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Теория множеств
Логика
Гедаль
Множество
Математика
Кризис
аксиомы
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00:00:02
you know the difference between mathematics and such sciences,
00:00:05
physics, chemistry, biology, is mainly
00:00:09
that mathematics does not strive to describe the
00:00:12
world, but rather develops an apparatus or
00:00:15
mental tool that can be
00:00:17
useful for describing the world, we know how
00:00:21
physics lost its unity after moving away from
00:00:23
Newtonian mechanics to the general theory of
00:00:26
relativity and quantum mechanics
00:00:29
and is now in search of a general
00:00:31
theory of everything in mathematics, there was also its own
00:00:35
theory of everything, the theory of sets, it
00:00:38
unified mathematics, then it split and
00:00:41
gave rise to many heated debates; it is
00:00:44
extremely convenient and powerful in solving
00:00:47
many long-standing problems, but at the same time it is
00:00:50
also huge number of critical questions
00:00:52
that led to the greatest
00:00:55
crisis of foundations in the history of mathematics resonance in
00:00:59
mathematics
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it is somewhat similar to the theory of
00:01:01
relativity and quantum mechanics
00:01:04
so let's study together how
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these outstanding events in the history of mathematics took place
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chapter 1 the rapid restoration of the
00:01:15
mathematical Atlantean
00:01:17
since the times of ancient Greece in mathematics
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or even in the philosophy of mathematics
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there were two types of infinity:
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potential and actual, for example,
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integers, infinite,
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you can potentially always add another unit and
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get a larger number, in
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other words, there is always the potential
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opportunity to continue a series of numbers,
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but among them there is not some
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infinitely large number, an actually
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infinite number, such an understanding the
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existence of potential
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infinity
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was preserved until the advent of mathematical
00:01:58
analysis; it required understanding both the
00:02:01
infinitely large and the infinitesimal,
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but mathematical analysis eventually
00:02:07
reduced these concepts to potential due to the
00:02:11
concept of la
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you can move from a
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sequence of regular
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polygons to a circle
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due to the limit transition of a special
00:02:21
operation, but the circle itself is not an
00:02:24
infinite third-party polygon,
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simply as a result of the operation we received
00:02:29
such an object; at the same time,
00:02:31
mathematical analysis
00:02:33
revived discussions about the existence of
00:02:36
actual infinity and here set theory comes onto the scene; the
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main prerequisite
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for the creation of set theory was the
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problem of actual infinity, although
00:02:48
mathematical analysis somewhat weakened
00:02:51
the problems associated with infinity but
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still left many questions that I
00:02:56
would like to solve. Set theory
00:02:59
appeared in the second half of the 19th century
00:03:02
mainly thanks to the genius of the German
00:03:04
mathematician Georg Kanter. What is a set?
00:03:08
Kanter defined a set
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elegantly and simply, it is literally any
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collection of objects absolutely any
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nature
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and belonging to this set
00:03:20
is subject to any logical rules example of
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sets all integers or
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real numbers greater than five
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all black cats or all residents of
00:03:31
Orenburg 13 years old and the like
00:03:34
components of this set are called
00:03:37
elements of this set-set
00:03:40
may even contain other sets
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as its
00:03:45
key elements the concepts of such a theory, in addition to
00:03:47
set
00:03:48
and element, can be considered the belonging of an
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element to a set
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and the concept of an empty set in which there
00:03:56
are no elements, there is also the concept of a subset, which is
00:04:00
a collection of all elements of which
00:04:02
are also elements of the
00:04:05
original set, as well as the concept of power, a set of
00:04:08
power is a concept that
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generalizes in quotes the number of
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elements of a set, say power a
00:04:16
set of five cows is just 5,
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however, what if there are
00:04:22
infinitely many elements in a set, how to calculate an
00:04:25
infinite number directly command,
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if we imagine a group of students and
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chairs in a class, how can we check
00:04:37
whether their number is the same? Well, first of all,
00:04:39
you can take and re-read both of them and
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compare quantity
00:04:44
and secondly, you can ask
00:04:47
students to sit on chairs and find out if there are
00:04:50
people who are missing a chair or if there are
00:04:54
extra chairs; such a correspondence
00:04:56
between chairs and students in mathematics
00:04:59
is called a one-to-one
00:05:02
correspondence; it
00:05:05
can be illustrated by using arrows
00:05:06
from one set to another and checking
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the properties of these arrows
00:05:11
can be verified that the set is equal or not equal,
00:05:14
so Kanter defined equal to a
00:05:18
powerful set,
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these are sets between which one can
00:05:22
put a one-to-one correspondence,
00:05:25
this definition made it possible to consider the
00:05:28
infinite set as a whole, it is
00:05:31
actually possible to even define an
00:05:33
infinite set without mentioning
00:05:36
infinity simply as a set equal to
00:05:39
its powerful part if students
00:05:43
sitting on chairs could be lifted and
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placed only on part of all chairs,
