Vasil Penchev. Continuity and Continuum in Nonstandard Universum

Continuity and Continuum in
Nonstandard Universum

Vasil Penchev
Institute of Philosophical Research
Bulgarian Academy of Science
E-mail: vasildinev@gmail.com
Publications blog:
http://www.esnips.com/web/vasilpenchevsnews

Content
1. Motivation
s:
2. Infinity and the axiom of choice
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity
between two infinitely close
standard points
6. A new axiom: of chance
7. Two kinds interpretation of

This file is only Part 1 of the
entire presentation and
includes:

1. Motivation

: 1. Motivation :
My problem was:
Given: Two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
Where a is the power of
countable set
The problem:
Do the two sequences  and 

At last, my resolution proved
out:
That the two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
coincide or not, is a new axiom (or
two different versions of the
choice axiom): the axiom of

For example, let us be given two
Hilbert spaces:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
An analogical problem is:
Are those two Hilbert spaces the
same or not?
 can be got by Minkowski space
 after Legendre-like

So that, if:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
are the same, then Hilbert space
 is equivalent of the set of all
the continuous world lines in
spacetime 
(see also Penrose’s twistors)
That is the real problem, from

About that real problem, from
which I had started, my
conclusion was:
There are two different versions
about the transition between the
micro-object Hilbert space  and
the apparatus spacetime  in
dependence on accepting or
rejecting of ‚the chance

After that, I noticed that the
problem is very easily to be
interpreted by transition within
nonstandard universum between
two nonstandard neighborhoods
(ultrafilters) of two infinitely
near standard points or between
the standard subset and the
properly nonstandard subset of


And as a result, I decided that
only the
highly respected scientists from
the honorable and reverend
department ‚Logic‛ are that
appropriate public worthy and
deserving of being delivered
a report on that most intriguing

After that, the very God was so
benevolent so that He allowed
me to recognize marvelous
mathematical papers of a great
Frenchman, Alain
Connes, recently who has
preferred in favor of sunny
California to settle, and who, a
long time ago, had introduced

Content
1. Motivation
s:
2. INFINITY and the AXIOM OF CHOICE
5. Nonstandard continuity between
two infinitely close standard
points

Infinity and the Axiom of
Choice

A few preliminary notes about
how the knowledge of infinity is
possible: The short answer is: as
that of God: in belief and by
analogy.The way of mathematics
to be achieved a little
knowledge of infinity transits
three stages: 1. From finite
perception to Axioms 2. Negation

Choice

The way of mathematics to
infinity:
1. From our finite experience and
perception to Axioms: The most
famous example is the
axiomatization of geometry
accomplished by Euclid in his
‚Elements‛

Choice

infinity:
2. Negation of some axioms: the
most frequently cited instance is
the fifth Euclid postulate and its
replacing in Lobachevski
geometry by one of its negations.
Mathematics only starts from

Choice

infinity:
3. Mathematics beyond finiteness:
We can postulate some
properties of infinite sets by
analogy of finite ones (e.g.
‘number of elements’ and
‘power’) However such transfer

Choice

A few inferences about the math
full-scale offensive amongst the
infinity:
1. Analogy: well-chosen
appropriate properties of finite
mathematical struc-tures are
transferred into infinite ones
2. Belief: the transferred

Choice

The most difficult problems of
the math offensive among
infinity:
1. Which transfers are allowed
by in-finity without producing
paradoxes?
2. Which properties are suitable

Choice

The Axiom of Choice (a
formulation):
If given a whatever set A
consisting of sets, we always can
choose an element from each
set, thereby constituting a new
set B (obviously of the same po-
wer as A). So its sense is: we

Choice

Some other formulations or
corollaries:
1. Any set can be well ordered
(any its subset has a least
element)
2. Zorn’s lema
3. Ultrafilter lema
4. Banach-Tarski paradox

Choice

Zorn’s lemma is equivalent to the
axiom of choice. Call a set A a
chain if for any two members B
and C, either B is a sub-set of C or C
is a subset of B. Now con-sider a
set D with the properties that for
every chain E that is a subset of
D, the union of E is a member of D.
The lem-ma states that D contains
a member that is maximal, i.e.

