Axioms for Positive Construction

Communication transcending culture or species has protocol, thus has some self-evident principles which I prefer we stop referring to as assumptions. These principles form our Axiom System which, via syntactic structure, let us realize our self-expression. Previous systems attempted to cause the facts of physical matter to agree with intuition formed by practising arithmetic responding to which, this Gary Larson cartoon provides exemplary definition of the cardinal number one, albeit with some misapplication of the word 'the'. (Define multiplication as fast addition; define addition combinatorially.)

In general, a formal system has its formal language which supplies variables, constructive syntax, axioms together with derivation guidelines. We work with the formal language L defined by three variables, inclusion, description, if, or, together with the existential quantifier. Classical mathematical axiom systems including Zermelo-Fraenkel (ZF) and Peano attempted to formulate (from fundamental assumptions intuitively given by arithmetic) a theory of sets upon which mathematics could model life or discern truth. Independent of the ZF and Peano systems and motivated by working with infinite sets, several mathematical logicians formulated variations of the Axiom of Choice to make infinite sets seem to behave as finite sets. (Encyclopedia69.com)

The Axiom of Choice was meant to express that whenever our life path meets with a container holding a hundred socks, we have the ability to reach in and pull out one sock, which seems obvious in an abstract sense however life differs from that Axiom of Choice. Equivalent to the well-ordering principle, the Axiom of Choice was employed to prove the Tarski Paradox that claims that a sphere finitely cut sphere could be moved through space to form two spheres of volume equal to the original sphere (from a pea, make a sun).

A. Tarski showed in 1924 that two equal-area polygons in the plane are equi-decomposable, and this led to the formulation of the Tarski paradox. [2, p501, V 5]

Our previous blog entry shows paradoxes have absence of existence where logic agrees with physics, therefore something in Tarski's proof of 1924 contains a cognitive error. The deducability of that Axiom of Choice from other set theoretic axiom systems might be meaningless for us since we question classical rules of deduction together with ZF or Peano systems, also for now we reject infinite. A classic definition of that Axiom of Choice was

For each family F of sets having something within, there seems to exist a function f which, for each input set S belonging to the family F returns as output an element of S, IE that axiom asserts existence of f(S) inside S. [2, p314, V 1]

A well-ordered set S containing something satisfies conditions:

1.  Each pair in S has a relation of comparison such as less than, more than or equal, 
2. If a pair's comparison is bidirectional then the two are equal, 
3. Comparisons are transitive, 
4. Each subset of S which contains something has a least element. 

The Rothschild family has agreement with the logic in our universe, this biosphere, together with living beings in our biosphere, plus contains something; however, rather than relate by comparisons, we relate. We have family loyalty without bidirectional relatedness causing equality, without transitivity, also without a least member. Constructive methods have bidirectional inclusion. For example, trees have inclusion in our biosphere, some of our biosphere's resources have inclusion in our trees, without thereby causing our trees to be equal to our biosphere. 

Discerning which axioms may be constructive starts with reconsidering relations. Classical mathematics presents relations in terms of a set theoretic axiom system rather than holding relations among the self-evident principles of construction; in that presentation, relations were considered to be maps. However, in logic, of two beings a relation may have absence of existence, positively constructive existence or a reversible negative bond. We may represent a relation R between two beings as xRy inside a representative set such as {-1, E, 1} with E denoting absence of relation. Traditional matrix representation of correspondences had integral values from [0, 1] which caused a lie of omission since a negative relation shows displacement of positive construction rather than removing positive construction. Reconsidering Boolean Algebra makes sense after revisiting axioms, meanwhile correspondences observably have domain, range plus influence. IE for each pair of beings x, y, xRy could influence these related beings' relation with a third being, z.

Positively Constructive Axioms

Having observed much of Graph Theory seems to work in relations, some events in 1993 inspired me to reconsider set theoretic axioms' intellectual honesty as a model of what happens in life. Set Theory meant to deal with paradoxes which I find could be best handled by logic together with physics, also known as Applied Mathematics. Hold in mind many of the paradoxes were formed inside binary logic constraints imposed by attempting to work on subsets of the evaluations {negative, null, absence, positive}. We mathematical logicians have had emptiness confused with absence. An emptiness requires a physical form to define its existence whereas absences merely are. From that confusion, we found the empty set difficult to define. Observe we refer to the empty set, E, as an existent entity, exemplified for instance by the food in an empty refrigerator. Observe we habitually omit reference to an absence of the empty set, exemplified for example by whatever could be in our blind spots. (Merely removing the referenced refrigerator is insufficient to exemplify absence since a missing refrigerator shows by its boundaries in a kitchen expecting it, or in a set of refrigerators expecting it.) The empty set has been defined as the (existent) set described by the variable which excludes all variables. I propose an alternative definition,

Let emptiness, E, be defined as the existent set described by the variable whose internal content (inclusion or exclusion) occurs as unknown.

