Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on \emph{ad hoc} axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic \emph{sustainability} of this design has been irremediably compromised by the underlying it peremptory, Reductionist mindset of the XIXth century's ideology of science.
Deep Dive into Halfway Up To the Mathematical Infinity: On the Ontological and Epistemic Sustainability of Georg Cantors Transfinite Design.
Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously – albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering – and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research – all insights and adjustments due to Kurt G"odel’s revolutionary insights and discoveries notwithstanding – has compliantly centered its efforts on \emph{ad hoc} axiomatizations of Cantor’s intuitive transfinite design. We demonstrate here that the ontological and epistemic \emph{sustainability} of this design h
arXiv:0812.3207v3 [math.GM] 9 Feb 2009
Halfw
a
y
Up
T
o
the
Mathemati al
In nit
y
I:
On
the
On
tologi al
&
Epistemi
Sustainabilit
y
of
Georg
Can
tor's
T
rans nite
Design
Edw
ard
G.
Belaga
No
v
em
b
er
3,
2018
Abstra t
Georg
Can
tor
w
as
the
gen
uine
dis o
v
erer
of
the
Mathemati al
In-
nit
y
,
and
whatev
er
he
laimed,
suggested,
or
ev
en
surmised
should
b
e
tak
en
seriously
alb
eit
not
ne essary
at
its
fa e
v
alue.
Be ause
along-
side
his
exquisite
in
b
eaut
y
ordinal
onstru tion
and
his
fundamen
tal
p
o
w
erset
des ription
of
the
on
tin
uum,
Can
tor
has
also
left
to
us
his
obsessiv
e
presumption
that
the
univ
erse
of
sets
should
b
e
sub
je ted
to
la
ws
similar
to
those
go
v
erning
the
set
of
natural
n
um
b
ers,
in lud-
ing
the
univ
ersal
prin iples
of
ardinal
omparabilit
y
and
w
ell-ordering
and
implying
an
ordinal
re- reation
of
the
on
tin
uum.
During
the
last
h
undred
y
ears,
the
mainstream
set-theoreti al
resear
h
all
in-
sigh
ts
and
adjustmen
ts
due
to
Kurt
Gö
del's
rev
olutionary
insigh
ts
and
dis o
v
eries
not
withstanding
has
omplian
tly
en
tered
its
eorts
on
ad
ho
axiomatizations
of
Can
tor’s
in
tuitiv
e
trans nite
design.
W
e
demonstrate
here
that
the
on
tologi al
and
epistemi
sustainability
of
th!
is
design
has
b
een
irremediably
ompromised
b
y
the
underlying
p
eremptory
,
Redu tionist
mindset
of
the
XIXth
en
tury’s
ideology
of
s ien e.
Our
analysis
and
prompted
b
y
it
syn
thesis
lead
to:
(i)
the
extension
of
the
w
ell-kno
wn
t
w
o-terms
foundational
opp
osition
CN : {existence by axiomatic consistency ⇒notational existence},
to
its
no
v
el,
four-term
axiomati
viabilit
y
riterion RSCN
:
{ontological relevancy ⇒onto −epistemic sustainability ⇔CN},
redu ing ZF
and
its
exten
tsions
to
the
status
of
in
tera tiv
e
program-
ming
languages
manipulating
ad
ho
on
triv
ed,
pure
notational
in nite
totalities,
(ii)
the
new
on
tologi al
insigh
ts
in
to
the
nature
of
the
on-
tin
uum
inspired
b
y
the
quan
tum-me
hani al
en
tanglemen
t
argumen
t,
and
(iii)
the
in
terpretation
of
Can
tor’s
lass
of
all
oun
table
ordinals
ω1
as
an
authen
ti ,
univ
ersal,
ev
er
emerging
and
nev
er
ompleted
ordi-
nal
s ale
of
the
p
o
w
er
and
sophisti ation
of
iterativ
e
logi al
argumen
ts.
1
I
was
b
eside
the
Master
r
aftsman,
delighting
him
day
after
day,
ever
at
play
in
his
pr
esen
e,
at
play
everywher
e
on
his
e
arth,
delighte
d
to
b
e
with
the
hildr
en
of
men.
