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.

Remove Paradoxes

Traditional paradoxes have been viewed as paradoxes due to our having been convinced of binary myopia as a valid and useful formal thought structure, which it sometimes is. How to counter-argue paradoxes is via mathematical logic and physics.

Paradox  A situation allegedly containing a contradiction, and supposedly contradictions cannot exist. Paradoxes show deficiencies in theories and cause attention to new effects and further studies.

Sophism  A deliberately presented wrong conclusion masking another error.

Zeno's Achilles and Tortoise  Achilles (A) and the Tortoise (T) go east at speeds v and v/100 respectively starting with 100 meters between them. Zeno's 5th century BC idea was that A cannot catch T due to having to go through an allegedly infinite sequence of points in a finite time, which Zeno imagined impossible. The construction in Zeno's argument is Achilles' midpoint sequence: C1 = AT/2; C2 = C1T/2; CK = C(K-1)T/2 for an integer K; and the Tortoise's corresponding midpoint sequence given by moments t(Ci) for i = 1, ... K.

Achilles' sequence of midpoints would be infinite were Achilles a physically existent human. A human has the property of omitting the hypothetical option of shrinking while navigating an infinite convergent sequence, therefore the limit is forced by physics to obtain since the human gets so close to his or her target that separation is clearly of zero distance. For example, if Achilles' speed is 200 meters per minute, then his sequence of midpoints converges to obtaining the 100 meter distance in half a minute. Slightly trickier is when Achilles catches his tortoise, which does happen since both omit shrinking into infinitesimals in our biosphere: equate 200t with 2t + 100 to obtain their collision place as 101.01... meters from where Achilles started, which happens at time t = 1/1.98 minutes.

At merely the 5th century BC, Zeno had every reason to know to avoid forming a cyclic argument, amphiboly and demonstrate ignoratio elenchi - missing the point. Presuming what he purports to prove, Zeno sets up Achilles in an infinite sequence approaching T and then claims A cannot obtain T plus the distance T displaced in the real time of real A catching real T. Zeno's idea of continuity had a structural defect the correction of which is: the limit point obtained has to be included in the path, which is what physics does. Zeno's era was sufficiently advanced, he could have repeatedly experimented with a variety of runners and turtles to form a theory bearing some cognitive connection with physical observation. Had Zeno constructed his theory with intellectual honesty and human integrity, then he would have derived the predictable conclusion from the premises rather than forming an unrelated conclusion, that of the limit of a convergent sequence in a path being (according to Zeno) unobtainable.

We may conjecture Zeno had absence of thought for the hydrogen atom, which does seem to avoid limit convergence. A hypothetical hydrogen atom having assigned quantum number so large that the atom's radius is half a centimetre tends to agree with classical physics in terms of conjectured light emission, yet lost its real existence agreement with the biosphere.

In A. G. Dragalin's response to Zeno's first paradox in the section on Antinomy, A. G. Dragalin acknowledges the Newtonian mechanics response, omits insight from quantum mechanics and challenges the Archimedes Principle which states, for a pair of real numbers a, b, > 0, there exists a natural number n such that an > b. A. G. Dragalin challenging this analytical construction surprises me since taking the ceiling of b/a and adding one suffices for selecting n. A. G. Dragalin further surprises me by subsequently claiming Zeno's proposed paradox presents a real problem by demonstrating the possibility of denying Newtonian mechanics inside the valid domain of Newtonian mechanics (I think possibilities are by definition unreal, since were they constructible they would be labelled as such). A. G. Dragalin and I are agreed as to infinity being a concept worth revisiting, however the pragmatic impulse for reading non-Archimedian ordered fields is non-obvious to me.

Zeno's Sandpile  is presented by A. G. Dragalin as:

One grain of sand does not form a sandpile. If n grains of sand still do not form a sandpile, it follows that they will not form a pile after another grain of sand has been added. Accordingly, no number of grains of sand can form a sandpile. 

 Obviously whatever we agree a small pile of sand is, it has a finite number of grains of sand, as do beaches. Disagreement about a definition's detail differs from disagreement regarding the meaning of words. For example, in my own life, rather than me being required to argue as though in an endless sequence of PhD defences and argue against an endless sequence of PhDs all the while having the experience of Jonathan Borwein unresponsively indicating each argument insufficient for him, instead, end that useless sequence and replace that with requiring Jonathan Borwein to defend his written assertion of May 1996 that I have an absence of mathematical skill, together with an acceptable grade in his analysis class in spring 1995, together with an honours mathematics degree from Dalhousie University. We could agree a pile of sand has two million grains, ten thousand grains or one hundred grains. Let some natural number m be the number upon which we agree (observe the absence of the axiom of choice in this construction since prerequisite to two of us agreeing, communication happens and communication resulting in agreement is constructive) is the defining minimum number of grains of sand in order to form a sandpile, disambiguated from a bunch of sand insufficient to pile. m > 1. Proof by induction is irrelevant to Zeno's paradox as presented and the entire paradox depends on having not yet agreed upon a definition among relevant participants in the conversation. The erroneous conclusion exemplifies forming a conclusion based on an absence of definition; due to that absence, equivocation and amphiboly are evaded however, leaping to conclusions based on full absences of relevant material exemplifies appeal to ignorance.

Adjusting mathematics and physics for mathematical unification and for a clear explanation of quantum gravity causes many of us to ask questions we would not normally ask. For example, A. G. Dragalin's response to the Sandpile Paradox questions the inductive proof method in mathematics by raising consideration of undefined volumes which makes sense when considering the quantum number associated with a hydrogen atom, and remains difficult to interpret in classical mathematics. A. G. Dragalin introduces indefinite volumes handled by mathematical logic by exact methods which differs from indiscriminate interpretations of classical analytical methods; I agree with A. G. Dragalin as to the absence of mathematical induction inside unification on the basis of constructive selection.

Russell's Paradox Construct the set T whose elements are sets each of which is not an element of itself. Is T an element of itself? In traditional binary logic without considering the meaning of the word 'not', each of the two available answers contradicts itself. However, 'not' could validly mean approximately, expected to be, asymptotically approaching, an inversion of, or a successor of.

