Division:

 

mn=o:

 

n=o/m; n/n=1; n≠0 (o can be divided into n pieces of m-parts (o divided by m is n)).**

 

Certainly the distributive law given, but because it is evident (as evident as for example p+p=2), the following proof, given division, of it is a proof (given division, "law of cancellation" and distributivity):

 

n(m+o)=nm+no:

 

n(m+o)/n=(nm+no)/n:

 

m+o=m+o.

 

Which of course is valid (given Up), with which the distributive law is valid (alternatively it can be defined with ≠, and consequently a contradiction arises in accordance with Kp (different is different in accordance with Kp, so consequently there is a contradiction if the same is defined to be different, if m+o≠m+o as then in this case, which of course also contradicts Ip, as well as Up), which defines that = is valid (there is no alternative in this case, if ≠ not is valid, then simpliciter = is valid, yes, the whole argument can of course be comple-tely false, neither ≠ nor = is valid, but that is of course per assumption excluded in this case. This kind of proof, given a contradiction/-adversariality with a clear alternative, is usually finely called reductio ad absurdum; If there is no clear alternative, then of course nothing can be inferred from the stated contradiction, other than of course that especially that which leads to the contradiction is false, given that the stated p-superpositionality is assumed to be absurd/false)).

 

Sets defined on the basis of this arithmetical p-basis are of course defined by p's:

 

Set={p}.

 

Which especially of course defines "intersection" (S) to be those p which different sets jointly possesses:

 

S={p}x,y,z,..; x,y,z,.. are sets (of p's).

 

xy of course defines subsets; xx given Ip, that x=x, in which case x of course not is a subset (in itself, >,<x), but just is itself ((=)x).

 

Set addition ("union"):

 

x+y-S.

 

And set subtraction:

 

x-y:

 

x-y=0; y=x.

 

Which simply defines the p-difference (the difference in number of elements=p) between x and y, in exactly the same way as in "ordina-ry" subtraction (n-m).***

 

The preceding versus N-logic (Classical logic)

 

There are similarities between the collection of formulas of N-logic and the mathematics defined above, especially regarding II and Dl, but II then follows from the (p-)assumption that n-n=0 and that -n+n=0, which not define (is defined by) N (n=-n=n, but given the "can-cellation law" it defines that n=n or that -n=-n (in accordance with Up)), which then defines Dl (Dl follows from), so it is consequently a matter of two completely different forms of logic. That the N-logic is the basis for the mathematics ("logicism") is nonsense. But as said, there are similarities between the formulas in these different forms of logic, the formulas of the mathematics (in accordance with the ab-ove) are though based on evident intuition, the formulas of the N-logic primarily on N (and Tp, which the mathematics also uses/presup-poses when it assumes the existence of superclones). N that not are present in the mathematics at all, that a mathematical expression (x) implicatively identically always defines another mathematical expression y that is true if x is false (and vice versa), that is nothing but nonsense.

 

__________

* If symmetry not is valid, it may be asked whether this "law of cancellation" (Lp-principle) also then is valid? Thus if [on=om]=[n=m]; m≠n, yes, at least for numbers can be stated, in percentage meaning, but it is of course commonly difficult to justify n=m; m≠n, but in some specific context it certainly rationally may be justified, thus that n implicatively identically (implies) is m(≠n).

 

** o, given natural numbers, not always goes even out in n pieces of m-parts, by which certain quotients of course not are defined, but for that n=o/m must be assumed to define a number, for all natural numbers, which (of course) defines rational numbers.^

 

*** Conventionally additional "axioms" are defined:

 

The Axiom of extensionality shall rationally (already indicated) be replaced by Up; Up defines identical x (with exactly the same proper-ties) to be a unique (one, and only one) x, while the Axiom of extensionality (superclonically) allows identical x to be different, more ex-plicitly the Axiom of extensionality defines that x=y; [{x}x]=[{x}y], which rationally shall be interpreted in accordance with Up, but by many (irrationally) are interpreted as that x and y can be different, is that desired, thus that different x can be identical, then the Axiom of extensionality should be adjusted to:

 

e) x=y; [{x}x]=[{x}y]; {x}≤x (if {x}=x, then this expression defines Up, thus that x=y=[unique x]).^^

 

The Power set, the set of all subsets in a set, can of course be defined, but why define something so abstract, such a possible myriad of sets and of (superclonical) p's (given the p-basis)?

