** Even if everything possible given E, existing as eternal possibilities in E (given T1), one with E, manifested as mx or not, are eternal possibilities, "ideas", in E, which of course for example makes the idea that the Earth is round as eternal as that the Earth is flat, that the former is a rational idea and the latter an irrational idea according to "the empiri" (in broader sense, for example seen by an astronaut who has travelled around the Earth, one who has lived in one and the same place all life may have harder to embrace that idea, but may then have to believe the astronaut if they would meet) hasn't the slightest significance concerning the eternity stamp, given that these thoughts exists, which of course constitute proof of that they exists, which, given E(/T1, of course) directly also defines their eternal status; That something, x, never will be thought or manifested as mx other than defining a thought, not exclude the existence of x as possibility, can well be added.
About N again
Given N then nonx then is a unique specific x(=y), not =0, and this then always, which then commonly not is valid:
Commonly nonx just only defines that x not is valid, or rather that x not is in focus, but it is nonx that is in focus, even if x can, or rather is there in the background, through x in nonx, and nonx not defines anything specific, before nonx maybe is defined to define something specific, directly (as then for example N does), or indirectly, through a context:
Is a context defined, nonx more specifically maybe at least can begin to give itself, for example, restated, if E and xÎE are defined, in which nonx closest defines Ex, but nonx commonly naturally also defines all specific nonx=yÎE, y which also can be clusters(/sets) of y, as well as maybe xÏE, as well as 0 (or Nothing) if x≠0.
Or take nontrue (x), in accordance with N, then false (x) is meant by nontrue, which commonly naturally not necessarily is valid, closest nontrue can mean decidable/undecidable (x), probable(/improbable) (x, with some probability), possible/impossible (x), etcetera, except of course everything else that not is true as a concept/word, such as mould (mould≠true as concepts, and consequently nontrue=mould is valid (interpreted) in that way), Sweden or E; Nottrue=false is an (irrational) definition, which further lowers Classical logic, especially in its truth table form.
Basic mathematics given E
Houses are built from the ground up, and the same shall (should) be valid for logic, trying to find axioms that fit some (that are seen as) results(/theorems) is commonly irrational. So if (rational) axioms do not lead to any (desired) result, then commonly the only alternative to not reject those "results", is to assume them ad hoc, as axioms simply. The foundation is thus the most important, theory is at least as loose as its foundation. "Commonly" because there may be exceptions, when it may be rational to fit the axioms to "results", an example of this is the "empirical" assumption that mx have attraction force, given the consequences of not assuming that (which at first sight is the most rational).
The geometrical basis
All continuous geometric shapes (where the pencil so to speak not is lifted) are defined by the curve (if p’=p, then either (a) p is defined, or that the curve ends in p, in the p in which the curve began to be "drawn"):
d(p,p’); p)=p]; p]=p) (given the continuity/homogeneity of E; Curves principally exists between all p's that not are identical, not Nothing, it is so to speak possible to draw a line between p and p’; p’≠p):
The smallest curve/distance:
dp=min[d(p,p’)]
The smallest area (triangle):
y=min[d(dp,p)]; pÏdp.
The smallest volume (tetrahedron):
v=min[d(y,p)]; pÏy.
The mathematical E:
E=∞’v.
The Fundamental logical E:
E=∞’mv; mv>,<,=v.
Given the reasoning in the Lp section, ∞’v=∞’mv, which may seem unintuitive if mv≠v, but so it (rationally) only is. Which has consequences for how it rationally shall be calculated with infinities, and rationally also t2 shall be taken into account:
∞’p=dp:
np=p; n<∞’.
Yes, the rationally best is simpliciter to try to avoid all calculus of infinity.
The arithmetical basis
Arithmetic commonly requires the existence of superclones, with which the simplest and best is to count with p's (points, nonextended positions), which especially are good in the sense that they can be seen to be superclones existing in the same p (to be superclones of one (first/original) p), as well as can be seen to be different p's (existing in different p), this with which fundamentally the following directly can be stated (given Up):
I) n±m=n±m, where n is n number of p, defining a natural number (according to conventional (Arabic) definition: 1,2,3,..), and m is m number of p, also defining a natural number:
p+p+p=3 for example.
And of course:
p=1.
I defines per se an Lpprinciple, thus that m can be added or subtracted from a (p)identity, in the meaning that the identity persists, of course in the meaning that there are equally many p's on both sides of =:
[n±m=n±m]=[n=n] (symmetrically valid).
This usually called cancellation law, but is of course not an (unprovable) "law" given the foregoing, but selfevident, an evident fact (given Up), given that it is about p's.
This "law" is usually defined: [n±o=m±o]=[n=m], but given that it is about p's, then m=n (m is not an implicative identity (n=m) that defines anything other than n, but m is then n (m=n), this definitely if it is a matter of superclonical p's, but it is also valid if it is about different p's, there are equally many p's on both sides of =, even if it is the matter of different p's then, not superclonically identical p's), symmetry is valid:
n=m; m=n.
And of course also "reflexivity" (given Up):
n=n.
And trivially also "transitivity" is valid (because symmetry is valid, and thus m,o=n):
n=m=o (or as it usually is written: n=m, m=o ® n=o).
"Commutativity" can further be stated to be valid for addition, it is only to line up the p's and it will be seen that it is valid:
n+m=m+n.
Commutativity for subtraction is not valid if []≠[+]:
nm≠mn; m≠n.
But if []=[+], it is valid, which also only is to line up the p's to see that it is valid; If m>n in nm (n excluding m), then the number of p>n defines "negative" p's, which can be defined by the absolute number sign:
nm=mn.
For nn is best defined:
nn=0, n+n=0, where 0 is idempotent void:
0n=0, 0/n=0, n/0=0, n±0=n (for multiplication and division see further below).
If n=n, then:
nn=0, which given that n+n=0 gives (given Up):
n=+n:
II) n=n.
It is also evident that "associativity" is valid, that order of addition doesn't matter, it also just is to line up the p's to see it:
(m+n)+o=m+(n+o).
The fact that p=1 makes it easy to define multiplication, by collecting m groups of n number of 1's or then p's:
n1+n2+n3+.. +nm=nm (n m number of times).
And the order of course doesn't matter, it is again just to line up the p's to see it, which defines "commutativity":
nm=mn (m n number of times is identically (to) n m number of times).
The "cancellation law" directly follows from this:
[on=on]=[n=n].*
And "associativity":
(nm)o=n(mo).
Which immediately can be realized by lining up p's, but also can be proven in this way:
Take nm o times:
nm1+nm2+..+nmo, which given commutativity is the same as taking o nm times:
(nm)o=o(nm):
(nm)o=on(m); (nm)=n(m):
(nm)o=n(om); on=no, o(m)=(om).
n(m)=(nm)=nm brings the thoughts to the "distributive law":
n(m+o)=nm+no.
Which quickly is realized to be valid, that q=m+o n number of times is the same as m n number of times and o n number of times, just because m+o=q; q n number of times is the same as each subsetÎq n number of times.