My refutation of Russell"s Paradox is A = {A} which sidesteps interpretations of the word 'not' and is supported by my two birth certificates issued by the government of Canada.

JES is not an element of JEO and JEO is not an element of JES, yet both are me thus I am.

Responding to A G Dragalin's response to Russell's Paradox, I disagree with the traditional response to Russell's Paradox of prohibiting sets which are members of themselves. Recursion happens via containing a defined entity within its definition. A G Dragalin guides us to consider the question whether an exactly-described set of properties therefore causes a set of objects to exist in possession of the described properties.

Inside binary myopia, contradictions fallaciously yield nothing. "Left + right = nothing. Say sing = nothing. Possible impossibility = nothing. Two birth certificates in semblance of disagreement = nothing. Us + them = nothing. You + me = nothing. Vancouver Canada + Logic = nothing. One red shift star + one blue shift star = nothing since shift cancels to 0." Inside honest ternary logic, the one which works as this biosphere works, "Left one step + right one step = two steps. Us + them = us all together. You + me = us. Vancouver Canada + Logic = a new ten billion dollar industry. One red shift star + one blue shift star = two stars. Two birth certificates = one legal name change without red tape."

Russell's Paradox is constructed with semblance of reasonableness due to double negatives cancelling in traditional binary logic, also due to the absence of self-inclusion in arithmetic sets normally studied in undergraduate mathematics. However, two 180 degree rotations in R^3 differs from absence of rotation. Consider the complementary definition, a set Y has property A iff Y is self-inclusive; let Z be the set of all such Y then ask whether Z is in Y - Z being self-inclusive suffices. 


Russell's Paradox raises the valid question of what a mathematician is. This mathematical logician excludes anyone relying on vacuous claims, lies, or harm. Formal fallacies I have observed among some mathematicians practising what could be interpreted as embezzlement include False Cause, Enthymeme, Composition, Accident, Existential Fallacies, Exhortation, Illicit arguments, Bandwagon argument, Appeal to Ignorance, Appeal to Authority, Ignoratio Elenchi, Red Herring, Petitio Principia, Appeal to Force, Suppressed Evidence, Ad Hominem Abusive, Ad Hominem Circumstantial, Amphiboly; Informal Fallacies I observed include blaming victims for what was done to victims of real crimes, insisting interpretations be dishonestly low, agreeing to semblances of fairness too little too late, stealing intellectual property in lieu of teaching, insistence upon blind faith within mathematics, or living inside a 24/7 hour party. 


Who belongs in this world? Some say the definition for existence is to be able to make a product which sells profitably. Some say our definition for existence is in being able to define matter or energy. I prefer inclusive capitalism in our dual economy provide our existence conditions, giving each of us supported liberty to relate within our range of skilled interests. I prefer standards we establish in our range of logical disciplines exclude anyone that relies on fallacies to obtain results, plus includes whom contributes positively to our discipline, our community, humanity, the people of Earth plus our biosphere. I prefer our fields stop being hideaways for anyone shy in this world. We would lack strategy were we to follow Russell's lead by defining mathematicians by a particular attribute since the logical complement of the definition qualifies, too; however, denying obvious presence of skill in the hope of secrecy could be assessed as a weak strategy. 


A G Dragalin seeks a domain protected from paradox. Consider a form G/H where H is being mapped to the empty set rather than to zero. In binary, (G, H) could have traditional truth values (T, T), (T, F), (F, T) or (F, F). Add a dimension for manoeuvrability of information, while mapping G/H to G (since H becomes empty) arrange truths strictly via honest communications to T. G at truth value T is a final solution, as A G Dragalin sought. Reference, my corrections of Aristotle in 1996. 


The Village Barber  The traditional form is, a village barber opts to shave only those villagers who do not shave themselves; does the village barber shave himself? Avoiding the traditional trap, form the positively constructive interpretations of "not" which do exist: 

• Some people who do not shave their heads are bald. 
• Some people who do not shave their heads avoid all haircuts with preference for long hair. 
• Some people prefer to keep some hair, thus go to hairdressers or neighbours for hair cuts. 
• Some people shave their own heads. 

Thus the village barber shaves the heads of the villagers who prefer having shaved heads without self-sufficiency. 

Contrary to one of A G Dragalin's assertions, such barbers may exist. Contrary to a second of A G Dragalin's assertions, most real-life situations could be exactly formulated or reliably defined. We agree as to the importance of internal consistency in reliable systems. 

Cantor's Paradox  The traditional form has been to ask whether the power set of the set of all sets could be contained inside the set of all sets, without examining whether the set of all sets may be constructed by what specific definition. (The power set of a set S is the set of subsets of S, generally denoted P(S).) Clearly in English, of course the power set of the set of all sets, being merely one set, must be in the set of all sets in order for the sentence to make sense; however there is more to mathematics than English. 

What constructability constraints have we? During autumn 2011, we specified formal language L (publicly distributed) has the existential quantifier without universal quantification. Adapting from binary logic to ternary, we place strong emphasis on the word some, an emphasis which may wane as we habituate to ternary. In the formal language L, Cantor's Paradox remains unconstructable meaningfully. 

For reference, the formal language L to which we refer is {x, y, z, inclusion, description, implication, or, existence}, with the empty set having a positively constructive definition as the description of whatever variables ostensibly exist which accept anything included in the variable. 

Richard's Paradox  

In the formulation of G. D. W. Berry (1906), consider the set of natural numbers each one of which can be uniquely defined by a meaningful text containing not more than 1000 syllables. Obviously, the number of such texts is finite, since the collection of all texts with 1000 syllables is finite. Consider the smallest natural number that is not a member of the set defined above.

The above paragraph is a meaningful text, containing not more than 1000 syllables, which uniquely determines some natural number which, by definition, cannot be characterized by a set of this type. Obviously, this paradox can be avoided if this text is declared to be not meaningful (or to be a text which does not define a natural number), but in such a case, as before, there arise difficult problems concerning the criteria for a text to be meaningful. 