 

That there exists infinite sets is also defined, yes, but Fundamental logically only one: E (E then seen as a set, which E a little sought can be seen as), which (already mentioned) implies that extreme caution should prevail in definition, assumption of infinities, because it then (of course) is a matter of pure abstraction (except in the case of E, given the Fundamental logic).

 

"Well-ordering" can also be mentioned, which given the p-basis in accordance with its name means to arrange p's, which of course can be done (in the mind), the p's in a set be structured, placed in different ways, so that they so to speak create different (p-)images, but what good does that? Another "functional" alternative is to define a p in a set (functionally) define n number of p's (a set (of p's)) in another set, well, it is useful in some context, ok, sure.

 

Regularity can finally be mentioned, which given the p-basis defines that in every set of p's (x) there is a p that not belongs to the set of p's (x), an absurd (irrational) p-superpositionality (px and px), so that axiom shall (rationally) of course simpliciter be deleted; It is a kind of geometrically continuity assumption (p]=p)) for sets, but geometry and set theory should strictly be kept apart.

^ Infinity problems immediately arises when numbers are defined, to take a little look at that, a smallest infinite natural number then is defined by ∞, which intuitively defines the existence of natural numbers >∞, by which it can be defined that all natural numbers are ∞ many (p's):

 

All n=∞.

 

How many dp's defines that? Well, if it defines a finite extension, then additional natural numbers can be defined, simply by adding espe-cially 1, or then 1 p, so ∞ is infinitely long, by which it (the number line) can be defined as consisting of ∞ many dp's:

 

∞=∞dp.

 

So all natural numbers are consequently ∞2 many, and this then only the natural numbers (1,2,3,..), the rational (positive) numbers are ∞4 many in accordance with the definition of division, and then of course also real numbers can be defined, which are problematical, be-cause they principally defines the existence of distances shorter than dp: Take any distance, whichever, for example (principally) between 0.1 and 0.2, then there always are shorter distances, given the real numbers, for example between 0.001 and 0.002, or given that distance for example between 0.000001 and 0.000002, etcetera. dp=∞p (then given that numbers are p's) is so to speak nowhere to be defined, yes is dp defined somewhere (between two numbers, on the line of numbers), then the number line (∞) ends there (the number line ends after dp, at least under distances <dp), which then contradicts ∞. That it exists smallest distances is intuitive, with which dp=∞p is a rational definition, although the existence of ∞ (a smallest infinite number) of course is pure abstraction, with which the dp-definition is difficult to abandon, but more rational seems to be to abandon the definition of numbers as points, when (if) the real numbers are defined, with which the only rational alternative is that they are 0*, thus E, an idea in E in short.

 

This that dp (smallest/shortest distances) exists (is assumed to exist) makes the mathematical definition of smallest objects problematic, especially for dp itself, because d can evident that in the thought for example be divided into two: dp/2, which of course not is possible if dp exists. But especially regarding mx, given the existence of dp, mx must so to speak from edge to edge through a centre be at least dp long in all directions. Yes, space itself must locally be at least dp long in all directions, which it of course is as ∞* "long" (T2), but a bit tricky is it still, for the mind, which equally wants to think that space distances shorter than dp can exist, which it of course can think, but given that the thought has defined dp, then they simpliciter not exists, which then as said is tricky for the mind to accept. This even if the mind rationally realizes, precisely as regarding the definition of the mx-"jumps", that a position different from another must be some distance from the first position, otherwise it of course is the matter of the same position, a distance, which then (of course) defines at least dp. This existence of dp, then excluding the existence of distances shorter than dp, is simpliciter adversarial to the thought (a paradox), even if the thought (rationally) realizes that so must be the case.^^^

 

^^ Yes, the e-condition can also be defined [{x}x]≠[{x}y], although it then is difficult to claim any form of identity.