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for example on every second, then the set of
00:05:52
students would be infinite because of
00:05:54
this, in particular, all sorts of
00:05:56
Hilbert’s hotels appear, it also turned out that the
00:06:00
set of natural numbers and the set of
00:06:02
rational numbers are equal to powerful ones, that is,
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fractions and of integers literally the
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same number can be all
00:06:12
infinite sets are the same no
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Kanter also proves that
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real numbers are
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numbers that can be represented as
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infinite decimal fractions
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these are large infinite sets
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whether proof of this Kanter
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used what is called
00:06:32
Kanter's diagonal argument, first let's take
00:06:34
natural numbers and write them down column,
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we cannot write
00:06:38
them down for real since there are infinitely
00:06:41
many of them, but we can imagine it, then
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we will take as an example the set of
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real numbers
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numbers on the interval from zero to one
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and for convenience we will convert them into the
00:06:53
binary system for the quantity this will not affect in any way
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each real number this an
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infinite decimal fraction which in
00:07:01
our case can be represented as a
00:07:03
simple infinite set of zeros and ones,
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let’s assume that the set is still equal to
00:07:10
powerful, which means that each natural
00:07:13
number can be associated with
00:07:15
its own set of zeros and ones, we
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get some kind of infinite table,
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let’s take the diagonal of this table and
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then change every 0 in it is by 1 1 by
00:07:27
0,
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so we get
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a sequence about which we can
00:07:32
say two things: firstly, it is a
00:07:34
real number on the interval from zero
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to one; secondly, this number,
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however, does not coincide with any of the columns of
00:07:43
all natural numbers as the chairs are already
00:07:46
occupied and we got an extra student
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who didn’t have enough chair,
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which means the set is still not equal
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powerful, note that infinity
00:07:57
does not interfere with the conclusion in any way, yes, we cannot
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write down all the natural numbers,
00:08:02
write down this table, get the diagonal,
00:08:04
but our reasoning is enough to
00:08:08
show that there is excess number there is a
00:08:11
set with the power of natural numbers
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mathematicians call countable because
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such sets can be counted in quotes
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or numbered by natural
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numbers and a set with the power of
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real numbers
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Kanter called a continuum the diagonal
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method is associated with another concept with a
00:08:32
telling name: the
00:08:34
set of all subsets if you take
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some finite set
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for example, the set pen pineapple apple
00:08:41
then its set of all subsets is an
00:08:44
empty subset three single
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subset 3 subset of two
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elements and the original set itself is
00:08:53
only eight elements of each of which the
00:08:56
set of 3
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we got 8 or two to the third power
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this relation holds for all
00:09:03
finite sets
00:09:05
with infinite sets everything
00:09:07
turns out to be more interesting because they
00:09:10
have infinite subsets, the
00:09:13
diagonal argument proves that the
00:09:15
set of all subsets is a way to
00:09:18
increase the power of an infinite
00:09:21
set, we looked at this for
00:09:23
specific sets, but it works for the
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general case; moreover, the set of all
00:09:30
subsets is the only known
00:09:33
way to increase the power
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any addition of sets of power no
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more than the initial one is not capable of increasing
00:09:41
infinity, for example, a line, a plane, and
00:09:44
even three-
00:09:45
dimensional three-dimensional space have the
00:09:48
same power - a continuum, in connection
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with this, Kanter put forward a rather logical
00:09:54
assumption
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since the set of all subsets is
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the only way to obtain more than infinity, that
00:10:00
set is less than the
00:10:03
continuum but more countable does not
00:10:06
exist this hypothesis is known as the
00:10:09
continuum hypothesis
00:10:11
at about the same time Julius dd Kent
00:10:14
forms the rules for the operations of
00:10:16
calculation and addition of sets this
00:10:19
leads
00:10:20
m100 wider hole to the idea of ​​the similarity of set
00:10:23
theory and logic
00:10:25
on the basis of set theory the
00:10:28
theory of real numbers is formed and the
00:10:31
axiomatics of arithmetic are created then named
00:10:35
after Josie Papa, at the very end of the 19th
00:10:38
century, Kanter published a series of works that
00:10:41
actually completed the formation of set theory,
00:10:45
which would later be called naive, despite
00:10:48
some opponents, set theory
00:10:51
penetrated everywhere and became one of the most
00:10:54
powerful tools in mathematics,
00:10:56
this was facilitated by major mathematicians
00:10:59
of that time at the first International
00:11:01
Congress of Mathematicians in 1897
00:11:05
showed how set theory advanced
00:11:09
mathematical analysis.
00:11:10
Not the least role was played by the school of David
00:11:13
Hilbert, which already at that time
00:11:16
had a great influence in the mathematical
00:11:18
community. In general, in about 30 years, set
00:11:21
theory captured mathematics as the
00:11:24
main
00:11:25
and most general theory, useful
00:11:28
practically everywhere Mathematical sciences are
00:11:32
now increasingly penetrating into various
00:11:34
fields of activity and their deep
00:11:37
knowledge is becoming more and more in demand.