Pro
v
erbs
8:30-31
Con
ten
ts
1
In
tro
du tion
3
1.1
Can
tor’s
Mission
and
His
P
eremptory
Amalgams
.
.
.
.
.
.
.
.
3
1.2
Ob
je tiv
es
and
Results
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
1.3
Bibliographi al
Note
and
A
kno
wledgmen
ts
.
.
.
.
.
.
.
.
.
.
7
2
Epistemi
Prequel
to
the
A
dv
en
t
of
Ordinals:
Plain
Con
tin-
uum
Standing
Alone
and
High
8
2.1
Prelude
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
2.2
An
tiquit
y:
an
In nit
y
Flash
ba
k
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
2.3
F
rom
F
ormal
to
Platoni
Existen e,
and
Bey
ond.
.
.
.
.
.
.
.
10
2.4
Putting
the
Epistemi
and
Pro
edural
Pri e
T
ags
on
the
Can-
torian
P
o
w
erset
Abstra tion
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
12
2.5
Con
tin
uum
Hyp
othesis:
F
rom
P
oin
t
Cen
tered
to
Subset
Cen-
tered
Set
Theory
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
15
2.6
Can
tor’s
Unrestri ted
Uses
of
the
P
o
w
erset
Constru tion
.
.
.
17
2.7
Can
tor’s
Univ
ersal
Cardinal
Comparabilit
y
Prin iple
and
T
w
o
On
tologi ally
Distin t
Sour es
of
Sets
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
3
Ordinals:
Their
A
w
e-Inspiring
Beaut
y
,
Their
Ne essary
Uses,
Their
P
eremptory
Abuses
18
3.1
The
Beaut
y
and
E ien y
of
Ordinal
Constru tion
Redux
on
the
Outside
the
Can
torian
Set
Theory
.
.
.
.
.
.
.
.
.
.
.
.
.
18
3.2
Can
tor’s
T
rans nite
Design
T
ak
es
Shap
e
at
a
Pri e:
a
Clear-
ut
Case
of
the
Abuse
of
the
Ordinal
Devi e
.
.
.
.
.
.
.
.
.
.
22
4
P
osterior
Axiomatizations
of
Can
tor’s
T
rans nite
Blueprin
t:
F
oundational
Challenge
24
4.1
Ptolemai -lik
e
Deadlo
k
of ZF −based
and
Ev
er
Extending
Axiomati s
of
the
Can
torian
Set
Theory
.
.
.
.
.
.
.
.
.
.
.
.
24
4.2
On
tologi al
and
Epistemi
Construals:
Consisten y
,
Relev
an y
,
and
T
ruth
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
25
2
4.3
Constativ
e vs
P
erformativ
e
Axiomati
P
aradigms:
P
erformativ
e-
Iterativ
e
Wishful
Axiomati
Thinking
.
.
.
.
.
.
.
.
.
.
.
.
.
.
26
4.4
Can
torian
Op
erational
and
Generativ
e
and
Hilb
ertian
Univ
er-
sal
and
Argumen
tativ
e
Redu tionism
.
.
.
.
.
.
.
.
.
.
.
.
.
.
28
4.5
Hilb
ertian,
P
ost-Hilb
ertian,
Gö
delian
Programs
and
Their
Gö
delian
&
P
ost-Gö
delian
Stum
bling-Blo
ks
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
29
5
Nostalgi
In
terlude.
F
rom
the
In nit
y
ab
o
v
e
to
the
In nit
y
b
elo
w:
Con
tin
uum
&
Ordinals
32
6
Lo
al
Causation
of
Man’s
Mathemati s
V
ersus
Non-Lo
alit
y
of
Classi al
Mathemati s
34
6.1
The
Con
tin
uum,
Suslin’s
Problem,
Lo
al
Causation,
Quan
tum
Non-lo
alit
y
,
and
Ch
ur
h-T
uring
Thesis
.
.
.
.
.
.
.
.
.
.
.
.
.
34
6.2
Ordinal
Constr
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