(Page 207, Volume 1, Encyclopaedia of Mathematics 2nd edition)

A finite subset of natural numbers which may be uniquely defined, each set member in one-to-one correspondence with one specific meaningful text containing up to 1000 syllables (as per Berry's formulation) may have distinct numbers corresponding to different meanings conveyed by the same n syllables for some natural number n defined in a traditional way. The smallest meaningful paragraph has one monosyllabic verb in the imperative form thus intuitively could correspond to the natural number one however each imperative monosyllabic verb makes a distinct meaningful paragraph of normal size one, therefore one such paragraph could define whichever natural number we select, including the option of selecting zero. The first quoted paragraph defines an unspecified natural number, say m, without contradicting the provided definition contrary to the claim in the second paragraph. Words have meaning. Pi exists. The text of Richard's Paradox conveys meaningful information. What criteria establish a text as meaningful?

In August 2003, I announced I hold life's clock, by which I referred to (a) a binary pair of opposites of a binary pair of opposites trapping semblances of nothing within, (b) relativity without time, (c) ternary logic in agreement with living entities in this biosphere coordinated with physics. I have since been surprised by some interpretations of my announcement, including the expectation of explaining where our universe comes from. The correct definition of if (the positive parts) forms part of part (c) of my 2003 announcement.


Texts which communicate in retrospect have some meaning. Landmark Education provides an interesting alternative response to Richard's Paradox in which words can be viewed as meaningless and results as possible to create from nothing. Landmark Education's technology works without belief yet also without disbelief. One could invite a representative of Landmark Education to explain the phenomenon of distance education in accredited universities inside a theory of meaningless text since distance education in accredited universities also works. 

Eubulides' Paradox  Euclid's student who was an adversary of Aristotle (4th century BC) announced, "What I say now is a lie." The paradox arises from assuming whatever is supposed by saying ABC is therefore true. We have a format for supposition in mathematics which lets us work with sensible hypothesis via logic to hold our work inside a range of sensible conclusions. While working, we implement logic without supposing a declared supposition be true or false - what would motivate us to implement decisions prematurely? Observe Eubulides reportedly avoided formats equivalent to, "I am lying now." Thus the alleged paradox in the supposition, "what I say now is a lie," has absence of existence as a paradox. Eubulides could equivalently have said, "Suppose I am lying now."

References

The Soviet Encyclopaedia of Mathematics, 2nd edition, chief editor I. M. Vinogradov. ISBN 1 55608 010 7.

Physics by Halliday and Resnick. LCCC 66 11527.

Some Mathematics

Further to my entry No Mathematics that iffy concept time has been on my mind, as has gravity. Since I have not yet read Physics, my physical insights have mainly been from information theory. For example, the stable wormhole I constructed in summer 2011, taking advantage of Vancouver-area's deceit, is my online communication network via medias and letters, and my Vancouver-area local, who are disassociated with each other yet pass some data via me. Clearly many have similar information space wormholes stabilized and some pass data via friendly environments. My 5 March 2012 claim that if = iff is of course based on my mathematical absence of reference to the physical universe. Electrons, humans, gamma rays, tossed tea cups, Darwinian selection and relations all have clear sense of direction and show discernment... insofar as tossed tea cups discern their future. In Physics and Mathematics (the latter does exist even though Jon Borwein put an interesting question forward, specifically, do a mathematician's mathematical skills disappear due to expedience and declaration by people with titles?), cancelling wave crests and troughs to get an illusion of zero seems acceptable, although we know we added two inputs. However, the same practice in trade relations causes enmity IE repulsion. Consider purchasing £100 of gold then receiving a 50-50 mixture of gold and tungsten and being told acceptance of the mixture in lieu of the purchased product is compulsory, and "no harm" was done since the sum of gold and tungsten is zero, yet conspicuously the £100 isn't returned for an alleged absence of product. That way of doing business, which I learned at Simon Fraser University in Vancouver-area Canada, is repulsive and causes enmity.

Since I am being held financial hostage in a hostile environment (hating is different from Heyting and the locals assure me they are unaware of the latter), I thought I might get some positive constructions and experiments in information theory handled. Richard Feynman imagined particles travel via every possible path. My results based on human correspondence and information transfer during 2011 to 2012 show a correct analogy to gravity exists in the information space of human relations, said space being a path connected manifold; and shows human minds are correctly analogous to particles, IE energy concentrations with dispersion and distribution options. The mechanical theory of kinetic and potential energy is an incorrect analogy for minds as particles in information space. Of course I extended my work to include my tribe of eight cats who, doing what they do, mindfully interact in the process. For example, while I was working with Mikhail Prokhorov on Finance and Logic in summer 2011, I was wondering what to do with the world and Saith coincidentally clawed my exercise ball which reminded me of slow deflation. Good Saith. In this information space model of what could be happening in a Physicist's experiments with particles, particles (minds) discern paths as do macroscopic entities observed in the real world. For example, animals who do real physical work while moving themselves from A to B disagree with the idea in Theoretical Physics that all work calculations are path independent and that return journeys take zero work; similarly businesses also demonstrate path awareness and discernment while relocating from State of Affairs A to State of Affairs B. Ohm's Law indicates electrons are opportunists, and slit experiments with photons indicate the same about photons. When is if = iff? The supply chain of existence, nature, business, relations and all subsequent activities indicate dropping the idea of 'only if' and working with if, sufficient and necessary conditions, IE supportive guidance, works. Darwin's individual to environment adaptation theories are for slow adaptation and are based on mutual positive relation (sufficiently analogous to gravity, different from repulsion). Slow adaptation to extreme putridity and hatred differs from what Darwin observed and set out to explain. Thus part of the global solution implemented in summer 2011 has anyone expressing putrid hatred toward logic do so on record, else remain silent; by putting themselves on record they volunteer for the equal opposite response to their own expressed volition.

Today I started reading about ammonia and external forces on systems of masses which have conserved momentum. I remember the n-body problem for n greater than 2 is said to be unsolved and yet am wondering why an n-body problem appears to have n external forces, each different, rather than being understood as sharing the one external net force which then reduces the n body problem to the n-1 body problem previously solved. IE  I think proof by induction could be applicable. For example, the 3 body problem is analogous to the ammonia molecule where the net external force is supplied the nitrogen atom, and since the nitrogen atom wants to bounce, it could reasonably be ignored from the perspective of the hydrogen atoms whose configuration is still given by the nitrogen regardless of anthropomorphic awareness. Similarly, people live their entire lives in our world totally unaware of the market, the invisible hand, the Rothschild Family, DARPA and advances in mathematical disciplines; I could be anthropomorphising those people when saying their configuration is given by a net external force regardless of the extent of their awareness of human intelligence.