00:11:39
In general, it seems to me that now the main goal of
00:11:42
mathematics is to successfully solve
00:11:43
fundamental and applied problems;
00:11:46
deep knowledge in a mathematical
00:11:48
profile cannot be obtained without
00:11:52
master's programs in higher
00:11:55
educational institutions where every student
00:11:57
will be confident as an
00:12:00
education to become a professional in
00:12:03
your field,
00:12:04
if you are soon graduating from
00:12:06
a bachelor's degree or have graduated earlier and
00:12:09
want to be a more advanced
00:12:11
specialist, then you will definitely be interested in the
00:12:13
master's program in applied
00:12:16
mathematics and computer science
00:12:18
at the Russian Peoples' Friendship University. The
00:12:20
mandatory part of the program includes
00:12:23
fundamental training in
00:12:25
mathematical disciplines, the study of
00:12:27
modern mathematical methods and
00:12:30
active research work, I
00:12:33
am sure that if you are a person interested not
00:12:36
only in such a fundamental topic as
00:12:38
set theory, but also in the methods of applied
00:12:41
mathematics and computer technology, as well as
00:12:44
interested in developing
00:12:46
software for solving
00:12:48
problems in science and technology,
00:12:51
you will definitely not go wrong with this
00:12:53
direction,
00:12:55
chapter 2, the first wave of crisis and paradoxes,
00:12:59
be that as it may, rosy times just
00:13:02
as quickly ended, mathematicians
00:13:05
discovered problems, or rather paradoxes, of
00:13:08
set theory and, in contrast to the theory of
00:13:11
relativity and quantum mechanics, in
00:13:13
which production is nothing more than a
00:13:16
discrepancy between intuition and reality the
00:13:18
paradoxes of set theory were
00:13:21
real contradictions in set theory itself, the
00:13:25
first product was discovered by the
00:13:28
office itself in its work, he decided to
00:13:30
consider the set of all sets
00:13:33
or the universe, a set can be given
00:13:35
any logical rules, including
00:13:38
any set, if we consider
00:13:41
the set of all subsets of the universe,
00:13:44
then since all possible sets are already
00:13:47
contained in the universe, the set of all
00:13:50
subsets of the universe
00:13:52
must be part of it or coincide with
00:13:56
it, but the diagonal argument says that
00:13:58
the set of all subsets must
00:14:01
only be greater, that is, the existence of
00:14:04
the universe is contradictory. At first this
00:14:08
contradiction did not seem like something
00:14:10
serious, it seemed that it was enough
00:14:12
to revise some of the evidence in
00:14:14
more detail and it will be resolved but no, the
00:14:18
paradox discovered by the outstanding mathematician
00:14:21
Bertrand Russell
00:14:22
was already a more serious question; by the
00:14:25
foundation of set theory,
00:14:27
it became clear that the problem is deeper and the
00:14:29
worst thing is that thanks to a small
00:14:32
change, the paradox easily passed into
00:14:35
logic; the
00:14:36
product concerns sets that
00:14:39
include themselves as an element, for example, the already
00:14:41
discussed set of all sets
00:14:44
must contain itself as an element but but
00:14:47
such is not the only one
00:14:49
we call the set unusual if it
00:14:52
contains itself as an element
00:14:54
ordinary if it does not contain consider once the
00:14:58
Slavic set which contains all the
00:15:01
ordinary ones the question is whether 1 Slavic set itself is usual
00:15:05
if a narrow
00:15:08
set grows ordinary then it contains
00:15:10
itself as one from ordinary but then by
00:15:13
definition it is unusual but if the
00:15:16
Rostov set is unusual it means
00:15:20
it does not contain itself since it contains
00:15:23
only ordinary ones
00:15:25
but the unusual set must contain
00:15:28
itself a contradiction means that the
00:15:31
Rostov set should not
00:15:33
exist but why since the Slavic
00:15:36
set is given by simple logical
00:15:38
rules all the ordinary ones do not even postulate
00:15:42
the existence of sets that contain
00:15:44
themselves, we simply ask a question and get
00:15:48
contradictions, except for the paradox of growing up, there are
00:15:51
several similar in meaning, more
00:15:53
understandable paradoxes, the liar's paradox,
00:15:56
which has been known since ancient times,
00:15:59
has a similar structure,
00:16:01
this statement is a lie, the question is whether
00:16:05
this statement will be true, I hope it has
00:16:08
become clear, that is this paradox
00:16:10
is imaginary
00:16:11
in fact there is no
00:16:13
paradox here the product of a barber a
00:16:16
barber shaves all those who do not shave
00:16:18
themselves and only their question Bradley barber
00:16:22
himself the
00:16:23
Middle Ages people liked to ask me
00:16:25
funny riddles for example in the village there
00:16:29
lived only one barber he shaved everyone
00:16:32
who did not shave himself who shaved the barber
00:16:36
meter children of the little man of all things this
00:16:39
question is meaningless and the answer is no
00:16:42
well noted also there are products similar in
00:16:45
meaning
00:16:46
take ghrelin by nelson richard and many
00:16:51
other products became a blow for many
00:16:54
researchers
00:16:55
especially after the Russell paradox dadykin
00:16:58
suspended his research because he
00:17:00
considered that the basics set theories
00:17:02
collapsed Frege had just finished his
00:17:05
main work of life when he informed
00:17:08
him about the discovery because of which Frege writes in his
00:17:11
afterword that all his work is
00:17:13
now doubtful
00:17:15
for the Russell Ankara paradox that he was one
00:17:18
of the disseminators of set theory and
00:17:21
completely abandoned it in general an
00:17:24
emotional situation in the mathematical
00:17:27
community it turned out to be suppressed, many
00:17:30
specialists ventured into safe
00:17:32
areas so as not to be in constant
00:17:35
doubt about the reliability of
00:17:37
their work
00:17:40
Chapter 3 split in the foundations of mathematics It must be
00:17:45
said that the question of the foundations of mathematics
00:17:48
was raised a long time ago
00:17:50
one of the oldest areas of
00:17:52
research in the foundation of mathematics
00:17:54
as a science was occupied by algaecide we are the essence of this
00:17:58
the idea of ​​reducing mathematics to logic
00:18:00
goes back to one of the fathers of
00:18:03
mathematical analysis,
00:18:05
Gottfried Leibniz, you might think
00:18:08
