Algebraic General Topology. Vol 1: Paperback / E-book || Axiomatic Theory of Formulas: Paperback / E-book

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Remark 1. It is a very rough partial draft. It is meant to express a rough research idea, not to
be correct, readable, or complete. First read the book:
http://www.mathematics21.o rg/algebraic-gene ral-topology.html upon which this formalistic is
based (especially about the definition of generalized limit).
See also http://planetmath.org/MetasingularNumbers
The idea is simple (for these who know funcoids theory). But to nd exact formulations abo ut thi s
is notoriously difficult. Below are attempts to formulate things about the theory of singularit ie s.
New theory
Definition 2. Singularity level is a transitive, T
2
-separable endofuncoid.
Let ν be a singularity level. Let be a filter.
Define SLA(ν) as:
Ob SLA(ν) = {ν f | f is a monovalued funcoid with domain }
X [SLA(ν)]
Y x X K GR xL Y : L K
[FIXME: It is probably not a funcoid .]
Remark 3. GR x is used despite of it is a funcoid not reloid.
Proposition 4. SLA(ν) is an endofuncoid.
Proof. ¬( [SLA(ν)]
Y ) and ¬(X [SLA (ν)]
) are obvious.
I J [SLA(ν)]
Y I [SLA(ν)]
Y J [SLA(ν)]
Y is obvious.
X [SLA(ν)]
I J x X K GR x L I J: L K x X K GR x :
(L I: L K L J: L K)??
??
Alternative definition:
[FIXME: It is probably not a funco id.]
Definition 5. X [SLA(ν)]
Y z Ob µK GR z x X , y Y : x, y K
Proposition 6. SLA(ν) is a funcoid.
Proof. X [SLA(ν)]
Y z Ob µ , x X, y Y K (GR z)
X ×Y
: x, y K
x,y
??
I J [SLA(ν)]
Y z Ob µK GR z x I J , y Y : x, y K z Ob µ K GR z:
( y Y : y K (x I: x K x J : x K)) z Ob µK GR z: (( y Y : y K x I:
x K) ( y Y : y K x J: x K)) ??
Proposition 7. SLA(ν) is T
2
-separable.
Proof. ??
Proposition 8. SLA(ν) is transitive.
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Proof. ??
Galufuncoids
Let A and B are Rel-morphisms. I will denote like (
A
) = GR A and (
B
) = GR B.
Definition 9. Galufuncoi ds between A and B is a quadruple (A; B; α; β) such that
x Ob A, y Ob B: (αx
B
y x
A
βy).
Definition 10. x [f] y x
Src f
βy.
Obvious 11. x [f ] y x
Src f
βy αx
Dst f
y.
Remark 12. Galuf uncoids are a gener alization of both (pointfree) funcoids and Galois connec -
tions.
Definition 13. The reverse galufuncoid is defined by the formula:
(A; B; α; β)
1
= (B; A; β; α).
Proposition 14. Composition of (composable) galufuncoids is a galufuncoid.
Proof. (α
2
α
1
)x y α
2
α
1
x y α
1
x β
2
y x β
1
β
2
y x (β
1
β
2
)y.
Obvious 15. Galufuncoids form a category (sim ilarly to the category of pointfree funcoids).
Definition 16. On the set of galufuncoids is defined a preo rder by the formula: f g [f ][g].
Galufuncoidal product
Functional galufuncoid
Definition 17. Functional galufuncoid ν/∆ of ν thr ough filter is the endo-galufuncoid defined
by the formulas:
Ob(ν/∆) = FCD(Base(∆); Ob ν)
hν/∆if = ν f and h(ν/∆)
1
if = ν
1
f
f
Ob(ν/∆)
g g
1
f id
FCD
[TODO: Restric t to the special case f = ν F to make i t T
2
.]
[TODO: X [SLA(f )] Y is defined as existence of x X such that for every entourage of x ther e is
y Y which is a subfilter of this entourage.]
Obvious 18.
Ob(ν/∆)
is a symmetric relatio n.
Proposition 19. This is really a galufuncoid and f [ν/∆] g g
1
ν f id
.