Answering Stephen Hawking's question(s) asked on page 224 of Illustrated Brief History of Time, is there a unified theory? Yes. Is the theory complete? Yes. Is it a collection of overlapping formulations? No. Is our understanding an infinite, asymptotic approach? No. Is the universe random, perhaps chaotic? No. From page 67, does the universe have a beginning? No. Whence energy? With reference to page 15, we disagree about what a good theory (or model) is, and we seem to disagree about the role of proof in Physics, a role which atrophied due to our understanding of gravity. A model does accurately describe observed phenomena. For example, Euclidean R^n does accurately describe some financiers' public communication patterns. However, prediction is another matter. From the perspective of living in state machines in ecosystems of, rather than plans, we do what we do while what we do works, then we switch. For example, financiers are at liberty to publicly communicate via knot theory rather than R^n per their preferences, provided communication continues. An alternative example is provided by my work with mice in 2002: Bridgit's garden was unpredictable. I provided Brigit with the same support as other mice, most of whom had names, and she's the only one who gardened. So the provisioner's role is to provide environments in which a wide range of responses may happen, and then see what does happen, which is different from top-down control techniques and different from typhoon (hurricane) prediction techniques. The recent sonic booms heard in Britain surprised the people due to the same principle of unpredictability of future events based on past trends having lurking variables. Traditional scientific methods make seeing lurking variables, such as untagged sharks, difficult. Prerequisite to developing the proper role of proof in Physics is Physicists adapting for including lurking variables in their work. IE the previous theories had difficulty obtaining the GUT of Physics due to an excessively restrictive logistical methods, due to rigorously applying Occam's razor too soon in development. This adaptation differs from what slows me down when reading Physics: your discipline has the habit of being lazy in notation describing value, difference and rate of change. Values, differences and rates of change are very different concepts, and have been inappropriately swapped with each other in some expositions of Physics.

How can I answer Stephen Hawking with certainty while I'm still discerning when Physicists intend to subtract, differentiate, or do neither? Physics and Mathematics have a relation; Physics and Logic have a relation; Physics and matter have the relation that Physicists observe what matter does; therefore Mathematical Logic exists; therefore the existence question posed by Jon Borwein on my graduate student record in 1996 is answered as (a) declarations by people with titles are insufficient to make skills disappear, and (b) the GUT of Physics is the unification of Mathematics, which suffices to bring Mathematics to market entirely and properly. In this context, asking me to state the GUT of Physics in terms of something material such as gamma rays almost makes sense, except, I haven't got a particle accelerator, so I'm doing this experiment on humanity modelling particles in information space modelling the universe, since I'm being held financial hostage thus found something to do with my time and resources - once again demonstrating the key to solving the n body problem.

From a Mathematical Logic perspective, subsequent to the work done in No Mathematics, the sensible next step is identifying axioms and relations which work and which hold true to the unification of mathematics, which is given in the positive definition of the empty set, rather than following the scientific approach of listing all types of anything we know exists and hoping to develop the theory from data while ignoring scientific progress of the past 300 years. To my surprise (the sonic boom of my life), there exists in our biosphere a set of Mathematical Logicians who do Constructive Mathematics and who have already reviewed several axiom system options and implications across the Mathematical disciplines. Whence my astonishment? During my recruitment to the Centre for Experimental and Constructive Mathematics (CECM) in 1993 and 1994, Peter Borwein let me know Constructive Mathematics is his and Jon Borwein's new mathematical revolution. During my graduate studies in 1994 and 1995, Jon Borwein spoke of introducing experimental practices from Physics into Mathematics, which readily made sense to me from my high school Physics course, and spoke of constructive mathematics being applicable in "industry", exemplified by selling Maple to companies and by solving a medical imaging problem. The aim of and basis of the CECM was presented to me as work entirely new to humanity. The courses of the graduate studies programme I was in were all irrelevant to the agendas of experimental mathematics and constructively bringing mathematics to a financially sensible position beyond academia, for bridging directly from mathematical academic programs into sensible jobs in industries for mathematical skills. Thus in January 1996, I returned Jon Borwein's intellectual property on multisectioning to him and requested the role of bringing mathematics to market as my thesis topic - his intellectual property which I agreed to develop and return with the results. Jon Borwein said yes. I negotiated with Bruce Clayman, VP Research and Dean of Graduate Studies, for a time stop on my graduate degree. I aimed to bring mathematics to market yet the work hadn't been previously done thus we couldn't pre-imagine the results apart from being clear, it places mathematicians in industries in positions to exert mathematical skills to advantage. Bruce Clayman and the Department Chair, Len Berggren, agreed in writing to give me two years, then extended the agreement in writing for two more years. In 1999 and in 2000, I got the results, and aimed for degree completion.

Throughout 1993 to 2000, local subject matter expert expressed awareness of Constructive Mathematical Logicians making progress since the 1950s was nil.

During my 1993 and 1994 recruitment to the CECM in Simon Fraser University, I expressed concern about our lack of similar mathematical interests and Peter Borwein assured me, between him and Jon Borwein, they read all of Mathematics thus I would have liberty to work on my selected topic. In 2012 looking back, Jon and Peter Borwein are arithmeticians who read all of arithmetics and who in the 1990s expressed disdain for the role of logic in mathematical disciplines. The field of experimental arithmetics has made progress since 1993 and Jon Borwein's work shows.

As my work develops and the bridge from academia to industries for mathematical minds is constructed upon my results, mathematical minds will be who drives entire industries, military organizations, governments, intelligencia, medias, transportation, information, waste and distribution systems, environmental clean-up, and education. Formalizing my work to date, rather than learning Physics, my best proper response to the questions Stephen Hawking asks in his work is to respond directly to Constructive Logicians who reach for description of reality without the gift to humanity from Physics: the discipline of observing real data and discarding ideas which are disproved by observation. (I disagree with the notion of energy being a free gift since so much work of mind went into putting the product together.) I think reading Physics could be a good second step. In summer 2011, Mikhail Prokhorov and I confirmed, at the core, Logic, Physics, Finance and the constructive (state machine) method in business and in natural existence are all identical. In autumn 2011, my seeing Baron Edouard de Rothschild's mind express displacement in media showed the same core insight, which agrees with my negation of the traditional square of opposition and correction of the definition of if expressed in summer 2003. What remains is an adjustment in Calculus.