that Katyuschik and Torvald are
00:18:10
modern Internet lags, but no,
00:18:12
they just don’t understand logic well; in fact, it was in
00:18:15
attempts to bring
00:18:17
mathematics to logic that famous paradoxes appeared;
00:18:21
one of the most prominent logisticians was Bertrand
00:18:25
Russell Russell, together with his colleague while he
00:18:28
dam, in his fundamental work
00:18:30
Mathematics Principle, developed the
00:18:33
logical foundations of mathematics and when
00:18:36
writing it came across the same
00:18:39
paradox of the name of himself before the Mathematician Principle, and
00:18:43
even in the first editions Russell was still
00:18:45
confident that logical laws are
00:18:49
a priori true conditioned
00:18:51
the nature of the
00:18:52
information from mathematics to logic was supposed to
00:18:55
prove
00:18:56
the consistency and truth of all
00:18:59
mathematics,
00:19:00
although at that time I already knew about different
00:19:03
geometries, but nevertheless I believed that
00:19:05
each of them is simply some
00:19:07
special image of reality and in
00:19:10
fact mathematics follows from the laws of
00:19:13
logic that represent part of
00:19:15
reality some foundation and walls,
00:19:19
however, Russell later realized that the laws of
00:19:22
logic are not necessarily true
00:19:25
but are just as arbitrary as the
00:19:28
axioms of Euclidean or other
00:19:30
mathematical axioms, which
00:19:32
was facilitated by criticism of the axioms of logic
00:19:34
that he used in the book, in the end,
00:19:38
once he came to the conclusion that it
00:19:41
makes no sense to reduce mathematics to logic, and
00:19:43
since logic itself is not
00:19:47
immutable, so formal logic is
00:19:50
not for show, but this is classical
00:19:54
economic logic, the one that
00:19:56
describes some immutable dependencies
00:19:59
in the universe, but the work of
00:20:01
eliminating products since he still
00:20:04
did the solution products 1 l
00:20:07
develops the so-called theory of types;
00:20:10
this theory, although it solved well-known
00:20:12
problems, turned out to be too complex and
00:20:15
confusing, which made it not very convenient
00:20:18
for using the
00:20:19
foundations of mathematics, since he even
00:20:22
admitted the delightful certainty
00:20:24
that I always hoped to find in
00:20:27
mathematics and got lost in the confusion of the
00:20:29
concept
00:20:30
and conclusions however, now the theory of types is in
00:20:34
some way useful for recording
00:20:36
data in computer technology,
00:20:38
the work of Russell and White Hod itself became one
00:20:42
of the most influential works on
00:20:44
mathematical logic and subsequent
00:20:47
research is somehow based on
00:20:50
it; honor of mathematicians had a
00:20:52
different opinion about the appearance of paradoxes on the
00:20:55
crowd of them at the idea that mathematics not on the
00:20:58
right path,
00:20:59
this direction was called
00:21:01
intuition from below, the founder of which
00:21:03
can be considered Ledza Brown, a
00:21:06
Scottish mathematician known for the
00:21:09
fixed point theorem, in addition to him, in
00:21:12
addition to
00:21:15
Cyrus Poincaré, Lebesgue and Barrel are also included in this camp,
00:21:18
the main reason for the appearance of intuition
00:21:21
nismo was criticism of lagi zisma
00:21:24
their path it was quite a radical rejection
00:21:27
of conventional logic,
00:21:29
especially the law of exclusion 3, the
00:21:32
main principle of mathematics is to make
00:21:34
fuzzy explicit logic and mathematical
00:21:37
intuition, it is necessary to bring logical
00:21:40
principles in accordance with this intuition
00:21:44
braun, the creator of this direction
00:21:46
opposed the formalization of the foundations of
00:21:49
mathematics, including intuition nismo,
00:21:53
but then all of them - in some
00:21:55
way they formalized the form of formal
00:21:57
intuition, low logic, its understanding
00:22:01
needed to mark conclusions that are
00:22:03
acceptable to the human mind,
00:22:06
intuitions are related to reality, and
00:22:09
other conclusions are unacceptable,
00:22:11
and in this the only way to build the
00:22:14
foundations of mathematics, intuitionism
00:22:17
presupposes the existence of only such
00:22:19
mathematical objects that are obtained by
00:22:22
direct logical by construction,
00:22:25
for example, the number 5 can be constructed
00:22:28
from four by adding one, a statement
00:22:31
in classical logic has only two
00:22:34
possible values, true and false, this
00:22:37
gives the law of exception 3, this law makes it
00:22:42
possible to prove that nothing
00:22:44
exists explicitly without obtaining the object itself,
00:22:47
the method of proof is called by
00:22:50
contradiction, let us need to prove that
00:22:52
not that is true let us assume that this is a lie
00:22:56
and come to a contradiction with some of the
00:22:59
premises, it will follow from this that the
00:23:02
original not that there can only be truth
00:23:05
in intuition of low logic the third besides
00:23:09
truth and lies is not excluded, but why does
00:23:12
intuition bottom there so not like this
00:23:15
Brouwer's law believed that historically and
00:23:18
this law was based on finite
00:23:20
sets and was unreasonably
00:23:23
extended to infinite ones. If we have a
00:23:26
finite set,
00:23:28
then we can check whether
00:23:30
some property is satisfied for each element
00:23:32
step by step, however, for an infinite set, it will
00:23:39
not be possible to check all elements. This led to besides,
00:23:41
intuition nist there had to essentially
00:23:43
rewrite mathematics and cross out
00:23:46
most of the theorems, since in
00:23:48
mathematics there
00:23:51
was an absolute majority of proofs by contradiction, which
00:23:54
means that they had to cross out the
00:23:56
entire
00:23:57
topic, which was already useful at that time, it
00:24:00
turned out that in practice
00:24:02
nist intuition was not used
00:24:05
intuitions rather simply proclaimed
00:24:07
their own types of rules and followed them; the third
00:24:11
movement was the school of the outstanding
00:24:13
mathematician David Hilbert, or
00:24:16
formalism, Gilbert created an entire
00:24:18
program to create strong
00:24:20
foundations of mathematics and resolve the
00:24:23
paradoxes and problems of developing logisticians
00:24:26
Hilbert, they were satisfied because they
00:24:28
required an
00:24:29
intricate theory of types and dubious
00:24:32
axioms proposed by racial at the same time he