Choice

Ultrafilter lemma: A filter on
a set X is a collection of
nonempty subsets of X that is
closed under finite
intersection and under
superset. An ultrafilter is a
maximal filter. The
ultrafilter lemma states that
every filter on a set X is a

Choice

Banach–Tarski paradox which
says in effect that it is possible
to ‘carve up’ the 3-dimensional
solid unit ball into finitely many
pieces and, using only rotation
and translation, reassemble the
pieces into two balls each with
the same volume as the original.
The proof, like all proofs

Choice

First stated in 1924, the Banach-
Tarski paradox states that it is
possible to dissect a ball into
six pieces which can be
reassembled by rigid motions to
form two balls of the same size
as the original. The number of
pieces was subsequently reduced
to five by Robinson

Choice

Five pieces are minimal, although
four pieces are sufficient as long
as the single point at the center
is neglected. A generalization of
this theorem is that any two
bodies in that do not extend to
infinity and each containing a
ball of arbitrary size can be
dissected into each other (i.e.,

Choice

Banach-Tarski paradox is very
important for quantum
mechanics and information since
any qubit is isomorphic to a 3D
sphere. That’s why the paradox
requires for arbitrary qubits
(even entire Hilbert space) to be
able to be built by a single qubit
from its parts by translations

Choice

So that the Banach-Tarski
paradox implies the phenomenon
of entanglement in quantum
information as two qubits (or
two spheres) from one can be
considered as thoroughly
entangled. Two partly entangled
qubits could be reckoned as
sharing some subset of an initial

Choice

But the Banach-Tarski paradox is
a weaker statement than the
axiom of choice. It is valid only
about  3D sets. But I haven’t meet
any other additional condition.
Let us accept that the Banach-
Tarski paradox is equivalent to
the axiom of choice for  3D sets.
But entanglement as well 3D

Choice

But entanglement (= Banach-
Tarski paradox) as well 3D
space are physical facts, and
then consequently, they are
empirical confirmations in favor
of the axiom of choice. This
proves that the Banach-Tarski
paradox is just the most decisive
confirmation, and not at all, a

Choice

Besides, the axiom of choice
occurs in the proofs of: the Hahn-
Banach the-orem in functional
analysis, the theo-rem that
every vector space has a ba-
sis, Tychonoff's theorem in
topology stating that every
product of compact spaces is
compact, and the theorems in
abstract algebra that every ring

Choice

The Continuum Hypothesis:
The generalized continuum
hypothesis (GCH) is not only
independent of ZF, but also
independent of ZF plus the axiom
of choice (ZFC). However, ZF plus
GCH implies AC, making GCH a
strictly stronger claim than
AC, even though they are both

Choice

The Continuum Hypothesis:
The generalized continuum
hypothesis (GCH) is: 2Na = Na+1 . Since
it can be formulated without
AC, entanglement as an argument
in favor of AC is not expanded to
GCH. We may assume the negation
of GHC about cardinalities which
are not ‚alefs‛ together with AC

Choice

Negation of Continuum Hypothesis:
The negation of GHC about
cardinali-ties which are not
‚alefs‛ together with AC about
cardinalities which are alefs:
1. There are sets which can not be
well ordered. A physical
interpretation of theirs is as
physical objects out of (beyond)

Choice

But the physical sense of 1. and 2.:
1. The non-well-orderable sets
consist of well-ordered subsets
(at least, their elements as sets)
which are together in space-time.
2. Any well-ordered set (because
of Banach-Tarski paradox) can be
as a set of entangled objects in

Choice

So that the physical sense of 1.
and 2. is ultimately: The mapping
between the set of space-time
points and the set of physical
entities is a ‚many-many‛
correspondence: It can be
equivalently replaced by usual
mappings but however of a