2
Proof. We need to prove
hν/∆if
Ob(ν/∆)
g g
1
ν f id
f
Ob(ν/∆)
h(ν/∆)
1
ig .
Really,
hν/∆if
Ob(ν/∆)
g ν f
Ob(ν/∆)
g g
1
ν f id
f
Ob(ν/∆)
h(ν/∆)
1
ig f
Ob(ν/∆)
ν
1
g g
1
ν f id
Remark 20. A way to come to the above formula
x atoms ∆: fx [ν] gx x atoms : x [g
1
ν f ] x g
1
ν f id
.
Hierarchy of singularities
Consider two endo-galufuncoids µ and ν. Values on Ob µ will be have like arguments of functions,
of Ob ν like values of functions.
I call SLA(Ob µ) singularity level above Ob µ the set of sets of funcoids ν f |
hµi
{x}
(or alternatively
of lim its xlim f |
hµi
{x}
) where f is a monovalued principal funcoid in FCD(Ob µ; Ob ν).
[TODO:
Maybe exclude the zero funcoid?]
Consider a galufuncoid ω defined by the formulas:
hωif = ν f and hω
1
if = f ν and f [ω] g x Ob ν: g
1
f id
hµi
{x}
FCD
.
We need to prove
hωix
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
and ??
The first is equivalent to (
ν
) f
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
FCD
.
Really, (
ν
) f
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
Lemma 21. Let ν ν ν and ν
1
ν ν. If x and y are ultrafi lters, then x [ν] y hν ix = hν iy.
Proof.
[TODO: Prove fo r the more general case of galufuncoids. It’s problematic.]
x [ν] y y hν ix hν iy hν ν ix hν iy hν ix. So taking symmetry into account we have
x [ν] y hν ix = hν iy.
Let now hν ix = hν iy, Then y
hν
1
ν ix; y hν
1
ν ix; y hν ix; y
hν ix; x [ν] y.
Theorem 22. f [ν/hµi
{x}] g ν f |
hµi
{x}
=ν g |
hµi
{x}
for f , g SLA(Ob µ).
[TODO:
Generalize it for any instead of hµi
{x}?]
Proof. It’s enough to prove hf ix [ν] hgix hν ihf ix = hν ihgix for every ultrafilter x.
hf ix [ν] hg ix hgix
Ob µ
hν ihf ix ?? hν ihgix
Ob µ
hν ihf i
??
Theorem 23. SLA(Ob µ) is T
2
-separable .
The reloid ν
on the set SLA(Ob ν) could be defined by one of the two formulas below:
3
For every point x get
σ
f
(x) =
u
[
f | dom u = hµi
{x}
.
q
f
=
F
xOb µ
σ
f
(x).
f [ν
] g x Ob µ:
G
q
f
[ν/hµi
{x}]
G
q
g
.
or
f [ν
] g ν q
f
= ν q
g
.
[TODO: How to define galufuncoids corresponding to the above formulas (if at all possible)?
x D: f [ν(x)] g x D: g
hν(x)if ?? g
F
xD
ν(x).
The ?? does not generally hold: our lattices are co-brouwerian not brouwerian!
]
The above formula holds if g is a discrete reloids. So repla ce every funcoid f SLA(Ob ν) with
(RLD)
in
f. Then continue for arbitrary reloids.
Another try: hν
ix = ν
FS
x
y
hν
ix y
ν
FS
x ν
y (
FS
x)
1
ν
1
(
FS
x) y
1
[FIXME: x and y are of
different types.]
x
hν
1
iy x
ν
1
FS
y
The rest
One more ot he r definition:
f [ν
′′
] g
G[
g
1
ν
G[
f
0
Yahoo! (i j) [ν
′′
] g i [ν
′′
] g j [ν
′′
] g etc.
Proof. (i j) [ν
′′
] g (
FS
g)
1
ν (
FS
(i j))
0 (
FS
g)
1
ν ((
FS
i) (
FS
j))
0
(
F S
g)
1
ν (
F S
i) (
F S
g)
1
ν (
F S
j)
0 (
F S
g)
1
ν (
F S
i)
0
(
FS
g)
1
ν (
FS
j)
0 i [ν
′′
] g j [ν
′′
] g
Proposition 24. ν
′′
is a galufuncoid.