00:24:35
read the necessary actual
00:24:37
infinity and many other things
00:24:40
accumulated in mathematics and thanks to
00:24:42
set theory, intuition did not become he
00:24:45
criticized for the rejection of the law of
00:24:47
excluded middle, which led to the rejection
00:24:50
of the most powerful means of mathematics,
00:24:54
proof by contradiction and all
00:24:57
proofs built on it
00:24:59
and for a minute, most of them are
00:25:02
Gilbert's first ideas, adding the axioms of
00:25:05
mathematics to the axioms of logic, Gilbert
00:25:08
also believed that it was more useful to consider
00:25:09
mathematics as formal knowledge, that is,
00:25:12
knowledge of how to correctly
00:25:14
transform symbols in isolation from their
00:25:18
meaning, and this was his main difference
00:25:21
from the logisticians since Herbert
00:25:24
did not have mathematics for some reason in
00:25:27
reality, it was just a specific
00:25:30
symbolic language like a
00:25:33
programming language, in order to eliminate
00:25:35
discrepancies without using intuitive
00:25:38
ideas and to achieve objectivity,
00:25:40
Gilbert considered it necessary to write down all the
00:25:44
statements of logic and mathematics and to the
00:25:46
symbolic form of the benefit of symbolism to
00:25:49
logic, we have already quite developed lags by
00:25:52
writing down all mathematical and logical
00:25:55
laws in this form
00:25:57
hyper prepared the main question what
00:26:00
should be understood by a proof according to
00:26:03
Hilbert, the proof was
00:26:05
the transformation of a formula with admissible
00:26:08
transformations,
00:26:10
the installation of one symbol in place of
00:26:12
another or a group of symbols, essentially a
00:26:14
simple rewriting of a formula according to the
00:26:17
formalism of mathematics is a set of
00:26:20
formal systems, each such system
00:26:22
can have its own logic Gilbert saw his set of
00:26:26
axioms, rules of inference and theorems as the main
00:26:30
task of studying the
00:26:32
properties of
00:26:33
formal systems, in particular consistency, the
00:26:35
absence of paradoxes
00:26:38
and contradiction; it was simply the properties of a
00:26:41
formal system as a list of symbolic
00:26:44
forms. After several years of research,
00:26:47
the consistency of almost all formal
00:26:50
systems of mathematics was reduced to the
00:26:52
consistency of ordinary arithmetic
00:26:55
and and consistency acquired
00:26:58
key importance,
00:26:59
but here a problem arose because
00:27:03
formalists were able to reduce systems to consistency,
00:27:06
but they could prove without reducing them to something
00:27:09
Gilbert, his students Ackerman Bernays
00:27:14
and John von Neumann developed a method
00:27:16
called proof theory or meta-
00:27:19
mathematics; the essence of the method was in the transition
00:27:22
to a new one language so that by studying the language of
00:27:25
mathematics to come to meta-mathematics in
00:27:29
meta-mathematics and Gilbert proposed
00:27:31
to use the least controversial axioms of
00:27:33
logic to exclude infinity the law of
00:27:36
excluded middle and all the controversial
00:27:39
points in mathematics
00:27:40
this led to the fact that meta-mathematics
00:27:43
essentially became low intuition
00:27:46
naturally all this caused criticism from
00:27:49
outside Russell criticized his rivals
00:27:51
for the fact that the axioms do not
00:27:54
clearly convey the meaning of natural numbers, and
00:27:56
in general, in his opinion,
00:27:58
the work of the formalist of was reminiscent of the work of
00:28:00
a watchmaker,
00:28:01
that he was concerned only with the appearance of the watch, they their
00:28:04
work, intuition and bottom also generally
00:28:07
perceived Hilbert’s program with hostility,
00:28:09
according to them, mathematics began to resemble a
00:28:12
game of symbols into meaningless
00:28:15
formulas, and most of all, both of them were
00:28:18
outraged by the concepts of existence,
00:28:21
formalism, both
00:28:22
for registrants and for intuition, without
00:28:25
becoming this was the biggest
00:28:27
stumbling block. The Olga cists believed that a
00:28:30
mathematical object exists if it is
00:28:33
deduced from logic and intuition.
00:28:38
stated that when
00:28:42
asked where to look for the truth of
00:28:44
intuition, they answer that in the mind
00:28:48
and formalists that on paper, Gilbert
00:28:51
responded by calling intuitionism a
00:28:53
betrayal of science with its desire to
00:28:56
throw everything overboard, in fact, indisputable
00:28:59
principles within meta-mathematics
00:29:01
were met with exactly the same principled
00:29:04
disputes everywhere it happens at about the same
00:29:07
time, quite imperceptibly, another group was formed,
00:29:11
its founder can be considered a German
00:29:14
mathematician and the rise in prices is nice, the approach of this
00:29:17
group is now called
00:29:19
multiple theories, initially they did not have
00:29:22
any philosophy or any program, all
00:29:25
this appeared in the process when
00:29:28
paradoxes appeared many mathematicians came to the
00:29:30
conclusion that the problem of set theory in
00:29:33
set theory simply needs to be more clearly
00:29:36
defined by the concept of sets and thereby
00:29:39
correct the problem, that is, to
00:29:41
deal with the axiom of these concepts of set theory, the
00:29:44
first attempt was made by Ernst Sir
00:29:47
Milo, unlike other terms, he only
00:29:50
sought to get rid of contradictions
00:29:52
and did not follow any philosophy, the
00:29:55
axiom system of cer milo contained one special
00:29:58
axiom, which many people heard out of the blue,
00:30:01
even if they were not particularly
00:30:03
interested in mathematics,
00:30:05
this is the axiom of choice, I am looking for, denoted as an
00:30:09
essay of the axioms oh choice, the axiom system of the goal of
00:30:13
milo and Frenkel, together with the axiom of choice, is
00:30:15
now called as z fc named after the 2
00:30:20
creators and the axiom of choice what is
00:30:23
remarkable about this axiom open goal
00:30:25
nice let’s say there is a certain set a
00:30:28
collection of sets the axiom says that
00:30:31
from each
00:30:32
set in this collection you can select
00:30:35
exactly one element and make up a
00:30:38
new set from them and everything sounds
00:30:42
quite obvious After all, does
00:30:46
this property of finite sets really follow from other axioms
00:30:48
for countable ones? This also doesn’t sound very
00:30:51
counter-intuitive.