Choice

Since the physical quantities have
interpreted by Hilbert
operators in quantum mechanics
and information
(correspondingly, by Hermitian
and non-Hermitian ones), then
that fact is an empirical

Choice

But as well known, ZF+GHC
implies AC. Since we have already
proved both NGHC and AC, the
only possibility remains also the
negation of ZF (NZF), namely the
negation the axiom of foundation
(AF): There is a special kind of
sets, which will call ‘insepa-

Choice

An important example of
inseparable set: When
postulating that if a set A is
given, then a set B always
exists, such one that A is the set
of all the subsets of B. An
instance: let A be a countable
set, then B is an inseparable
set, which we can call

Choice

The axiom of foundation: ‚Every
nonempty set is disjoint from
one of its elements.‚ It can also
be stated as "A set contains no
infinitely descending
(membership) sequence," or "A
set contains a (membership)
minimal element," i.e., there is an

Choice

The axiom of foundation
Mendelson (1958) proved that the
equivalence of these two
statements necessarily relies on
the axiom of choice. The dual
expression is called
º-induction, and is equivalent to

Choice

The axiom of foundation and its
negation: Since we have accepted
both the axiom of choice and the
negation of the axiom of
foundation, then we are to
confirm the negation of º-
induction, namely ‚There are sets
containing infinitely descending
(membership) sequence OR

Choice

The axiom of foundation and its
negation: So that we have three
kinds of inseparable set:
1.‚containing infinitely descending
(membership) sequence‛ 2.
‚without a (membership) minimal
element‚ 3. Both 1. and 2.
The alleged ‚axiom of chance‛

Choice


The alleged ‚axiom of chance‛
concerning only 1. claims that
there are as inseparable sets
‚containing infinitely descending
(membership) sequence‛ as such
ones ‚containing infinitely
ascending (membership)
sequence‛ and different from

Choice

The Law of the excluded middle:
The assumption of the axiom of
choice is also sufficient to derive
the law of the excluded middle
in some constructive systems
(where the law is not assumed).

Choice


A few (maybe redundant)
commentaries:
We always can:
1. Choose an element among the
elements of a set of an
arbitrary power
2. Choose a set among the

Choice


A (maybe rather useful)
commentary:
We always can:
3a. Repeat the choice choosing the
same element according to 1.
3b. Repeat the choice choosing the
same set according to 2.

Choice

The sense of the Axiom of Choice:
1. Choice among infinite elements
2. Choice among infinite sets
3. Repetition of the already
made choice among infinite
elements
4. Repetition of the already

Choice

The sense of the Axiom of Choice:
If all the 1-4 are fulfilled:
- choice is the same as among
finite as among infinite
elements or sets;
- the notion of information being
based on choice is the same as

Choice

At last, the award for your kind
patience: The linkages between
my motivation and the choice
axiom:
When accepting its negation, we
ought to recognize a special kind
of choice and of information in
relation of infinite entities:

Choice

So that the axiom of choice
should be divided into two parts:
The first part concerning
quantum choice claims that the
choice between infinite elements
or sets is always possible. The
second part concerning quantum
information claims that the
made already choice between


Choice
My exposition is devoted to the
nega-tion only of the ‚second
part‛ of the choice axiom. But
not more than a couple of words
about the sense for the first
part to be replaced or canceled:
When doing that, we accept a new
kind of entities: whole without
parts in prin-ciple, or in other
words, such kind of


Choice
Negating the choice axiom second
part is the suggested ‚axiom of
chance‛ properly speaking. Its
sense is: quantum information
exists, and it is different than
‚classical‛ one. The former
differs from the latter in five
basic properties as following:
copying, destroying, non-self-
interacting, energetic


Choice

Classical
Quantum
1. Copying, Yes No
2. Destroying, Yes No
3. Non-self-interacting, Yes No
4. Energetic medium, Yes No
5. Being in space-time Yes No

Choice

How does the ‚1. Copying‛
(Yes/No) descend from
(No/Yes)?