Casimir

Physics is poetic. Wrote Lucretius, "Things cannot be born from nothing, cannot when begotten be brought back to nothing." Wrote Lavoisier, "Nothing is created; and nothing takes place beyond the changes and modifications in the combinations of these elements." [1]

"Alba Goya, Solo Goya," wrote the painter in his portrait of a duchess. Art's skill outlasts the frigid, cruel, self-made head of the plebians. Her feet too small to stand upon in paint removed in life after death. What love announces itself to deafness tortured by nightmares in standing sitters? [2]

Georges Seurat invented pointellism. Prior to Georges Seurat, pointellism had absence of existence, of which we were unaware, and then he invented it for us (him included) thus we developed awareness of pointellism, everywhere discontinuous paintings. [3] The concepts with which we construct our world disobey conservative energy ideas, such as the notion of situations having pre-formed kinetic energy to spend, imagined as existent potential. Some situations are that obedient. A phoenix keeps her turtle safe.


References:

[1] Physics, David Halliday and Robert Resnick, page 171. LCCC 66-11527.

[2] What Great Paintings Say V 2, Rose-Marie and Rainer Hagen, pp 530 - 535. ISBN 3-8228-4790-9.

[2] What Great Paintings Say V 2, Rose-Marie and Rainer Hagen, pp 685. ISBN 3-8228-4790-9.

Free Energy and Social Responsibility

The question of social responsibility in its socialist form was introduced to me by Friends in Bath, England, during 1987-8. I subsequently read snippets of the social thoughts of Bertrand Russell and Albert Einstein. Prior to my announcement of the definition of if in 2003, and prior to writing Biological in 2008, I revisited the events of my life of 1994 to 2003 in which I gave each person a chance to be who he or she is, and each did, whence my repeating assessment of Bertrand Russell's correctness. We mathematical logicians do the work we do, correctly, once, for humanity, and for ensuring the problems we remove have permanent removal. Said human to human with emotional empathy for ones who have suffered, there is incorrectness in assigning suffering to the ones who provide the most while assigning frivolity to the ones unable to discern they are in a constructed system. There is incorrectness in letting the most heinous crimes continue while having the means to stop them and feeling lazy, preferring to sip a latté instead. The problem which put itself, uninvited, into my life in expression in 1994 strikes upon the core question of what our species is. Are we mere flesh spreading ourselves across our planet, mostly oblivious to where money comes from, or do we have brains whose synapses produce thoughts discernible across distance and time via correlated actions. Why not do as we have been doing for over ten thousand years, collapse society, have a power grid failure across a geographic area, and start over from the survivors? That way, the problem would repeat. In the future, another mathematical logician could be interrogated by brutal physical force with survival seriously threatened, interrogated most dumbly as expressions of creatures of flesh and bone less civilized than our domestic animals. Amoeba and snails do logic, so when human semblances refuse logic in preference for brutality without gain, something is wrong in their synapses. Therefore, the natural selection method of crashing civilizations and breeding from the survivors was dysfunctional. Therefore the proper solution, for the prevention of physical and economic torture of mathematical logicians as an expression of capriciousness, and for gene pool corrections to replace the urge to capriciously bite the hands who feed us with sensible and civilized urges, this life I am living is lived once, by one human, by me, ending across our entire species the boring debate between the ones among us who prefer words having meaning inside sensible sentences corresponding to reality, and the ones among us who prefer random gibberish in grammatically correct form. Therefore, I sent this email 





from:  Jennifer Prokhorov prokhorov.jennifer@gmail.com
to:  accountingdepartment@hillaryclinton.com,  AskDOJ@usdoj.gov,  info@tutu.org,  Assembly of First Nations <mgoogoo@afn.ca>,  Bill and Melinda Gates Foundation <info@gatesfoundation.org>,  Canada <pm@pm.gc.ca>,  Eliana Araujo  <eliana.araujo@mc.gov.br>,  Federal Reserve <publications-bog@frb.gov>,  Government of Croatia Public Relations Office <press@vlada.hr>,  "Government of India, National Portal Secretariat" <indiaportal@goc.in>,  Government of Iraq <info@goi-s.com>,  Government of Portugal <gabinete.ministro@mf.gov.pt>,  Government of Venezuela  <dggcomunicacional@presidencia.gob.ve>,  Hungary <ugyfelszolgalat@nfm.gov.hu>,  International Court of Justice <webmaster@icj-cij.org>,  International Financial Club <tp@mfk-bank.ru>,  "Library, George Bush" <Media.Bush@nara.gov>,  "MOFA Taiwan, One China" <eyes@mofa.gov.tw>,  Navy SEALs <motivators@navsoc.socom.mil>,  Nelson Mandela Centre of Memory <nmf@nelsonmandela.org>,  NSA <nsaarc@nsaarc.net>,  NSA <smallbusiness@nsa.gov>,  President of Finland <presidentti@tpk.fi>,  "Prime Minister's Office, Israel" <pm_eng@pmo.gov.il>,  Rothschild Patrimoine <ContactRothschildPatrimoine.dl@fr.rothschild.com>,  Saudi Arabia <americas@sagia.gov.sa>,  The British Royal Family <press@royalcollection.org.uk>
cc:  feedback@hindustantimes.com,  golpira@tehrantimes.com,   Jason.Burke@guardian.co.uk,  lheard@gulfnews.com,  Chronicle Herald <letters@herald.ca>,  National Geographic <Newsdesk@ngs.org>,  "Seattle Times, business" <business@seattletimes.com>,  Taipei Times <letters@taipeitimes.com>,  Tulsa World <web@tulsaworld.com>,  Washington Post <letters@washpost.com>,  Bank of Canada <info@bankofcanada.ca>,  Bank of Israel <statistics@boi.org.il>,  Bank of Russia <webmaster@www.cbr.ru>,  Banque centrale du Brésil <imprensa@bcb.gov.br>,  Banque Privée Edmond de Rothschild <contact@bper.ch>,  Europe Central Bank  <info@ecb.europa.eu>,  International Monetary Fund <ethics@imf.org>,  People's Bank of China <webbox@pbc.gov.cn>,  Royal Bank of Canada <mfrp@rbc.com>,  Saudi Arabian Monetary Agency <info@sama.gov.sa>,  World Bank <delnaggar@worldbank.org>
date:  Sun, Mar 11, 2012 at 9:09 AM
subject:  sensibility
mailed-by:  gmail.com