00:30:53
The problem is that this action can be
00:30:56
performed for any collection of
00:30:58
sets and this gives the axiom of choice a
00:31:01
special superpower. Because of this axiom,
00:31:04
sometimes such completely non-
00:31:07
intuitive theorems are given as
00:31:10
Banahatarian's theorem on the doubling of the ball, but 1 of the
00:31:15
theorem that made one doubt the
00:31:17
axiom of choice was proved by the term la itself
00:31:20
that any set can be completely
00:31:24
ordered; ordering means assigning to
00:31:27
each pair of elements of the set either
00:31:29
more or less and completely adds
00:31:33
to this condition that all subsets
00:31:37
have the smallest element of the problem is
00:31:40
that mathematicians have no idea how to
00:31:42
specifically completely order at least the
00:31:45
set of real numbers, not to mention
00:31:49
all the sets, there was even
00:31:51
evidence that real numbers
00:31:54
cannot be completely ordered, although they are
00:31:57
erroneous, another example of an unpleasant
00:32:00
consequence developed by the French
00:32:02
mathematician Henry Lee Beg the
00:32:05
immeasurable set of measure theories generalizes ban and
00:32:07
volume on arbitrary sets
00:32:10
using the concept of measures of a numerical function are given
00:32:13
on sets; however,
00:32:16
this concept cannot be generalized to all possible sets;
00:32:19
using the axiom of choice,
00:32:22
it can be proven that there is a set
00:32:25
whose volume cannot be measured at all, and
00:32:28
measurability is a very important property for
00:32:31
many theories, including those those
00:32:34
close to practice, such as
00:32:36
differential equations of intuition,
00:32:39
nesty saw the problem in the fact that the axiom
00:32:41
does not explicitly indicate the methods of choice and the
00:32:44
set of elements, it simply says
00:32:47
that there is a method,
00:32:48
although it is impossible to detect it due to
00:32:51
infinity, because of this, it is impossible to explicitly somehow
00:32:55
indicate a way to completely order
00:32:58
for the set of real numbers or
00:33:00
present at least one
00:33:03
immeasurable set or a way to specifically
00:33:05
divide a ball
00:33:06
to get 2, however,
00:33:08
Frenkel notes that the axiom of choice is
00:33:11
still important,
00:33:12
for example, it is important in mathematical
00:33:15
analysis when we use the
00:33:17
definitions of continuity both through the
00:33:19
epsilon-delta language and through limits
00:33:22
then this is a direct consequence of the axiom of choice and
00:33:25
its rejection promises great confusion and not
00:33:29
only in mathematics because by 1930 z fc
00:33:33
became a classical axiomatics and
00:33:36
became the basis for the work of a group of
00:33:38
French mathematicians working under the
00:33:41
pseudonym Nicolas Bourbaki Bourba
00:33:43
cysts were guided by zfc
00:33:47
and logic to show that all
00:33:49
mathematics can indeed be built
00:33:52
strictly from these axioms; their principles
00:33:55
combined the multiple theorist
00:33:57
and formalist approach; it must be said that,
00:34:00
unlike the term
00:34:02
and Frenkel, or the fishermen, becoming logic was
00:34:05
just another list of
00:34:07
axioms; they said that logic in
00:34:09
mathematics is nothing more than the grammar of the language
00:34:12
they use mathematics, the
00:34:15
subsequent development of mathematics may
00:34:17
require modifications to logic, which by the
00:34:21
way has been confirmed at the moment,
00:34:23
there are many different logics, including
00:34:26
low intuition, and one must understand that
00:34:29
from the very first law of logic it follows that no
00:34:31
other
00:34:33
alternative logics can
00:34:36
exist, in principle, Barba cysts
00:34:38
calmly used the actual
00:34:40
infinity axioms choice and the
00:34:43
law of exclusion 3
00:34:45
they were not too concerned about
00:34:47
the consistency of mathematics, they were simply
00:34:49
engaged in trying to express all
00:34:52
mathematics in a
00:34:53
certain formal style that
00:34:56
involved updated
00:34:57
axiomatics, showing that
00:35:01
mathematics actually comes from it chapter 4
00:35:06
second wave of crisis consistency
00:35:09
and completeness, although the products were
00:35:12
defeated by the question about the proof of
00:35:15
consistency still remained that the fact
00:35:18
that known products were eliminated does not
00:35:21
mean that they were not hidden somewhere else. In
00:35:23
addition to consistency, Gilbert drew
00:35:26
attention to one more property
00:35:28
in which mathematicians were convinced:
00:35:31
completeness, completeness means that in the
00:35:35
axiom system, any statement that makes sense for an axiom is
00:35:38
either true
00:35:40
or incorrect and the third was not given for
00:35:44
mathematicians of that time, it was an
00:35:46
intuitive understanding of how
00:35:49
mathematics and logic work in general; every clearly
00:35:52
formulated question should have an
00:35:55
answer; however, at that moment there were
00:35:58
quite a lot of riddles that
00:36:01
remained unanswered.