It is obviously: ‚Copying‛ means
that a set of choices is
repeated, and
consequently, it has been able to

Choice

If the case is: ‚1. Copying – No‛
from
- Yes,

then that case is the non-cloning
theorem in quantum information:
No qubit can be copied

Choice

How does the ‚2. Destroying‛
(Yes/No) descend from

(No/Yes)?

‚Destroying‛ is similar to
copying:
As if negative copying

Choice

How does the ‚3. Non-self-
interacting‛ (Yes/No) descend
from
(No/Yes)?

Self-interacting means
non-repeating by itself

Choice

How does the ‚4. Energetic
medium‛ (Yes/No) descend
from
(No/Yes)?

Energetic medium means for
repeating to be turned into
substance, or in other words, to

Choice

How does the ‚5. Being in space-
time‛ (Yes/No) descend from
(No/Yes)?

‘Being of a set in space-time’
means that the set is well-
ordered which fol-lows from
the axiom of choice. ‘No axiom of
chance’ means that the well-

Content
1. Motivation
s:
3. NONSTANDARD UNIVERSUM
5. Nonstandard continuity between
two infinitely close standard
points

Nonstandard universum

Abraham Robinson
(October 6, 1918
Leibnitz – April 11, 1974)


Abraham Robinson
(October 6, 1918
His Book (1966) – April 11, 1974)

‚It is shown in this
book that Leibniz
ideas can be fully
vindicated and that
they lead to a
novel and fruitful
approach to
classical Analysis
His Book (1966) and many other
branches of

‚…G.W.Leibniz argued that
the theory of infinitesimals
implies the introduction of
ideal numbers which might
be infinitely small or
infinitely large compared
with the real numbers but
which were to possess the

The original approach of A.
Robinson:
1. Construction of a nonstandard
model of R (the real continuum):
Nonstan-dard model (Skolem
1934): Let A be the set of all the
true statements about R, then:  =
A(c>0, c>0`, c>0``…): Any finite
subset of  holds for R. After

2. The finiteness principle: If
any fi-nite subset of a (infinite)
set  posses-ses a model, then
the set  possesses a model too.
The model of  is not isomorphic
to R & A and it is a nonstandard
universum over R & A. Its sense is
as follow: there is a
nonstandard neighborhood x

The properties of nonstandard
neighborhood x about any
standard point x of R: 1) The
‚length‛ of x in R or of any its
measurable subset is 0. 2) Any x
in R is isomorphic to (R & A)
itself. Our main problem is about
continuity and continuum of two
neighborhoods x and y
between two neighbor well

Indeed, the word of G.W.Leibniz
‚that the theory of infinitesimals
implies the introduction of ideal
numbers which might be
infinitely small or infinitely
large compared with the real
numbers but which were to
possess the same properties as
the latter‛ (Robinson, p. 2) are

Another possible approach was
developed by was developed in
the mid-1970s by the
mathematician Edward Nelson.
Nelson introduced an entirely
axiomatic formulation of non-
standard analysis that he called
Internal Set Theory or IST. IST is
an extension of Zermelo-

In IST alongside the basic binary
membership relation , it
introduces a new unary predicate
standard which can be applied to
elements of the mathematical
universe together with three
axioms for reasoning with this
new predicate (again IST): the
axioms of

Idealization:
For every classical relation R, and
for arbit-rary values for all other
free variables, we have that if for
each standard, finite set F, there
exists a g such that R(g, f ) holds for
all f in F, then there is a particular G
such that for any standard f we have
R (G, f ), and conversely, if there
exists G such that for any standard
f, we have R(G, f ), then for each

Standardisation
If A is a standard set and P any
property, classical or
otherwise, then there is a
unique, standard subset B of A
whose standard elements are
precisely the standard elements
of A satisfying P (but the
behaviour of B's nonstandard

Transfer
If all the parameters
A, B, C, ..., W
of a classical formula F have
standard values then
F( x, A, B,..., W )
holds for all x's as soon as
it holds for all standard xs.