 

^^^ That that E can start E-contractions (given the possibility of x/mx) and that thrusted mx (in accordance with the "empiri") "jumps" fairly in thrusting mx "jump"-directions (by that thrusting mx "passes directional information" to thrusted mx) and that mx (according to the "empiri") have attraction force are also paradoxes, albeit weaker than this with dp, the thought finds it easier to agree to these three "facts" than the "fact" that there are no distances shorter than dp. Perhaps just because the former are in accordance with the "empiri", the latter not, at least to some extent, in some (inscrutably) way.

 

 

Properties and platonism

 

Nothing=[propertieslessness, propertiesless x]:

 

Nothing=[x with x (properties)](; Kp).

 

And x(≠Nothing) is thus implicatively identically, at one and the same time, its x (properties), x is x (x=x), for if not, then of course it not is a matter of x having(/being) x, but perhaps of another x having (owning) other x.

 

Given the E theory, mx-manifested x is properties-wise in narrowest sense {mx} (one, "its" cluster of mx, x={mx}), {mx} (including mx properties) is x properties (in narrowest sense). In broader sense, x can properties-wise be defined as having properties in relation to its environment, x can have the property of (being capable of) affecting xE(; x≠x): ((x x))(; x=x, where x defines the property x that defines how x affects x), or if x can be affected by xE, then x have the property of being able to be affected by x (in some speci-fic meaning, which defines the property xx, then defining how x≠x affects x): ((x x)).

 

x which has nothing to do with x, not defines any properties xx, but are there such x:

 

Given that all xE (T2), then all x have the property of belonging to the same World (E), so in that sense all x have something in com-mon, something to do with each other, but the E-theory must of course be known in order for that (E-theoretical) property to have any meaning, the following can commonly be stated:

 

For x conscious x have properties-wise (greater or lesser, but not 0) significance for x (depending on how x is interpreted/perceived) just because x is conscious, exists in x sense-sphere.

 

For x unconscious x have properties-wise 0 significance for x if x only is a thought, if x on the other hand exists beyond x thought, sense-sphere, and positively or negatively affects x, then x (of course) properties-wise have significance (≠0) for x, if x so are uncon-scious or conscious of x; If (for x unconscious) x not affects x, x have 0 significance for x; For x unconscious x which (of course) perhaps can become conscious for x (primarily depending on x thinking capacity, the higher the more x can become conscious for x, and vice versa).

 

Is this a platonistical(/empirical) truth, eternal truth? Can be asked as introduction to the following:

 

Are there "thoughts"/phenomena that cannot be thought, especially regarding any empiri? Yes, the E-theory defines phenomena incom-prehensible for the thought, for example the (mx-)attraction, that thrusting mx pass directional information over to thrusted mx and the "jump"-property (that mx not are (flat not exists) in any positions between two positions). Because the thought simpliciter cannot see how that can be possible, it can only reason its way to it, for example then through the reasoning that leads to the "jump"-conclusion. From this, from a theory, specifically here then the E-theory, to conclude that there exists (beyond theory) platonistical phenomena, or here then empirical phenomena (which the thought is incapable of thinking) is though to take it too far in accordance with Up/FT, which defines that the thought, theory cannot reach outside itself, principles which of course can be considered to be false, seriously through some kind of rigorous argumentation for it, thus for that the thought can reach beyond/outside itself. The whole can be boiled down to arguments for or against Up/FT, which per se shows that it is only a matter of thoughts, assumptions.

 

With which it is a matter of having the best, most rational, argumentation, and regarding that it can especially be referred back to the sec-tion: A little more about the meaning of Fundamental logic, in Addition II.