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ZFC Axioms in j-space

April 2015

    Previously 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.  Obsi 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 Obsm, they were negated by Obsi 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 Obsi criterion.

Axiom of Empty Set: "There exists a set {}, with no element".  Again the definition of null measurement by 
Obsm is not true for Obsi.  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 Obsm, 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 Obsm 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 Obsm could do that then Obsm 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 Obsm as an infinite set, the information provided is still finite per Obsi.  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 Obsm but per Obsi criterion the measurement of the predicate by Obsm 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 Obsm would be determined to be of a temporary nature by Obsi. (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 Obsm, so is the definition of the relation f.  While the axiom will hold for Obsm, the relationship f cannot be necessarily measured with same accuracy each time as each member of X is mapped into F(X), per Obsi 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 = , there exists  such that  є X. And for every z є X, z = ∈ where {z1,z2,....,zn} = X.  Therefore z1 =, z2 =,........zn =.  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 .  Obsi has no apparent conflict with the Axiom of Foundation which was formulated to eliminate the concept of dictatorship in sets.  The concepts behind 0j 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


Knots in j-space



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

Visual Complex Analysis

The Einstein Theory of Relativity