The sense of the unary predicate
standard:
If any formula holds for any finite
standard
set of standard elements, it holds
for all the universum. So that
standard elements are only those
which establish, set the
standards, with which all the
elements must be in conformity: In

So that the suggested by Nelson IST is
a constructivist version of
nonstandard analysis. If ZFC is
consistent, then ZFC + IST is consistent.
In fact, a stronger statement can be
made: ZFC + IST is a conservative
extension of ZFC: any classical
formula (correct or incorrect!) that
can be proven within internal set
theory can be proven in the

The basic idea of both the version
of nonstandard analysis (as
Roninson’s as Nelson’s) is
repetition of all the real
continuum R at, or better, within
any its point as nonstandard
neighborhoods about any of
them. The consistency of that
repetition is achieved by the

That collapse and repetition of
all infinity into any its point is
accomp-lished by the notion of
ultrafilter in nonstandard
analysis. Ultrafilter is way to be
transferred and thereby
repeated the topological
properties of all the real
continuum into any its point, and
after that, all the properties of


What is ‘ultrafilter’?
Let S be a nonempty set, then an
ultrafilter on S is a nonempty
collection F of subsets of S having
the following properties:
1.   F.
2. If A, B  F, then A, B  F .
3. If A,B  F and ABS, then A,B  F
4. For any subset A of S, either A  F


Ultrafilter lemma: A filter on
a set X is a collection of
nonempty subsets of X that is
closed under finite intersection
and under superset. An
ultrafilter is a maximal filter.
The ultrafilter lemma states
that every filter on a set X is a
subset of some ultrafilter on X

A philosophical reflection: Let us
remember the Banach-Tarski
paradox: entire Hilbert space can be
delivered only by repetition ad
infinitum of a single qubit (since it is
isomorphic to 3D sphere)as well the
paradox follows from the axiom of
choice. However nonstandard
analysis carries out the same idea as the
Banach-Tarski paradox about 1D sphere, i.e.
a point: all the nonstandard universum can

The philosophical reflection
continues: That’s why nonstandard
analysis is a good tool for quantum
mechanics: Nonstandard universum
(NU) possesses as if fractal structure
just as Hilbert space. It allows all
quantum objects to be described as
internal sets absolutely similar to
macro-objects being described as
external or standard sets. The best
advantage is that NU can describe the

Something still a little more: If
Hilbert spa-ce is isomorphic to a
well ordered sequence of 3D
spheres delivered by the axiom of
choice via the Banach-Tarski
paradox, then 1. It is at least
comparable unless even iso-morphic
to Minkowski space; 2. It is getting
generalized into nonstandard
universum as to arbitrary number
dimensions, and even as to fractional

And at last: The generalized so
Hilbert space as nonstandard
universum is delivered again by the
axiom of choice but this time via
Zorn’s lemma (an equivalent to the
axiom of choice) via ultrafilter
lemma (a weaker statement than the
axiom of choice). Nonstandard
universum admits to be in its turn
generalized as in the gauge
theories, when internal and

Thus we have already pioneered to
Alain Connes’ introducing of
infinitesimals as compact Hilbert
operators unlike the rest Hilbert
operators representing transfor-
mations of standard sets. He has
suggested the following
‚dictionary‛:

Complex variable Hilbert

The sense of compact operator: if it
is ap-plied to nonstandard
universum, it trans-forms a
nonstandard neighborhood into a
nonstandard neighborhood, so that
it keeps division between standard
and nonstandard elements. If the
nonstandard universum is built on
Hilbert space instead of on real
continuum, then Connes defined
infinite-simals on the Cartesian

I would like to display that Connes’
infinitesimals possesses an
exceptionally important property:
they are infinitesimals both in
Hilbert and in Minkowski space: so
that they describe very well
transformations of Minkowski space
into Hilbert space and vice versa:
Math speaking, Minkowski operator
is compact if and only if it is
compact Hilbert operator. You might

