Heinz-Dieter Ebbinghaus is a German mathematician and logician. Ebbinghaus wrote various books on logic, set theory and model theory, including a seminal work on Ernst Zermelo, his book Einführung in die mathematische Logik, joint work with Jörg Flum and Wolfgang Thomas, first appeared in 1978 and became a standard textbook of mathematical logic in the German-speaking area. It is in its sixth edition. An English edition Mathematical logic was published in the Springer-Verlag Undergraduate Texts in Mathematics series in 1984. Heinz-Dieter Ebbinghaus, Volker Peckhaus. Ernst Zermelo: An Approach to His Life and Work, 2007, ISBN 3-642-08050-2. Heinz-Dieter Ebbinghaus, Jörg Flum. Finite Model Theory, 2005, ISBN 3-540-28787-6. Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas. Einführung in die mathematische Logik, five editions since 1978. Home page
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He created set theory. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities", he defined their arithmetic. Cantor's work is of a fact he was well aware of. Cantor's theory of transfinite numbers was regarded as so counter-intuitive – shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed; some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected.
The objections to Cantor's work were fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense", "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder; the harsh criticism has been matched by accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.
David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created." Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist, his grandfather Franz Böhm was a well-known soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg stock exchange. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt. In 1862, Cantor entered the Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer, he spent the summer of 1866 at the University of Göttingen and a center for mathematical research. Cantor was a good student, he received his doctorate degree in 1867. Cantor submitted his dissertation on number theory at the University of Berlin in 1867.
After teaching in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory, which he presented in 1869 upon his appointment at Halle University. In 1874, Cantor married Vally Guttmann, they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for. Kronecker, who headed mathematics at Berlin until his death in 1891, became uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.
Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work since he intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, it involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him to leave Halle. In 1881, Cantor's Halle colleague Eduard Heine died. Halle acc
Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university entrance examinations in 1905, he entered Det Kongelige Frederiks Universitet to study mathematics taking courses in physics, chemistry and botany. In 1909, he began working as an assistant to the physicist Kristian Birkeland, known for bombarding magnetized spheres with electrons and obtaining aurora-like effects. In 1913, Skolem passed the state examinations with distinction, completed a dissertation titled Investigations on the Algebra of Logic, he traveled with Birkeland to the Sudan to observe the zodiacal light. He spent the winter semester of 1915 at the University of Göttingen, at the time the leading research center in mathematical logic and abstract algebra, fields in which Skolem excelled. In 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet.
In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters. Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities, his notional thesis advisor was Axel Thue though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold. Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. Michelsen Institute in Bergen; this senior post allowed Skolem to conduct research free of administrative and teaching duties. However, the position required that he reside in Bergen, a city which lacked a university and hence had no research library, so that he was unable to keep abreast of the mathematical literature. In 1938, he returned to Oslo to assume the Professorship of Mathematics at the university.
There he taught the graduate courses in algebra and number theory, only on mathematical logic. Skolem's Ph. D. student Øystein Ore went on to a career in the USA. Skolem served as president of the Norwegian Mathematical Society, edited the Norsk Matematisk Tidsskrift for many years, he was the founding editor of Mathematica Scandinavica. After his 1957 retirement, he made several trips to the United States and teaching at universities there, he remained intellectually active until his unexpected death. For more on Skolem's academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory, most of all, set theory and mathematical logic, he published in Norwegian journals with limited international circulation, so that his results were rediscovered by others. An example is the Skolem -- Noether theorem. Skolem published a proof in 1927. Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by n elements.
In 1919, he showed that every implicative lattice is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse'Verbände' oder'Lattices'", surveying his earlier work in lattice theory. Skolem was a pioneer model theorist. In 1920, he simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model it has a countable model, his 1920 proof employed the axiom of choice, but he gave proofs using Kőnig's lemma in place of that axiom. It is notable that Skolem, like Löwenheim, wrote on mathematical logic and set theory employing the notation of his fellow pioneering model theorists Charles Sanders Peirce and Ernst Schröder, including ∏, ∑ as variable-binding quantifiers, in contrast to the notations of Peano, Principia Mathematica, Principles of Mathematical Logic.
