         2. the phrase sphere
         3. the sentence sphere

      This ws-sphere is governed (or delineated) by two operators, "Operator" of composite (Opc) and "Operator" of dot (Opd).

      With these definitions, the words, the phrases and the sentences are all members of the set Vx. And, the set Vx can be re-written as:


      Set Vx = {syx; syx is a symbol in Lx, words, phrases, sentences}. Thus,
      set Wx = {syx; syx is a word in Lx}
      set Px = {syx; syx is a phrase in Lx}
      set Sx = {syx; syx is a sentence in Lx}
      And, set Vx = Wx U Px U Sx; (union of Wx, Px and Sx).

      We now can prove some theorems.

      Theorem two -- In ws-sphere (context free), Vx = Lx
      Note: traditionally, we call Vx is the set of syntax.

      Theorem three -- (Lx, Vx) and (Ly, Vy) are two different natural languages, then,
      Vx (=F=) Vy
      That is, the syntax sets of two natural languages are functionally equal.

      Corollary 3.1 -- Lx and Ly are mutually translatable.

      Postulate 4 -- the Transitive Property holds for the (=F=), the functional equal.

      Now, we can re-write the set Vx.

      Let P is a process, the combination of Opc (operator of composite) and Opd (operator of dot), then,

      P ({wx}) = Sx U Px = P (Wx), the process P generates the Px (phrases) and Sx (sentences).

      So, Vx = Wx U P(Wx) , and I will re-write this set equation with a new convention,
      Vx = (Wx, P), the Vx can be constructed by having Wx (set of words) and P (process of constructing phrases and sentences). This new convention is, in fact, an "equivalent transformation".

      Now, (Lx, Vx) and (Ly, Vy) are two different natural languages, and,
      Vx = (Wx, Px) and Vy = (Wy, Py)
      Per theorem 3 -- Vx (=F=) Vy, the syntax sets of two natural languages are functionally equal. and we can prove a new theorem,

      Theorem 4 -- Wx (=F=) Wy and Px (=F=) Py, the word sets of two natural languages are functionally equal.

      Corollary 4.1 -- Wx (Chinese) (=F=) Wy (English).
      Wx (Chinese) has only about 60,000 characters and Wy (English) has about one million words. Yet, Wx (Chinese) is functionally equal to Wy (English).

      Seemingly, this corollary 4.1 is a commonly known old fact. Yet, when it becomes a theorem, a new logic is opened up. It, in fact, says that every English word can be encoded (or ciphered) with Chinese characters, one million words being encoded with a few thousand characters.

      If we can find a PB set, and PB (=F=) Wx (Chinese); PB is functionally equal to the entire Chinese character set. With the "postulate 4", the transitive of (=F=),


      Wx (Chinese) (=F=) Wy (English)
      PB (=F=) Wx (Chinese)
      then, PB (=F=) Wy (English)
      That is, Wy (English), all English vocabulary, can also be encoded with PB.

      Now, we have reached the starting point for PreBabel.

Signature --
PreBabel is the true universal language, it is available at
http://www.prebabel.info

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