Minkowski operator is compact if
and only if it is compact Hilbert
operator. Before a sketch of
proof, its sense and motivation: If we
describe the transformations of
Minkow-ski space into Hilbert space
and vice versa, we will be able to
speak of the transition between the
apparatus and the microobject and
vice versa as well of the transition
bet-ween the coherent and

Before a sketch of proof, its sense
and motivation: Our strategic
purpose is to be built a
united, common language for us to
be able to speak both of the
apparatus and of the microobject as
well, and the most impor-tant, of
the transition and its converse bet-
ween them. The creating of such a
language requires a different set-
theory foundation including: 1. The

and motivation: The axiom of
foundation is available in quantum
mechanics by the collapse of wave
function. Let us represent the
coherent state as infinity since, if the
Hilbert space is separable, then any
its point is a coherent superposition
of a countable set of components.
The ‚collapse‛ represents as if a
descending avalanche from the

and motivation: If that’s the case, the
axiom of foundation AF is available
just as the requirement for the
wave function to collapse from the
infinity as an avalanche since AF
forbids a smooth, continuous, infinite
lowering, sinking. It would be an
equivalent of the AF negation. A
smooth, continuous, infinite process
of lowering admits and even

A note: Let us accept now the AF
negation, and consequently , a
smooth reversibility between
coherent and ‚collapsed‛ state.
Then: P = Ps - Pr, where Ps is the
probability from the coherent
superposition to a given value, and
Pr is the probability of reversible
process. So that the quantum
mechanical probability attached to

A Minkowski operator is compact if
and only if it is a compact Hilbert
operator. A sketch of proof:
Wave function Y: RR  RR
Hilbert space: {RR}  {RR}
Hilbert operators:
{RR}  {RR}  {RR}  {RR}
Using the isomorphism of Möbius and
Lorentz group as follows:


{RR}  {RR}  {RR}  {RR}
 (the isomorphism)
{RR  R}R  {RR  R}R:
i.e. Minkowski space operators.
The sense of introducing of
nonstandard infinitesimals by
compact Hilbert operators is for
them to be invariant towards
(straight and inverse)

A little comment on the theorem:
A Minkowski operator is compact if
and only if it is a compact Hilbert
operator
Defining nonstandard infinitesimals
as compact Hilbert operators we
are introducing infinitesimals being
able to serve both such ones of the
transition between Minkowski and
Hilbert space (the apparatus and the

A little more comment on the
theorem:
Let us imagine those infinitesimals,
being operators, as sells of phase
space: they are smoothly decreasing
from the minimal cell of the
apparatus phase space via and
beyond the axiom of foundation to
zero, what is the phase space sell of
the microobject. That decreasing is

theorem:
Hamiltonian describes a system by
two independent linear systems of
equalities [as if towards the
reference frame both of the
apparatus (infinity) and of
microobject (finiteness)]
Lagrangian does the same by a
nonlinear system of equalities [the

theorem:
Jacobian describes the bifurcation,
two-forked direction(s) from a
nonlinear system to two linear
systems when the one united,
common description is already
impossible and it is disintegrating to
two independent each of other
descriptions

A few slides are devoted to
alternative ways for
nonstandard infinitesimals to be
introduced:
- smooth infinitesimal analysis
- surreal numbers.
Both the cases are inappropriate
to our purpose or can be
interpreted too close-ly or even

‚Intuitively, smooth infinitesimal
analysis can be interpreted as
describing a world in which lines are
made out of infinitesimally small
segments, not out of points. These
seg-ments can be thought of as being
long enough to have a definite
direction, but not long enough to be
curved. The construction of
discontinuous functions fails because
a function is identified with a curve,

‚We can imagine the intermediate
value theorem's failure as
resulting from the ability of an
infinitesimal segment to
straddle a line. Similarly, the
Banach-Tarski paradox fails
because a volume cannot be
taken apart into points‛
(Wikipedia, ‚Smooth infinitesimal