The problem is we think we exist.
http://www.smh.com.au/entertainment/tv-and-radio/brave-new-world-with-stephen-hawking-sunday-march-11-20120308-1ulpx.html


What started in the 1980s as my desire to teach people how to think turned swiftly into twenty years of two repeated debates,

• whether sentences and physical reality correlating is advantageous, and
• whether physical reality measurably exists.

In the back of my thoughts while many national leaders have shared their lives and thoughts with me, I struggled to explain coincidental earthquakes, http://www.ctv.ca/CTVNews/TopStories/20120311/japan-earthquake-tsunami-one-year-anniversary-commemorations-120311/


Coincidental solar flares are easy to explain,
http://fr.wikipedia.org/wiki/Initiative_de_d%C3%A9fense_strat%C3%A9gique


Incorrect evaluations of the situation include:

• let's pretend we do not have free energy sitting on the table for negotiated entry into the market,
• let's pretend anything in our lives matters more than this negotiation,
• let's pretend Jon Borwein invented a code with the word not in 1996,
• let's pretend Stephen Hawking could be naive,
• let's pretend that first we handle debts, settle with central bankers, negotiate a price, get everyone to agree on something, force logical people into submission to False Cause and somehow get something positive out of holding everyone hostage, and
• let's be Canadian by insisting we are not supposed to talk about that and we are not supposed to say people who disconnect from physical reality are being stupid.

If I were Stephen Hawking, I would have free energy rigged physically and socially for my death or disappearance by whatever means causing the end of our existence; without threat, the analysis is correct, the position is correct, the debate between sense and random word generation is done. Yes, let's do take a moment to feel sad this biosphere produced via evolution a species who made the mistake of giving free food to a subset incapable of feeding itself, incapable of being cognitively up to a level of living which may be supported within the constraints of our biosphere. We made this mistake at the expense of the biosphere, at the expense of life here. I think this situation could be rigged for worse, however, Stephen Hawking's thoughts have precision. My estimate is merely the existence of Earth is being negotiated.
If we do agree to the continued existence of Earth, life and our continued species (IE via the physical and measurable result of Stephen Hawking's recovery to full health via DARPA's bionanotechnology, free energy, and anyone who hoped to get in the way sitting down and shutting up), then subsequently we may discuss some of the questions national leaders and central bankers and Stephen Hawking have brought to my attention, all related to population management, education, distribution, economic design and related matters.


Basic Logic

• False: there is an afterlife.
• True: While you are alive, you are not yet dead.
• True: The type of dead we are talking about is different from the types of assessed deaths declared by medical doctors in which the body seems to recover for a while and then dies again.
• True: The type of dead we are talking about is dust without reference to Chinese poetry, with reference to lasers, solar power and renewable energy.
• True: Right now, I think life on Earth is dead with probability one,

http://www.bbc.co.uk/news/uk-17168984


The unique solution through this situation is to set aside all other priorities, and have DARPA's bionanotechnology healing Stephen Hawking now, without debate; without senseless random sentences; without misunderstandings; without avoidance; without hope of getting free energy after Stephen Hawking dies since I think we die via putting that option to Stephen Hawking, rightly so; without financial leverage; without distraction from those who still wonder whether physics is, whether neurons are, and whether words correlate with physical reality; without further detection of what sick freaks of nature borrowed this moment in evolution to emote "porn causes permission, yuk yuk."; and without waiting for me to repeat the 2003 intro to basic logic in which I spoke the declaration that I was singing.


Anyone who needs any assurance about his or her personal interests which he or she fantasizes could have higher priority than continuing life on Earth, and we do not know whether life exists elsewhere yet could hope so since this tedium is hopeless, can be reassured that yes, we do have a viable economic system outlined and ready for initiation, albeit perhaps without anyone whose priorities are clearly confused. Anyone still feeling unsure about the real role of physics in our lives, hardly a question for political debate, may look at the pictures in Stephen Hawking's Illustrated Brief History of Time. I have the message from many who hope I may negotiate population and life terms with Stephen Hawking and with some bankers; however, that hope arises from absence of understanding the situation. After I see Stephen Hawking recover his health by whatever news reports work, and after I see this physical situation physically including me and Stephen Hawking in the honest financial market which has successfully circumvented the WTO and has the capacity to break away, then I could willingly discuss filtration with him. Until this world groks and adjusts to acceptance and inclusion of physics and logic, the point in any other conversation is moot.

No Mathematics

The Correct Definition of if in Binary

0 implies 0 and 1 implies 1. Finite state machines replace plans and linear time suits some situations.


The biosphere exists. Our neurons exists. Snails whose behaviour Joseph LeDoux observes and describes in Synaptic Self exist, and behave in agreement with the correct definition of if. Our neurons are active in ways which agree with the correct definition of if. Our biosphere agrees with the correct definition of if. Therefore I also call this Nature's if. It seemed money was invented from nothing due to ambiguities in language inviting cognitive errors. Removing ambiguities and cognitive errors lets us earn real money in this real world, and keep our real lives in this biosphere.



Corrections in Basic Logic


With the incorrect definition of if, A ior A is in agreement with Not-A => A. However, with the correct definition of if, A => A is in agreement with A and A.

Also due to the correct definition of if, the following Axioms of Replacement are invalid due to they are mistaken consequences of the incorrect definition of if.

Transposition is invalid. Instead the correct statement is if A => B then B => A and iff is redundant.

Material Implication is invalid. Instead A => B is similar to A AND B with the only difference being a factor of time.