Optional viable reinterpretations of E exist within the range of interpretations of negation, however, Positively Constructive Axioms reject negation. For example, maybe E could be defined as

• approximately known, 
• not yet known, 
• expected, 
• inverted, 
• reduced, 
• refined, 
• rotated, 
• translated, 
• displaced, 
• concealed, 
• imagined, or 
• private.

A property P exists within the expressions of language L provided we observe some variable x in L has property P(x) hold true.

Replace union with disjoint union; work with sets discretely. IE x union y is the z which have inclusion in x or y. Obviously traditional union of x has absence of formation due to rejection of transitivity. Discard the notion that each variable self-relates.

Do we need sets to exist? From the perspective of particles, sets are an extra layer of construction misrepresenting reality. From the perspective of people tallying in standard business applications, sets are a useful tool for counting or for partitioning data, thus keeping sets seems tempting, however, the axioms of mathematics which express existence (IE have agreement with the unified theory of physics) work without sets. Assert existence by way of demonstration. Black swans exist including for who has not yet seen a black swan, since many black swans are visible in Perth, Australia, or equally visible in the Calgary Zoo in Canada as this one photographed here. However, can we assert unicorns do not exist? Have we observed the universe sufficiently? Conjecturing the absence of unicorns in our universe makes sense.

Review construction of an Axiomatic Theory of Sets in traditional binary logic with the wrong definition of if, provided by V. N. Grishin with A. G. Dragalin in [2, pp 318 to 320, V 2].

Rule 1: of two variables or terms, declared relations of inclusion or equality are formulae. 

Rule 2: Of two formulae, declared relations of equivalence, implication, disjunction, conjunction, negation or universal or existential instantiation are formulae, and descriptions of the variables for which a specific formula holds as a property are terms.  

 Both rules are syntactic in the formal language L, however, for mathematical insight informed by behaviour particles do, we omit declaration of a formula in L as sufficient cause for existence; we omit conjunction, negation and universal instantiation. Hold in mind, mutual inclusion differs from establishing equality.

The empty set is defined as the description of variables x which contain not-y for all y. 

This could be how to construct a hologram. Certainly this construction forms a hologram in information space, however, to what extent to real particles correspond with packets of information?

The set of all variables for which property A holds is defined by the description of all variables for which each of what these variables contains itself has property A. IE property A belongs homogeneously inside a population in order for the whole population to claim property A. 

This works for homogenized milk, however, this specification proves impractical when engaged in a simple activity with people familiar with high school mathematics, such as reporting a repeating, algorithmic criminal pattern in a (maybe unique) city: let T be the property of being a thief. City V could have property T in tangible business reality without the entire population in city V all exhibiting property T homogeneously. Similarly, particles each behave as its own entity, without visible homogeneity in a collection of particles.

An unordered pair, (x, y), is defined as the variables z which equal x or y. Ordered pairs <x, y> are defined as the set construction {{x}, {x, y}}. A set of size one is defined in terms of variable x as being equivalent to {x, x}. 

Order matters in many applications of mathematics since time (aka radiation) goes one way, however, such a set construction is outside the scope of our formal language L (our language identified for working with particles). What sequence of events puts photons where photons are prior to release of photons happening?

Since an electron seems to be itself together with the gamma photons it contains, an electron may be understood as itself together with the space it takes. What volume of space does an electron take, in being an electron (or, in becoming an electron)? We could define ordered pairs suitable for formal language L inspired by electrons with their resident photons: containment expressing order. Subsequently, an unordered pair could be interpreted for formal language L in terms of pairs of particles, optionally existing within one another. However, the definition of one could be questioned in view of electrons which produce photons seeming to be one (each) prior to production. Post-production, an electron together with the two photons it may produce form three particles; pre-production, one.

The union of variables x with y is all variables z which are included in x or in y. The intersection of variables x with y is all variables z which are included in both of x, y. The union over a variable x is each variable which is contained in an existent second variable, or the declared-existent second variable is contained in x. 

The definitions of unions, intersections works with information packets emulating particle behaviour, however, the intersection leads to slow counting methods. Thus, we work with disjoint unions of variables. The definition of a union over a variable insufficiently disambiguates the interior content of a variable from the exterior content of a variable (the former traditionally referred to as the variable's domain, the latter as beyond the variable's domain). The definition of union over a variable x I find sufficient is the defined union of the contents of a second variable with the contents of x, in which the second variable is contained in x. To handle the existence question of such a second variable, we may restrict our attention to variables on domains having content.

The Cartesian Product of two variables, x, y, is all variables z for which there exists an ordered pair of variables, <u, v>, such that z equals <u, v>, x = u, y = v. 

The purpose of the Cartesian Product may be to add a dimension to the domains of variables x, y. In contexts in which Newtonian mechanics suffices to explain or predict observed phenomena, the definition suffices as given, including with the proposed definition of containment expressing order since a generational indicator suffices to indicate axis. In terms of information packets emulating observed particle behaviour, contents transfer from one variable's domain to another's as a functional expression of the relation between the variables. IE exchange due to variable interaction changes the domains, thus we could question our concept of dimension.

The constructive algorithm for additive population increase is

• start with one, 
• an added one has independent co-existence, else links with a previous one. 

