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ZFC Axioms in j-space 4 ^{th} April 2015Previously we discussed the requirements for the measurement space whose members could be represented as the members of a Set. We had noticed that VTS was required i.e. quanta was defined for sets. Also implicit were the presence of source or non-zero divergence and the rotational properties for the entities described in such space. Further to form subsets, we would require at-least one symmetry property. These were the basic prerequisites for the discussion of ZFC axioms. Obs _{i}
would invalidate the axiom of choice right away, as without the
assumption of symmetry the definition of "exact one" could not be
formulated in j-space. In-fact we would notice that while the
axioms held true for a macroscopic observer Obs_{m}, they were negated by Obs_{i} except one.Axiom of Extensionality: "Two sets are the same iff they have the same elements." The property of same elements is possible only with a symmetry assumption, hence it would not hold true per Obs _{i} criterion.Axiom of Empty Set: "There exists a set {}, with no element". Again the definition of null measurement by Obs _{m} is not true for Obs_{i}.
Actually the whole mess started because null could not be accurately
measured. Hence {} can not exist in j-space. Even if the set {} with no numbers is measured by Obs_{m}, it is not unique
in j-space. Axiom of Pairing: "Given two sets X and Y, a set Z can be formed such that Z = {X,Y}". As noted previously X, Y and {X,Y}, can not exist simultaneously without the assumption of a symmetry. Axiom of Union: "If X is a set, there exists a set Y whose elements are precisely the elements of X." Can Obs _{m}
repeat the exact measurements of the set X to form another set Y
identical to X? Such precision is not possible in j-space. If Obs_{m} could do that then Obs_{m} could also form an unique null set {}.Axiom of Infinity: "There exists a set I such that {} belongs to I, and for every y є I, yU{y} є I." The set I represents an infinite set and includes all natural numbers. We note that even though the set I exists for Obs _{m} as an infinite set, the information provided is still finite per Obs_{i}.
Also {} can not exist without VTS. We also need to remember that in
real-life situations we replace infinite values in the limits of
integral, by experimentally determined values to achieve convergence.Axiom schema of Separation: "For each set X and predicate P, there exists a set of elements x є X such that x's are bound by P. The subset formed by such elements is a set itself." If we consider the predicate existence for example, it's measurement is not possible without VTS. Therefore the axiom holds for Obs _{m} but per Obs_{i} criterion the measurement of the predicate by Obs_{m} would not be precise enough. For example the measurement of the predicate existence, will alway represent a temporary existence. Similarly other predicates as measured by Obs_{m} would be determined to be of a temporary nature by Obs_{i}. (This statement is a conjecture.) Axiom schema of Mapping: "For each set X the image set F(X) corresponding to the binary relation f, is also a set." The members of set X are defined based on the capacity of Obs _{m}, so is the definition of the relation f. While the axiom will hold for Obs_{m},
the relationship f cannot be necessarily measured with same accuracy
each time as each member of X is mapped into F(X), per Obs_{i} criterion. The exact definition of inverse becomes a problem in this case, without which X can not be verified from F(X).Axiom of Power Set: "Every set X has a power set P(X), whose elements are all the subsets of X." At least one symmetry is required to perform this construct. Axiom of Foundation: "Every non-empty set X contains an ∈ -minimal element, that is, an element such that no element of X belongs to it." We can also state that for every set X such that ¬X = Ø, there exists ∈ such that ∈ є X. And for every z є X, ¬z = ∈ where {z _{1},z_{2},....,z_{n}} = X. Therefore ¬z_{1} = ∈, ¬z_{2} = ∈,........¬z_{n} = ∈. Since ∈ є X we can also write, ¬∈ = ∈. This is a rather extraordinary situation. Essentially what can not be measured in j-space, is equal
to what is measured at the maximum capacity of the observer. This is
the definition of quantum in the measurement space. We also note that ¬ ....¬¬∈ = ¬¬∈= ¬∈ = ∈. If a set is properly defined in j-space then the empty
set has no proper definition and the null set defines quanta ∈. Obs_{i}
has no apparent conflict with the Axiom of Foundation which was
formulated to eliminate the concept of dictatorship in sets. The
concepts behind 0_{j} and ∈ are
comparable, and both essentially represent the limitations of the
observer. The Axiom of Foundation, truly is a fundamental axiom. The definitions are taken from Wikipedia, The Encyclopedia of Mathematics, and Stanford Encyclopedia of Philosophy. |
Previous Blogs: Set Theory Nutshell-2014 Knots in j-space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-field and lower mass bound Incompleteness II The supersymmetry The cat in box The initial state and symmetries Incompleteness I Discrete measurement space The frog in well |