Material Equivalence is invalid.

The Tautology A = A AND A is valid.

The Tautology A = A IOR A is valid.

The Tautology A = A XOR A is invalid since the statement A XOR A is false when A is true, and is false when A is false.

DeMorgan’s Rule is valid for the inclusive-or (IOR) and invalid for the exclusive-or (XOR).

Commutativity, Associativity and Distribution are valid.

Cause is normally thought of (in Basic Logic) in terms of sufficient conditions and necessary conditions.

However, visible in human trade relations and in Nature, the actual conditions for cause are threefold: sufficiency, necessity and something extra. The something extra prerequisite to cause is quantified as a function of and in proportion to the consequent of cause, and this may be new work not previously considered.

I prefer replacing Syllogisms with Predicate Logic since the latter is able to state all of the former without confusion.

I prefer teaching everyone to reject fallacies as their initial cognitive development training. Yes, advanced logicians do see the practical value of the fallacies, however letting beginners see and reject fallacies could strengthen their cognitive self-defense.

Reference: Introduction to Basic Logic by Patrick J Hurley Return to Persuasive Logic.

Probability of B Given A


With conditional probability we show the unviability of the old definition of if.

The conditional probability of event A occurring if condition B is met is defined by the formula:
P(A|B) = P( A intersect B) / P(B).

Paraphrased from source:Encyclopaedia of Mathematics, Volume 7, Kluwer Academic Publishers, Managing Editor M. Hazewinkel

I've reversed A and B from their traditional roles to deliberately test the old 'only if'. Thus suppose we have A => B with the old definition of if:

=> 0 1
0 1 1
1 0 1

The conditional probability that is worth testing is in the reverse direction: A only if B, and be clear to use the same truth table without flipping it over for convenient escape of the question. So A is on the left-hand side and B is on the top of the truth table, same as for A => B, and what we are testing is B => A with conditional probability and both definitions of if.

Observe that P(B) = 1 in any situation in the old definition of if. Therefore, the statement that B is given is meaningless. Therefore, in the old definition of if, P(A|B) = P(A) in all four cases: A and B have empty intersection; all three of (A intersect B), A\B and B\A have non-empty intersection; or either is a proper subset of the other.

Observe the new definition of if has in some cases the only if direction being identical to the if direction, preserving the formula from the Soviet Encyclopaedia of Mathematics.

Infinity


I prefer clearly disambiguating finite from countably infinite from more than countably infinite.

Prove we have more than countably infinite real numbers without reference to Cantor's Diagonal Argument:

Let N be the set of all natural numbers, 0, 1, 2... and let R be the set of all real numbers. Both N and R are infinitely large. The size of N is called countable infinity and the size of R was traditionally called uncountable infinity, now known as more than countably infinite. The set Q of rational numbers is countably infinite.

Proof of there being something more in R than in Q:

Let Q be the set of rational numbers. Write Q as the union over each n in N of {m/n | m is each integer}. Proof that the set of all real numbers, R, is beyond countably infinite may be done by showing that the proper subset of irrational numbers is uncountable. For this proof, hold N as N without 0. Suppose that the set of irrational numbers, R\Q, is countable. Then there is a one-to-one correspondence between Q x Q and Q x (R\Q). List Q x Q with the m values ordered 0, +1, -1, +2, -2, ... and with the n values 1, 2, 3, ... such that our matrix has 0 in the upper left corner the top row is 0, +1, -1, +2, -2,..., and the second row is 0, +1/2, -1/2, +2/2, -2/2,..., and so on.) According to our supposition, all real numbers are now in our matrix. Remember Pi. We know Pi is transcendental and therefore is without a rational representation. Therefore R\Q is beyond enumerable and therefore R is beyond countably infinite.

I confess the proof contains a contradiction.

Multiply-Interpretable Present


Now exists, and is always happening. Multiply-interpretable present situations have different optional futures.

General Comment about malevolence, how to see and remove that: 

Due to the correct definition of if, some mathematicians shall re-write some mathematics proofs which were proofs by contradiction, and I think it makes sense to rewrite also proofs which relied on Modus Tollens and other now inoperable Axioms of Replacement. I accept arguments which clearly prove existence of some things while holding open constructive options per positive motivation.

Friedman's operating assumption is the universe appears the same as interpreted via any direction from any point inside the universe. However, insofar as human information systems may be analogous to space manifolds, clearly the reported local area data stream differs rather than as intrinsic fluctuation, instead as an expression of differences of the quality of measuring and reporting devices. Thus, does anyone have any right to say anything is positive and constructive, or that could be relative? Instead, I prefer retaining reference to the biosphere and to ancient financial and trade systems which work together with ternary logic and the biosphere since I think this might suffice to remove malevolence from anyone pretending to do good while merely mugging the ones who construct. To what extent does the analogy of human minds in societal data generalize to particles in space? Parasites?

Could division be fast subtraction?

The expression multiplication is fast addition suits the ones for whom an interesting insight is expressed. I admit the expression had me wonder during the late 1980s why we do not say division is fast subtraction. Rather than wondering how the calculations work, I was wondering why the inverses were not lining up.

Alrighty then, let us review how the calculations work. Matrices and Cartesian coordinates are nice introductions to what multiplication could be. Essentially, handed a finite set of compartmentalizable stuff, we partition stuff, and when we partition stuff as though with a cookie-cutter, then we make available to ourselves relatively fast computation of the volume of stuff as the quantity of cookies cut times the volume of stuff in each cookie. Inside such contexts, some dual concepts are introduced together, including variables + constants, addition + subtraction, multiplication + division, and = seeming to be the verb to be, and seeming to be a full declaration of sameness in all attributes. The 1965 edition of Mathematical Logic by R. L. Goodstein distinguishes in the introduction The Function of Mathematical Logic the conceptual differences between knowing how to compute the partitioning of r by q for some real number r and some fraction q versus describing what has the methodology work, versus winning a dispute about that. Familiar to anyone is the locution, to divide by a fraction, multiply by the reciprocal; however, who has the capacity to say what has this work without algebraic reliance on the calculi and without the presumption of duality being complete includes mathematical logicians. The answer to my question, how division differs from fast subtraction, is division is a partition whereas subtraction is displacement, a translation from observed-event-space to private-event-space. Pardon me for inventing names for some of these concepts while working through this.

