текст 2 (английский) магистратура матфак ВГУ - методичка

The development of formal logic, together with concerns that mathematics
had not been built on a proper foundation. led to the development of axiom
systems for fundamen1al areas of mathematics such as arithmetic. analysis, and
geometry.
In logic, the term "arithmetic" refers to the theory of the natural numbers.
Giuseppe Peano ( 1888) published a set of axioms for arithmetic that came to
bear his name, using a variation of the logical system of Boole and Schroder but
adding quantifiers. Peano was unaware of Frege's work at the time. Around the
same time Richard Dedekind showed that the natural numbers are uniquely
characterized by their induct ion properties. Dedekind ( 1888) proposed a
different characterization. which lacked the formal logical character or Peano's
axioms. Dedekind's work, however, proved theorems inaccessible in Peano's
system, including the uniqueness of the set of natural numbers (up to
isomorphism) and the recursive definitions of addition and multiplication from
the successor function and mathematical induction.

In the mid-19th century. flaws in Euclid's axioms for geometry became
known (Katz 1998, p. 774). In addition to the independence of the parallel
postulate, established by Nikolai Lobachevsky in 1826 (Lobachevsky 1840).
mathematicians discovered that certain theorems taken for granted by Euclid
were not in fact provable from his axioms. Among these is the theorem that a
line contains at least two points or that circles of the same radius whose centers
are separated by that radius must intersect. Hilben ( 1899) developed a complete
set of axioms for geometry. building on previous work by Pasch ( 1882). The
success in axiomatizing geometry motivated Hilbert to seek complete
axiomatizations of other areas of mathematics, such as the natural numbers and
the real line. This would prove to be a major area o f research in the first half of'
the 20th century.

The 19th century saw great advances in the theory of real analysis. including
theories of convergence of functions and Fourier series. Mathematicians such as
Korl Weierstrass began to construct functions that stretched intuition. such as
nowhere-differentiable continuous functions. Previous conceptions of a function
as a rule for computation, or a smooth graph, were no longer adequate.
Weierstrass began to advocate the arithmetization of analysis. which sought to
axiomatize analysis using properties of the natural numbers. The modern «эпсилон-сигма»
definition of limits and continuous functions was developed by Bolzano und
Cauchy between 1817 and 1823 (Felscher 2000). In 1858, Dedekind proposed a
definition of the real numbers in tem1s of Dedekind cuts of rational numbers
(Dedekind 1872). a definition still employed in contemporary texts.

Georg Cantor developed the fundamental concepts of infinite set theory. His
early results developed the theory of cardinality and proved that the reals and the
natural numbers have different cardinalities (Cantor 1874). Over the next twenty
years, Cantor developed a theory or transfinite numbers in a series of
publications. In 1891. he published a new proof of the uncountability o f the real
numbers that introduced the diagonal argument. and used this method to prove
Cantor's theorem that no set can have the same cardinality as its powerset.
Cantor believed that every set could be well-ordered. but was unable to produce
a proof for this result. leaving it as an open problem in 1895 (Katz 1998, p. 807).

In the early decades of the 20th century. the main areas of study were set
theory and formal logic. The discovery of paradoxes in informal set theory
caused some to wonder whether mathematics itself is inconsistent , and to look
for proofs of consistency.

In 1900. Hilbert posed a famous list of 23 problems for the next cenl11 ry. The
first two of these were to resolve the continuum hypothesis, and prove the
consistency or elementary arithmetic, respectively: the tenth was to produce a
method that could decide whether a multivariate polynomial equation over the
integers has a solution. Subsequent work to resolve these problems shaped the
direction of mathematical logic. as did the effort to resolve Hilbert's
" Entschuldigungs program". posed in 1928. This problem asked for a procedure
that would decide, given a formalized mathematical statement. whether the
statement is true or false.