The infinitesimals x in smooth
infinitesimal analysis are
nilpotent (nilsquare): x2=0
doesn’t mean and require that x
is necessarily zero. The law of
the excluded middle is denied:
the infinitesimals are such a
middle, which is between zero
and nonzero. If that’s the case

The smooth infinitesimal analysis
does not satisfy our
requirements even only because
of denying the axiom of choice or
the Banach - Tarski paradox. But I
think that another version of
nilpotent infinitesimals is
possible, when they are an
orthogonal basis of Hilbert
space and the latter is being

By introducing as zero divisors,
the infinitesimals are interested
because of possibility for the
phase space sell to be zero still
satisfying uncertainty. It means
that the bifurcation of the initial
nonlinear reference frame to
two linear frames
correspondingly of the

The infinitesimals introduced as
surreal numbers unlike
hyperreal numbers (equal to
Robinson’s infinitesimals):

Definition: ‚If L and R are two
sets of surreal numbers and no
member of R is less than or
equal to any member of L then {

About the surreal numbers:They
are a proper class (i.e. are not a
set), ant the biggest ordered
field (i.e. include any other
field). Comparison rule: ‚For a
surreal number x = { XL | XR } and
y = { YL | YR } it holds that x ≤ y if
and only if y is less than or
equal to no member of XL, and no

Since the comparison rule is
recursive, it requires finite or
transfinite induction . Let us now
consider the following subset N
of surreal numbers: All the
surreal numbers S  0. 2N has to
contain all the well ordered
falling sequences from the
bottom of 0. The numbers of N
from the kind

For example, we can easily to
define our initial problem in
their terms:
Let  and  be:
 = {q: q  {N | 0}}
 = {w: w  {0 | 0  N}}
Our problem is whether  and 
co-incide or not? If not, what is
power of   ? Our hypothesis
is: the ans-wer of the former

That special axiom set includes:
the axiom of choice and a
negation of the generalized
continuum hypothesis (GCH). Since
the axiom of choice is a
corollary from ZF+GCH, it implies
a negation of ZF, namely: a
negation of the axiom of
foundation AF in ZF. If ZF+GCH is the
case, our problem does not arise

However a permission and
introducing of the infinite
degressive sequences , and
consequently, a AF negation is
required by quantum
information, or more
particularly, by a discussing
whether Hilbert and Minkowski
space are equivalent or not, or
more generally, by a considering

Comparison between ‚standard‛
and nonstandard infinitesimals.
The‚standard‛ infinitesimals
exist only in boundary
transition. Their sense
represents velocity for a point-
focused sequence to converge to
that point. That velocity is the
ratio between the two neighbor
intervals between three discrete

More about the sense of ‚standard‛
infinitesimals: By virtue of the axiom
of choice any set can be well
ordered as a sequence and thereby
the ratio between the two neighbor
intervals between three discrete
successive points of the sequence in
question is to exist just as before: in
the proper case of series. However
now, the ‚neighbor‛ points of an
arbitrary set are not discrete and

Although the ‚neighbor‛ points of an
arbit-rary set are not discrete, and
consequently, the intervals between
them are zero, we can recover as
if ‚intervals‛ between the well-
ordered as if ‚discrete‛
neighbor points by means of
nonstandard infini-tesimals. The
nonstandard infinitesimals are
such intervals. The representation

But the ratio of the neighbor
intervals can be also considered
as probability, thereby the
velocity itself can be inter-
preted as such probability as
above. Two opposite senses of a
similar inter-pretation are
possible: 1) about a point
belonging to the sequence: as
much the velocity of convergence

2) about a point not belonging to
the sequence: as much the
velocity of convergence is higher
as the probability of a point out
of the series in question to be
there is less; i.e. the sequence
thought as a process is steeper,
and the process is more
nonequilibrium, off-balance,
dissipative while a balance,