Subsequently define products of variables as fast addition in the counting methods introduced in childhood. In an arithmetic example, three bowls containing four apples each yields twelve apples (and three bowls). Disambiguating the product from the content union of variables, keep track of order. 

A variable has the property of being a function (in our formal language) when the variable is contained in the Cartesian product of its domain with its domain, with each initial domain element mapping to at most one domain element by way of the variable. 

 A variable maps content from its domain to the domain of a related variable (perhaps itself); such a map is a function by definition when mapping content from its domain to at most one recipient domain content instantiation. In information space with information packets behaviourally emulating particles, variables which map content from their domains to the domains of recipient variables experience an operation cost, which is a business reality not previously considered in mathematical construction of maps and functions. The values of the images of domain content mapped by a variable are known by their descriptions (post map-action, IE after the cost of doing business, the ROI becomes known).

The standards infinite set (aka variable) is defined as containing E, the empty set, and for each alternative variable contained in the infinite variable, the alternative variable in union with itself as the defining element of a set of size one also has inclusion in the infinite variable. 

The traditional definition of the infinite is clearly informed by a traditional proof of the countable infinity of the natural numbers: think of the biggest number you can; add one. In terms of information packets emulating particles' behaviour, transmissions happen by way of relations without transitivity - IE inclusion of a specific variable happens without implying inclusion of a variable's successor (or reconstruction as the defining element of a set of size one). Inclusion of a specific variable contains inclusion of the variable's domain. (Similarly, decisions contain their consequences.) A finite state machine without a stop condition may (without repetition) emulate our concept of infinite with its finite states. Therefore, we work without assertion of the existence of infinite, yet may fine the limit infimum or limit supremum concepts pragmatic in some situations.

This sufficies to start construction of an axiomatic theory of mathematics.

The axiom of extensionality claims for each variable x, the equivalence of x's inclusion in variable y with x's inclusion in variable z implies y equals z.

The idea of equality expressed by extensionality is content-based, with the intention being, whichever pairs of entities share the same interior content therefore have equality. This idea suffices in many material applications. In the two applications we consider most frequently, (particles, humanity), the notion of particle equality may be undefined, and human equality may challenging to define. Personally, I interpret human equality as our giving ourselves equal opportunity in similar circumstances, rather than as our giving equal pay on a theory of labour as though quality of results were irrelevant.

The axiom scheme of comprehension asserts of an arbitrary formula A, there exists a variable y whose domain (content, interior) is exactly the content of A.

The authors construct a contradiction in the system provided by these axioms with the formula, not-x is contained in x, for some variable x. A positively constructive rephrasing could be, approximately-x is contained in x. Since x is contained in its near approximations, the axioms of extension with comprehension provide for x being equated to its approximations, a notion understandably rejected by the steel industry and more. Content-based equality has obvious motivation, however, the motivation for comprehension could be elusive to readers of Real Analysis familiar with the construction of the interior of a set. (link to wiki set interior) We henceforth reject the axiom of comprehension.

The axiom of comprehension was introduced by E. Zermelo (1908), which inspired A. Fraenkel to propose the axiom of replacement (1922), with the resulting axiom system denoted ZF.

What is the axiom of replacement?

Resistance to, or explanations of, the paradoxes we handle in blog entry ... provide context for some subsequent axiom system development of set theory, including B. Russell's theory of Types. Attempting to overcome concept stratification in the theory of types, W. Quine responded with his NF axiomatic system (1937), subsequent to finitization of axioms provided by the Neumann-Godel-Bernays system (1925). The derivation rules - first order applied predicate calculus with equality, with description - in ZF and NF are identical.

The pair axiom asserts a pair of variables exists.

However, construction of the declaration of existence happens by way of the asserted existence of a third variable whose interior is contained in one or more of the pair of variables being thereby shown to exist, which admittedly leaves something to be desired. The existence axiom we prefer straightforwardly declars existence without explanation, with demonstration sufficing to communicate the phenomenon of existence.

Subsequent to development of NF, the axiom of equality may be restated as

If a pair of variables have equality, properties independent of the variables held by one variable could be held by the other of the pair.

Some mathematical logicians view declared unique existence as a quantifier, in contrast with which, we view uniqueness as a property an existent entity may have. The standard expression of uniqueness states, for each variable y which would replace a variable x (in terms of having a property), y = x establishes the uniqueness of x.

The union of the contents of a variable exists.

Asserting existence of the contents in the domain of a variable could be the constructive intention, however, the construction in the union axiom states containment is equivalent to the existence of a third variable in between variables in the domain of the variable being operated on by union. We could instead construct the union of the contents of a variable by forming the union of the contents in its domain.

The power set axiom asserts the set of all subsets of a given set exists. IE, a variable y containing  all instantiations of what variable x could contain exists. 

To be continued.

References
1. Encyclopedia69.com
2. The Soviet Encyclopaedia of Mathematics, chief editor I. M. Vinogradov, Edition 2,
ISBN 1 55608 010 7.