00:36:02
By the way, based on these riddles, a
00:36:05
list of Hilbert’s problems was created among of them,
00:36:08
for example, the continuum hypothesis, the
00:36:10
Riemann hypothesis, and other leagues of cysts
00:36:14
and representatives of the theorist of the
00:36:15
multiple approach were convinced of
00:36:17
completeness and consistency, but
00:36:20
could not prove them to intuition, they did not
00:36:23
take this seriously at all,
00:36:25
considering that intuition is
00:36:27
consistent and strong enough
00:36:30
to determine the truth or falsity of
00:36:33
any meaningful
00:36:36
formalists, led by Hilbert, were engaged in solving these problems;
00:36:39
they managed to construct somewhat
00:36:41
artificial arithmetic and prove its
00:36:44
completeness and consistency,
00:36:46
as well as make proofs for
00:36:48
other small
00:36:50
axiomatics, but there was not a single
00:36:52
result for serious powerful
00:36:55
axiomatics; the
00:36:56
first significant result was obtained by the
00:36:58
Austrian mathematician Kurt Gödel,
00:37:00
namely he proved the completeness and
00:37:02
consistency of first-
00:37:04
order logic or predicate logic, this logic
00:37:08
still remains the main one
00:37:10
in logic for mathematics, the formalist had already
00:37:13
opened champagne and were going to
00:37:15
celebrate, but Gödel’s next work
00:37:18
was a shock not only for pharma list of
00:37:21
but for all mathematicians and philosophers and
00:37:24
this generation,
00:37:25
his work contained two results: 1,
00:37:29
the consistency of any sufficiently
00:37:31
powerful mathematical system like
00:37:34
arithmetic or set theory
00:37:37
cannot be established by means of
00:37:39
this system itself,
00:37:41
strangely enough, although a contradiction
00:37:43
can follow from the theory, the absence of
00:37:46
contradictions does not depend on it, the second
00:37:49
result, which turned out to be more
00:37:51
sudden, incompleteness the
00:37:54
case of consistency of a powerful
00:37:57
mathematical theory is also not complete,
00:37:59
in other words, if the theory does not have
00:38:03
contradictions, then there is a formula
00:38:05
that is meaningful for the theory but is not
00:38:08
provable in it, and adding this formula
00:38:11
as an axiom
00:38:13
does not solve the problem, the theory simply
00:38:16
acquires a new unprovable formula and the formula
00:38:19
Godlev pair of theorems has become one of
00:38:22
very counter-intuitive facts not only to
00:38:25
mathematics
00:38:26
but also to logic and thinking as such, and
00:38:29
philosophers of logic and mathematics talk about it
00:38:31
until now, first of all,
00:38:34
the most powerful blow was dealt to the
00:38:37
views of Gilbert and his school, however,
00:38:40
other schools suffered as
00:38:42
mathematicians had no doubt that at
00:38:44
least axiomatic in mathematics you can
00:38:48
do full of
00:38:49
hips; after the publication of the theorems, he became a
00:38:52
celebrity not only in mathematics
00:38:54
but also in science in general; by the way, his friend
00:38:57
was Albert Einstein; the most important conclusion
00:39:01
follows from the theorems of the maximal
00:39:04
method; such a good and reliable method is
00:39:07
still definitely there some
00:39:09
irremovable disadvantages
00:39:11
can be noted that the theorems on incompleteness are to
00:39:14
some extent denied by the law of the
00:39:16
excluded middle;
00:39:18
no, of course, the classical
00:39:20
logic of predicates was used where this law and,
00:39:23
however, when adding a system of
00:39:26
axioms to it, it turns out that a statement
00:39:29
can be not only true or false,
00:39:32
but also take other meanings
00:39:34
vale did not stop there and in
00:39:37
the forties he proved that the negation of the
00:39:40
continuum hypothesis is unprovable z
00:39:42
function if z fc is consistent then
00:39:46
in the sixty-third Paul Joseph Kony
00:39:49
proved that this hypothesis is not provable,
00:39:52
which means that the hypothesis does not depend on the
00:39:56
system axiom z and in c and you can safely
00:39:59
build an axiomatic y with both a positive
00:40:02
answer to this question and a
00:40:04
negative one, and in both cases you can build
00:40:06
your own system of concepts,
00:40:09
which by the way was done; it was also
00:40:12
shown that the axiom is independent of
00:40:15
choice in the axiom of cer milo Frenkel,
00:40:18
which allows you to replace the
00:40:20
axiom of choice with another without problems with
00:40:23
inconsistency, in the sixty-second year,
00:40:25
Polish mathematicians proposed to the rural and
00:40:28
barbell ties the axiom of
00:40:31
determinism of the Iliad,
00:40:33
it is formulated much more complex than the
00:40:35
axiom of choice using a
00:40:37
theorist of game concepts,
00:40:39
but somewhat more specifically
00:40:42
defining the concepts of choice, this axiom,
00:40:45
although more complex in formulation, gives a
00:40:48
very good consequence
00:40:50
when adding south z with the axiom of the
00:40:53
love ring term or without the axiom of choice the new
00:40:56
axiomatic y is called z fd z
00:40:59
fd the term theorem is not true there is no
00:41:02
doubling of the Tarski ball there
00:41:04
are no immeasurable sets and the
00:41:07
continuum hypothesis is correct except that almost
00:41:10
all classical mathematical analysis
00:41:13
remains in place the
00:41:15
difference is mainly in topology and also
00:41:18
in the impossibility of constructing a cool
00:41:20
non-standard analysis and of course z ft is not
00:41:24
without the shortcomings discovered by the module, has it been
00:41:26
accepted, well, it has aroused interest and a
00:41:30
certain community of mathematicians is developing, however,
00:41:33
it cannot be said that it has become
00:41:35
more popular than z fc, it’s just that now a mathematician
00:41:39
can say that, for example, an
00:41:41
immeasurable set exists but not dat is
00:41:44
armed with it, that is, it does not correspond to the
00:41:47
more specific axiom
00:41:49
of determinism, in addition to z and sheep and z
00:41:53
fd, there are other ax mother and zations
00:41:56
one of them was developed by Neumann Bernays
00:41:59
and Hidel this is an extended version of the
00:42:02
axiom
00:42:03
called Angie its advantage is that
00:42:06
it includes z fc
00:42:09
plus allows you to work with sets of
00:42:12
all sets without the risk of paradoxes, they
00:42:16
simply prohibit them, finally the period from 22
00:42:20
to 30, the fourth year, Leuven Game and Skule
00:42:23
Moms showed another unexpected
00:42:26
result, it turns out that if we are going to
00:42:29
describe an axiomatic specific
00:42:31
mathematical model,
00:42:33
we will not succeed any axiomatics
00:42:36
leaves room for non-covalent
00:42:39
interpretations,
00:42:40
no matter how hard we try to narrow and define their scope.
