The theory of co∼events in brief questions and brief answers with illustrations

THE XVI CONFERENCE ON FAMEMS AND THE WORKSHOP ON HILBERT’S SIXTH PROBLEM, KRASNOYARSK, SIBERIA, RUSSIA, 2017




The theory of co∼events in brief questions and brief answers
with illustrations
Oleg Yu. Vorobyev
Institute of mathematics and computer science
Siberian Federal University
Krasnoyarsk
mailto:
http://www.sfu-kras.academia.edu/OlegVorobyev
http://olegvorobyev.academia.edu

Very briefly on the new axiomatics of co∼events [1], it is intended for rigorous mathematical description
of an entanglement between observer and observation. Here we try to consider the new axiomatics in the
normally human non-mathematical language, although it uses a new but quite natural terminology: “a
Abstract: co∼event occurs (as an entanglement of observer and observation), a ket-event happens (as chance of
observation), a bra-event is experienced (as an experience of observer)”. And finally, it seems to me that
the new theory in a not very rigorous mathematical language is most conveniently set out in the form
of questions and answers that my students, graduate students, and colleagues often asked me during
numerous internal and external discussions.
Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, measurable binary relation,
co∼event, bra-event, ket-event, believability, certainty, believability theory, certainty theory, theory
Keywords:
of co∼events, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event
axiomatics, dual sets of events, random set, experienced set.




1 Examples and brief discussion


The expression “A choosing (by the 15 foresters) the trees for felling (from the 387 trees)” describes
a result of “15 foresters × 387 trees” experienced-random experiment in which each forester (from 15)
independently chooses the trees for felling from the 387 trees (putting a label on each tree chosen by
her/him for felling).
What objections and questions can arise to this description?
Objections and questions Answers
Definitely! “A choosing (by foresters) the trees
1 for felling” is not an event (in Kolmogorov sense)
For some taste this is not an event (in the classical because it is co∼event (in the new theory sense)
sense) that can have a probability. as the result of the “15 foresters × 387 trees”
experienced-random experiment (see Tables).
No, it’s not! An event ”A randomly chosen expert
2
labels and a randomly chosen tree” is not the event
Maybe, the event here is: ”A randomly chosen
that describes a result of our experienced-random
expert labels and a randomly chosen tree”.
experiment.
If you make 15 x 387 independent trials of the
3
event ”A randomly chosen expert labels and a
Why the event ”A randomly chosen expert labels
randomly chosen tree” then you may get on
and a randomly chosen tree” does not describe
some tree multiple labels from the same expert.
the result of our experienced-random experiment
Moreover, some trees (from 387) may remain as
“A choosing (by foresters) the trees for felling” ?
trees un-chosen by all 15 experts.



c 2017 O.Yu.Vorobyev
○
This is an open-access article distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Oleg Vorobyev (ed.), Proc. of the XVI FAMEMS’2017, Krasnoyarsk: SFU, ISBN 978-5-9903358-7-5

, OLEG YU VOROBYEV. THE THEORY OF CO∼EVENTS IN BRIEF QUESTIONS AND BRIEF ANSWERS WITH ILLUSTRATIONS 135



|ω⟩ ∈ |ter(X//XR )⟩
|x′ ⟩=R
⟨ω *′ |
⏞ ⏟
⎫
⏟ ⏞ ⎪
|x⟩=R⟨ω * | ⎪
⎪
⎪
⎪
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⎪
⎪
⎪
⎪
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⟨ω * ′ | ∈ ⟨x′ | ⟨ω *′ |ω⟩ ⎪
⎪
⎪
⎪
⎪
′
/ X ⇐⇒ ⟨ω * ′ |ω⟩ ∈
x ∈ /R ⎪
⎪
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R ⊆ ⟨Ω|Ω⟩
⎪ ⎪
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*
⟨ω | ∈ ⟨x|
⎪
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⎪ ⟨ω * |ω⟩ ⎪
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x ∈ X ⇐⇒ ⟨ω * ′ |ω⟩ ∈ R ⎪
⎪ ⎪
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⟨Ω|
⎬
⟨TerX//XR | = R||ω⟩
⎪
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⟨Ω|Ω⟩
⎪
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⎭
⏟ ⏞
|Ω⟩
Figure 1: Venn diagram of the binary relation R ⊆ ⟨Ω|Ω⟩ (red) on Cartesian product ⟨Ω|Ω⟩ (aqua) with the R-labelling ⟨XR |SXR ⟩ and of three
cross-sections of the binary relation: R||ω⟩ , R|⟨ω* | , R|⟨ω*′ | by the ket-point |ω⟩ ∈ |Ω⟩ and the bra-points ⟨ω * | , ⟨ω *′ | ∈ ⟨Ω|, where the following
membership relations: ⟨ω * |ω⟩ ∈ R ⇔ x ∈ X and ⟨ω *′ |ω⟩ ∈ / R ⇔ x′ ∈ / X are equivalence.




2 Very briefly on the new axiomatics of co∼events which describes an
entanglement between observer and observation

The lines below describes the new axiomatics of co∼events [1] in the normally human non-mathematical
language, although it uses a new but quite natural terminology: “a co∼event occurs (as an entanglement
of observer and observation), a ket-event happens (as chance of observation), a bra-event is experienced
(as an experience of observer)”. (see Fig. 11 )


2.1 Ket-event (a chance of observation)

An observation chance is described by a ket-event from |Ω⟩ (an event in Kolmogorov sense).
Each ket-event can or cannot happen. A ket-event happens if there is an only elementary ket-event
belonged to it that happens. And a ket-event doesn’t happen if there isn’t any elementary ket-event
belonged to it that happens.


2.2 Bra-event (an experience of observer)

An observer experience is described by bra-events from ⟨Ω| (in the new axiomatics sense).
Each bra-event can or cannot be experienced. A bra-event is experienced if all elementary bra-events
belonged to it are experienced. And a bra-event doesn’t happen if there is an elementary bra-event
belonged to it that doesn’t happen.

1
︀
︀ Here |x⟩ = x∈X |ter(X//XR )⟩ ⊆ |Ω⟩ is a ket-event in the new axiomatics (an event in the Kolmogorov axiomatics), ⟨TerX//XR | =
x∈X ⟨x| ⊆ ⟨Ω| is a terraced bra-event in the new axiomatics.