The same about a cell of phase
space:
The same can be said of a cell of
phase space: as much a process is
steeper, and the process is more
nonequilibrium, off-balance,
dissipative as the probability of
a cell belonging to it is higher
while a balance, equilibrium,

Our question is how the
probability in quantum
mechanics should be interpre-
ted? A possible hypothesis is: the
pro-babilities of non-
commutative, comple-mentary
quantities are both the kinds
correspondingly and
interchangeably.
For example, the coordinate

The physical interpretation of
the velo-city for a series to
converge is just as velocity of
some physical process. If the
case is spatial motion, then the
con-nection between velocity and
probability is fixed by the
fundamental constant c:

The coefficients ,  from the
definition of qubit can be
interpreted as generalized,
complex possibilities of the
coefficients ,  from relativity:
Qubit: Relativity:
 2+2=1
 = (1-) 1/2

|0+|1 = q =v/c

The interpretation of the ratio
between nonstandard
infinitesimals both as velocity
and as probability. The ratio
between ‚stanadard‛
infinitesimals which exist only in
boundary transit

But we need some interpretation
of complex probabilities, or,
which is equi-valent, of complex
nonstandard neigh-borhoods. If
we reject AF, then we can
introduce the falling, descending
from the infinity, but also
infinite series as purely,
properly imaginary nonstandard
neighborhoods: The real

After that, all the complex
probabilities are ushered in
varying the ties, ‚hyste-reses‛
‚up‛ or ‚down‛ between two
well ordered neighbor standard
points. Wave function being or
not in separable Hilbert space
(i.e. with countable or non-
countable power of its
components) is well interpreted

Consequently, there exists one
more bridge of interpretation
connecting Hilbert and 3D or
Minkowski space.
What do the constants c and h
inter-pret from the relations
and ratios bet-ween two
neighbor nonstandard inter-
vals? It turns out that c

And what about the constant h?
It guarantees on existing of: both
the sequences, both the
nonstandard neighborhoods ‚up‛
and ‚down‛. It is the unit of the
central symmetry transforming
between the nonstandard
neighborhoods ‚up‛ and ‚down‛
of any standard point h като площ

And what about the constant h? It
gua-rantees on existing of: both
the sequen-ces, both the
nonstandard neighbor-hoods
‚up‛ and ‚down‛. It is the unit of
the central symmetry
transforming between the
nonstandard neighborhoods ‚up‛
and ‚down‛ of any stan-dard

One more interpretation of h: as
the square of the hysteresis
between the ‚up‛ and the ‚down‛
neighborhood between two
standard points. Unlike standard
continuity a parametric set of
nonstandard continuities is
available. The parameter g =
Dp/Dx = Dm/Dt =

One more interpretation of h:
The sense of g is intuitively very
clear: As more points ‚up‛ and
‚down‛ are common as both the
hysteresis branches are closer.
So the standard continuity turns
out an extreme peculiar case of
nonstan-dard continuity, namely
all the points ‚up‛ and ‚down‛
are common and both the

By means of the latter
interpretation we can interpret
also phase space as non-
standard 3D space. Any cell of
phase space represents the
hysteresis between 3D points
well ordered in each of the
three dimensions. The connection
bet-ween phase space and Hilbert


What do the constants c and h
interpret as limits of a phase
space cell deformation?

c.1.dx  dy  h.dx
Here 1 is the unit of curving
[distance x mass]

Forthcoming in 2nd part:
1. Motivation

5. Nonstandard continuity
between two infinitely close
standard points

CONTINUITY AND CONTINUUM
IN NONSTANDARD UNIVERSUM
Vasil Penchev
Institute for Philosophical Research
Bulgarian Academy of Science
E-mail: vasildinev@gmail.com
Professional blog:
http://www.esnips.com/web/vasilpenchevsnews

That was all of 1 st part
Thank you for your

Vasil Penchev. Continuity and Continuum in Nonstandard Universum

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Vasil Penchev. Continuity and Continuum in Nonstandard Universum