Proof. ??
An attempt of an alternate definition:
f [ν
] g xlim f ν xlim g
0 [FIXME: Does this make sense?] [TODO: Differe ntiate generalized
limit as a set of funcoids or its variation as a funcoid-value function.]
Proposition 25. xlim f SLA(Ob ν) if f FCD(Ob µ; O b ν) \
0
FCD(Ob µ;Ob ν)
.
Proof. ??
Proposition 26. τ (x) SLA(O b ν).
Proof. ??
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Proposition 27. τ (x) [ν
] τ (y) x [ν] y.
Proof. ??
Metasingular numbers
Let y SLA(Ob µ). I will denote r(y) such x Ob ν that τ (x) = y, if such x exis ts.
I will call base singular numbers the set BSN = Ob ν SLA(Ob ν) SLA(SLA(Ob ν))
.
[TODO: Set that this union is di sjoint.]
I will call meta-singular numbers the set MSN = {y SLA(Ob ν) | x BSN: y = τ (x)}.
Definition 28. I call reduced BSN its corresponding MSN (that is r applied to our BSN a natural
number of times while possible).
Definition 29. I call reduced limit the reduced generalized limit.
Functions with meta-sin gular numbers as arguments
Let f is an n-ary (n is an arbitrary possibly infinite index set) function o n O b ν. Then define
function f
on SLA(Ob ν) as:
f
(b) =
(
g
Y
(A)
b | g f
)
.
We c an’t use cross-composition product instead of above sub-atomic product because cross -compo-
sition product is not a funcoid (just a pointfree fun coid). We can replace sub-atomic product with
displaced product, but as about my opinion displaced product seems more weird an inconvenient.
The above ind uc e s a trivial definition of functions on MSN but only for functions of finite arity
(because having a finite set of MSN we can raise them to the same (maximum) level).
On differential equations
Replacing limit in the definition of derivative with the a bove defined reduced limit, the base set
Ob µ with MSN and operations f on the set Ob µ with corresponding operations on MSN, we get
a new interpretation of a differential equation (DE) (ordinary or partial).
Let call such (enhance d) differential e quations m eta-singular equations (as opposed to non-singular
equations that is customary differential equations).
There arise the following questions:
Definition 30. I call a solution of a DE a trivia l restriction if it is a restr ic tio n (to t he set of non-
singular points) of exactly one enhanced DE.
We need to find when there are solutions which are not trivial restrictions.
Then we can split such non-trivial solutions into follow ing classes:
“added solutions” are solutions whose restriction t o non-singularity points is not a non-
singular solution;
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“alternate solutions” is when an non-singular solution is a restriction of more than one meta-
singular solution;
“disappearing solutions” when a non-singular solution is not a restriction of a meta-singular
solution.
Special case of general relativity
I am not a expert in general relativity (I am not even a professional mathematician).
But it looks like that the equations of general relativity can be converted (as described above) into
meta-singular equations. For the special case of general relativity equa tions, the above classes are:
“added solutions” would possibly characterize a “world above” described not with real num-
bers as our world but with singularities. This may or may not be of physical interest.
“alternate solutions” would characterize black (or white) holes with additional information
hidden inside. This a dditional information may probably solve the well k nown pa radox of
information disappearing when it fa lls into a black hole.
“disappearing solutions” would mean that the laws of nature are possibly more restrictive
than c onsidered in more traditional physics. Could it resolve time-machine related p ara-
doxes?
I again repeat that I am not an expert in general relativity. I seek collaboration w ith general
relativity experts to solve the problems I’ve formulated.
I think (except of the case of the negative result that is there are no non-trivial solutions) this
research is destined to receive Nobe l Prize and/or Fundamental Physics Prize. I want my half.
Note that the g roup G (see the definition of gene ralized limit in my book) for general relativity
can be defined in two different ways: as the group of homeomo rpisms of the curved space or as the
group of only uniformly continuous (in both directions) bijections. This gives us two new theories
of general relativity.
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