In Alfred Tarski's "Logic, Semantics, Metamathemaitcs" the set of all meaningful sentences is described and the ordinary two-valued system of sentential calculus L is shown to be a subset of all decidable systems that is consistent and complete. That is amazing!
I would like to compare the algebra of subsets of S that is closed to the set=theoretic operations of union, intersection, and complimentation to the set of sentences of S that is closed to semantic the operations of implication and detachment.
I was thinking that since Tarski shows that the Set of all sentences is a subset of all linear orders, that there is a relation of sentences as linear sequences to the general projective plane of all linear operations. I think that is the classifying space.
I wonder if I am in correct in assuming that the set of all sentences implies there is a subspace topology in which every element is a singleton with indentically zero.
Is it possible that there is a way to connect the sentences and the algebra of prime ideals by an elemental formula. Could I used the Serpinski dual operators? The standard model? Maybe the relation is tautologically obvious and every one already knows?
What I realy want is the classifying space for music theory.
If you don't have Tarski's book here an outline of the definition of L with substitution of variables for music:
The ordinary two-valued system L of sentential calculus is the set of all sentences that satisfy the matrix M = [A, B, f, g] where A = {0}, B = {1}, and the function f and g are defined by the implication and detachment formulas: 1) f(0, 0)= f(0,1)=f(1, 1) = 1, f(1, 0) = 0; and 2) g(0) = 1, g(1) = 0. From this definition, it immediately follows that ordinary sentence calculus is a completely defined and axiomatizable system. The function f is the implication operation used to make new sentences, since {f(0, 0)= f(0,1)=f(1, 1) = 1, f(1, 0) = 0} is the propositional calculus for implication. The f is the pitch-position equivalence relation given by f(x, y)= 0 where x = y are any two harmonic values defined by the value function, described by Tarski, so that the proposition “if a, then b” is interpreted to signifying the implication “if pitch then position.” The pitch values are frequency values related to positions which are a state of system values in a finite state machine. The pitch value set is defined by integers i indexing points in the frequency domain, while the position value set is defined by integers j that index points that are not defined directly by frequency values (that is positon values have no frequency attached except by the pitch-position equivalence map which is the primitive equivalence relation defining membership in the semantic system).
The primitive detachment function g in music is the intonation function, which is a function of pitch (that is, frequency) only. Because intonation means to sound a fundamental mode of vibration at a specific pitch value, the intonation function g is a witness function that quantifies the musical object for all pitch values. Since tuning is by definition a pitch-position equivalence relation between pitch value index numbers and fret value index numbers, it is already clear in Tarski’s definition that pitch-positon function f and the intonation function g in music are inverse homeomorphisms.
For instance, if we have a musical string with 20 fret positions then we have the string f(x, y) = 0 defined by the pairwise disjoint relation of the pitch value set and the fret value set, which run together like railroad tracks that vanish at infinity. The pitch-position relation is a constant: the same for every string regardless of pitch. When a second string is added, the interval between the two string is the only determinant of the system formed by the intersection and union of the strings.
Tablature notation of guitar music is an infinite data strip that passes through the guitar and is output (like a player piano) as the tablature notation of the musical sounds. The numeric sequences in the tablature are a subset of the algebraic closed field that is formed over the guitar by the tuning vector which is the k = 6 point-wise restriction on the continuous function of pitch.
So we have the algebra of subsets of the musical key (using the prime ideals of the tuning space) that do not contain the music key itself; and we have the set of all sentences of the same set that are written in the algebraic language that satisfies the guitar tuning theory (x1, ... x6) tuple.
I know this is correct and sooner of latter, some one will get it. If you can't answer the question, maybe you can explain why music is not a valid topic in modern mathematics. The trash that passes for the mathematics of music in the literature is ridiculous. How come no mathematicians critque these authors like Mazzola "Topos of Music" and Tymoczko "Geometry of Music."
This is a great question.
At first glance, Tarski's set of meaningful sentences and the power set (set of all subsets of a set) are quite different. The members of Tarski's set each has semantic content. By contrast, members of the power set of a set have no semantic content.
Of course, if we choose to imbue the members of a power set with semantic content, then Tarski's sentential calculus is applicable.
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