The distinction in a nutshell is, when we see a mature elephant who very convincingly seems to weigh 10 pounds on Earth, rather than concluding we have found our first light elephant, instead we conclude the scale was miscalibrated. Ditto with time; if/when anyone wakes inside 1990s time management systems near the 49th parallel on Earth with all geographically local time declaration devices claiming 10:00 am yet the night sky is observed, then the time declaration devices are miscalibrated, regardless of how much they agree with each other. In grade school when teaching subtraction, we traditionally teach take away 3 from 5 rather than teaching translate 5 to have 3 displaced to the other side of 0 from which we agree to look away since the other side of 0 is private. 

I have been considering how I prefer to have us get the most out of my review of logic, calculi and related concepts without umbrage. Sure, we have the question of binary and ternary calculations however, most striking to me is the defense (against what conversation?) displayed in expositions of mathematical logic, leading me to wonder how straightforwardly these same concepts might be eloquently expressed were we to have mathematical logicians conversing in safety, without experiencing our existence nullified as though at war in every step we take. Who put such semblance of war in our conversation space I do not know, yet, Goodstein visibly expresses the entire self-defense structure, as does Bertrand Russell in The Principles of Mathematics. Let us examine the self-defense of Goodstein's presentation.

I confess some familiarity with Sociology and Dramaturgy introduced to me by my adoptive father Michael Overington whose books I omitted reading, and with whom I debated reality partitions and reality management for 16 years. These fond memories include my fascination with observing a Phi Beta Kappa mind who claims and produces convincing evidence of absence of mathematical and basic logic skill yet does so with algebraic calculations of basic logic and fallacy formulations perfectly intact. Aristotle gave us a bit of a conundrum when he rejected India's concept, the number zero, thereby handing our species our traditionally most coveted and most lucrative construction: nested binary debates. Our persuasions seem convincingly of two types: logic else fallacies. Do logical and fallacious people belong together else apart? Are fallacious arguments right else wrong? Let us bridge the conceptual gap between logic and fallacies as persuasive devices, for world peace. See the economic driver in the algorithm: scan the world, select two examples of whatever, hire lawyers to debate via a series of binary locutions which example of whatever is correct; repeat. Dick Cheney in In My Time reveals the related economic driver of top-down control applied to the masses as the cooking process of punching down risen dough in yeast breads, necessitating massive job creation for supervision of arbitrary control devices, pp 58 to 62. Ostensibly Aristotle wanted to prove God's existence thus by coincidence gave Galileo a challenge. Thank goodness Aristotle's accidental refutation of the physical existence of zero coincidentally drives money creation in western civilization. Examples of valid interpretations of physical instantiations of zero include our synapses which are nor axons nor dendrites; the spaces on roads into which we drive our vehicles else collide; empty food dishes; empty bank accounts; space available in tax-free savings accounts; absence of inclusion in society; the gap between bones in our joints; and for anyone still counting fingers, the spaces in between our fingers really does have measurable, observable existence. Matryoshka dolls solve Zeno's Paradox; the point being, the limit infimum of [0, 1] does exist and is zero regardless of any so-called mathematician's opinion otherwise, and by reverse direction of theoretical proof and physical observation, the limit supremum of [0, 1] does exist and is one, again regardless of any so-called mathematician's opinion otherwise. Incorrect calibration during physical observation combined with Appeal to Authority yields the marriage of Appeal to Ignorance with Appeal to Force, which exemplifies that sort of persuasive technique Aristotle establishes differs from logic together with correctly calibrated physical observation. A positive outcome of this exercise includes the convincing observable semblance of the force of gravity per se in Economics, admittedly a conceptual re-categorization which may have surprised Newton and which has me curious about the theory of gravity.

History informs us humanity apparently felt like spontaneously generating logical definitions of numbers, for communicating with unambiguous clarity what a number is thus also nailing what a number ain't, in the 1890s, exemplified by Gottlob Frege's introduction in 1894. (Goodstein, p2) Specifically, prior to Frege's publication, there were social conversations providing sensible context for Frege's publication, without which Frege's publication could have occurred as irrelevant, nonsensical, disobedient, nonconformist, nit-picky. Prior to our human minds exerting control are indicators of some absence of and some necessity for control, whether the inspiration to exert control comes from fear, desire to drive money, desire to have power, and alternative quests. Frege sacrificed some opportunities expressing his preference and his priorities among his options; he opted to exert his control of the definition of numbers. Something relevant happened prior, for example, something which could have questioned the reality construct of numbers, such as the seemingly benign and semi-competent random locution of numbers and real life being allegedly unrelated, and perhaps the social context in which Frege's publication occurs as acceptable has threats upon the reality construct of numbers posed as jokes. Without knowing Frege's societal context, from being human and knowing humans, clearly we normally have absence of spontaneously feeling like logically defining numbers unambiguously without any motivation, merely on a whim. IE minds in societal contexts are analogous to particles in waves, cf deBroglie. For example, my 2010 solution to Wikipedia's public claim of the openness of Waring's problem has been silently ignored since 2010, showing my absence of inclusion in peer review publication has been driven by societal context.  If Frege had been interpreted as suitable exclusively for slave labour, a forced idea generator for the benefit of the work on paper of lesser minds, empty-headed, off-kilter and irrelevant, then his logical definition of numbers could have remained unpublished and thus unread by Bertrand Russell in 1903. Via defining numbers, Frege and Russell disambiguate equality from similarity and establish the importance of one-to-one conceptual correspondence as a valid and real constructive method. How tight, how strong this conceptual, logical insistence on clear and honest correlation between physical observation and words for human communication expresses the existence of the unnamed unwelcomed uninviteds infiltrators in human conversation, existence proven by shifted demand curves of mathematical logicians responding, demonstrating the will and importance of this human defense.

References
Mathematical Logic, R. L. Goodstein, Leicester University Press.
Conversations en famille from 1986 to 2003.

Several classes in the honours mathematics stream, Dalhousie University, Halifax, Canada.
Zero, Charles Seife.