Home 
Contact us 
ZFC Axioms in jspace 4^{th} 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 nonzero divergence and the rotational properties for the entities described in such space. Further to form subsets we would require atleast one symmetry property. Entropy was inherent to the system. And all the measurements made by the macroscopic observer were of temporary nature i.e. they were of finite lifetime. (We note that the lifetime of the measurements may far exceed the life of Obs_{m} itself). These were the basic prerequisites for the discussion of ZFC axioms. The observer 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 jspace. Infact we would notice that while the axioms held true for a 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 jspace. Also even if the set {} with no numbers is measured by Obs_{m}, it is not unique in jspace. 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 jspace. 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 yєI, 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 reallife 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 xєX 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 nonempty 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 jspace, 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 jspace then the empty
set has no proper definition in jspace 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 consequence of the entropy inherent in the system is in
full force. This truly is a fundamental axiom. There is also an
interesting implications for how we define "negation". Similarly what
should we make of Cantor's theory? More on these some
other time.The criteria of Obs_{i} is very important. It allows us (Obs_{m}'s) to isolate the underlying framework for a given structure (universe) as measured by us. We are always in a state of measurement whether we are consciously aware of it or not. The finestructure constant symbolizes the limitation of a physical observer. We will be discussing the Hamilton's Anharmonic Coordinates next. The definitions are taken from Wikipedia, The Encyclopedia of Mathematics, and Stanford Encyclopedia of Philosophy. Information on
www.ijspace.org is licensed under a Creative Commons
Attribution 4.0 International License.

Previous Blogs: Set Theory Nutshell2014 Knots in jspace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, bfield 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 