00:42:43
Previously, mathematicians believed that
00:42:46
any interpretation of one axiomatics is
00:42:49
actually a wrinkle of the other, that is,
00:42:52
they differ only in concepts, but no
00:42:55
judge, it turned out that the interpretation of the theory
00:42:58
can be very unpredictable,
00:43:01
doomed to this
00:43:02
indefinable concepts that exist in
00:43:05
any axiomatics of force in
00:43:07
Newtonian mechanics.
00:43:09
and straight lines in Euclidean geometry and the
00:43:12
like, we can say that at that moment,
00:43:14
for many mathematicians, this theorem
00:43:16
finally buried the axiomatic
00:43:19
approach because it left no visible way
00:43:22
out of the situation; he came with a
00:43:24
sack a name, he wrote this
00:43:27
circumstance, it
00:43:29
seems to me, in favor of intuition nismo
00:43:32
chapter 5 modern mathematics at the
00:43:36
moment set theory
00:43:38
has ceased to be the most unifying theory, a
00:43:41
more general new theory has appeared,
00:43:44
category theory, a path that opened a
00:43:47
general algebra and the mentioned
00:43:49
and g, despite all the upheavals,
00:43:52
axiomatics have not gone anywhere and
00:43:54
are still a fairly classical and
00:43:57
reliable tool, but mathematics has not
00:44:00
been reduced to the axiomatic am of classical
00:44:02
set theory
00:44:03
remains zfc although axioms are used
00:44:07
when controversial and ambiguous
00:44:10
questions arise, in general, most theories not
00:44:12
directly related to the
00:44:14
research of set theory itself
00:44:16
continue to use naive
00:44:19
set theory, adjusted for the lack of
00:44:21
self-reflection and other problematic
00:44:24
issues, rigid positions have been replaced
00:44:27
a certain pluralism of different
00:44:29
views and systems of mathematicians intuition
00:44:32
became the main part of constructive
00:44:34
mathematics with the advent of computer
00:44:36
technology it became clear that
00:44:38
the study of such objects is very
00:44:41
useful for algorithms because in
00:44:43
them it is precisely non-constructive objects that are of
00:44:46
no use formalism energy qism
00:44:49
also influenced computer
00:44:51
technology there are fewer contradictions between them, we
00:44:54
can say that mathematics
00:44:56
has divided into several branches, that although it
00:44:59
comes from different premises, they are
00:45:02
nevertheless interconnected,
00:45:04
the search for unshakable foundations has led to the fact
00:45:07
that mathematics is as changeable as
00:45:10
other sciences,
00:45:11
old proofs may become
00:45:13
irrelevant, too not strict, they may be
00:45:17
born new theories that were not
00:45:19
previously assumed knowledge may
00:45:22
become obsolete some general inviolable
00:45:24
laws become special cases
00:45:27
therefore the foundations cannot remain
00:45:30
in place and it is impossible to find some
00:45:33
inviolable list of rules for all times
00:45:36
modern mathematics has leaned towards
00:45:38
convenience and the practical result is the breakdown of
00:45:41
mathematicians and has occurred in fact, where
00:45:44
in the natural sciences people previously
00:45:46
believed that they were working on indestructible
00:45:49
truths, but now they realized that their
00:45:51
work rather concerns various models,
00:45:55
except that for mathematicians such a blow
00:45:57
was more unexpected because it
00:46:00
seemed that it was possible to restore order in one’s head once and
00:46:03
for all, it
00:46:06
was possible think that the shock
00:46:09
had a bad effect on mathematical discoveries,
00:46:11
but no, the sixties began a new renaissance in mathematics,
00:46:16
which perhaps continues to this day, a
00:46:19
huge number of very different theories of both
00:46:22
applied and fundamental
00:46:24
significance began to appear by leaps and bounds;
00:46:27
mathematics became even more complex,
00:46:29
even richer; perhaps it was precisely this motivation
00:46:33
for development and progress
00:46:35
is the main merit of those people who
00:46:38
decided to understand such foundations of
00:46:41
mathematics, although the attempt was not entirely
00:46:44
successful. Anyone who wants to get acquainted with this topic, in
00:46:47
more detail in the description there will be a link to
00:46:49
additional explanations for this issue, do
00:46:51
not forget to like,
00:46:53
write comments and click on the
00:46:55
bell if you want to support my
00:46:58
channel on patreon or in other ways
00:47:00
the link is also in the description and I also wish
00:47:04
you happiness and proportional mass and yes yes yes
00:47:06
see you soon
00:47:10
[music]
00:47:20
[music]
00:47:28
my name is David Hilbert
00:47:30
I make a list of problems my name is Kurt
00:47:32
Gödel
00:47:33
I prove incompleteness theorems

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