Skolem pioneered the construction of non-standard models of set theory. Skolem refined Zermelo's axioms for set theory by replacing Zermelo's vague notion of a "definite" property with any property that can be coded in first-order logic; the resulting axiom is now part of the standard axioms of set theory. Skolem pointed out that a consequence of the Löwenheim–Skolem theorem is what is now known as Skolem's paradox: If Zermelo's axioms are consistent they must be satisfiable within a countable domain though they prove the existence of uncountable sets; the completeness of first-order logic is an easy corollary of results Skolem proved in the early 1920s and discussed in Skolem, but he failed to note this fact because mathematicians and logicians did not become aware of completeness as a fundamental metamathematical problem until the 1928 first edition of Hilbert and Ackermann's Principles of Mathematical Logic articulated it. In any event, Kurt Gödel first proved this completeness in 1930. Skolem was one of the founders of finitism in mathematics.
Skolem sets out his primitive recursive arithm
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of one object from each bin if the collection is infinite. Formally, it states that for every indexed family i ∈ I of nonempty sets there exists an indexed family i ∈ I of elements such that x i ∈ S i for every i ∈ I; the axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, such a selection can be made without invoking the axiom of choice. An illustrative example is sets picked from the natural numbers. From such sets, one may always select the smallest number, e.g. in the smallest elements are. In this case, "select the smallest number" is a choice function. If infinitely many sets were collected from the natural numbers, it will always be possible to choose the smallest element from each set to produce a set.
That is, the choice function provides the set of chosen elements. However, no choice function is known for the collection of all non-empty subsets of the real numbers. In that case, the axiom of choice must be invoked. Bertrand Russell coined an analogy: for any collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection. For an infinite collection of pairs of socks, there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice. Although controversial, the axiom of choice is now used without reservation by most mathematicians, it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice. One motivation for this use is that a number of accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced. A choice function is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f is an element of A. With this concept, the axiom can be stated: Formally, this may be expressed as follows: ∀ X. Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no choice function; each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; the axiom of choice asserts the existence of such elements. In this article and other discussions of the Axiom of Choice the following abbreviations are common: AC – the Axiom of Choice. ZF – Zermelo–Fraenkel set theory omitting the Axiom of Choice.
ZFC – Zermelo–Fraenkel set theory, extended to include the Axiom of Choice. There are many other equivalent statements of the axiom of choice; these are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains one element in common with each of the sets in X; this guarantees for any partition of a set X the existence of a subset C of X containing one element from each part of the partition. Another equivalent axiom only considers collections X that are powersets of other sets: For any set A, the power set of A has a choice function. Authors who use this formulation speak of the choice function on A, but be advised that this is a different notion of choice function, its domain is the powerset of A, and
Helsinki is the capital and most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of Uusimaa in southern Finland, has a population of 650,058; the city's urban area has a population of 1,268,296, making it by far the most populous urban area in Finland as well as the country's most important center for politics, finance and research. Helsinki is located 80 kilometres north of Tallinn, Estonia, 400 km east of Stockholm, 390 km west of Saint Petersburg, Russia, it has close historical ties with these three cities. Together with the cities of Espoo and Kauniainen, surrounding commuter towns, Helsinki forms the Greater Helsinki metropolitan area, which has a population of nearly 1.5 million. Considered to be Finland's only metropolis, it is the world's northernmost metro area with over one million people as well as the northernmost capital of an EU member state. After Stockholm and Oslo, Helsinki is the third largest municipality in the Nordic countries.
The city is served by the international Helsinki Airport, located in the neighboring city of Vantaa, with frequent service to many destinations in Europe and Asia. Helsinki was the World Design Capital for 2012, the venue for the 1952 Summer Olympics, the host of the 52nd Eurovision Song Contest. Helsinki has one of the highest urban standards of living in the world. In 2011, the British magazine Monocle ranked Helsinki the world's most liveable city in its liveable cities index. In the Economist Intelligence Unit's 2016 liveability survey, Helsinki was ranked ninth among 140 cities. According to a theory presented in the 1630s, settlers from Hälsingland in central Sweden had arrived to what is now known as the Vantaa River and called it Helsingå, which gave rise to the names of Helsinge village and church in the 1300s; this theory is questionable, because dialect research suggests that the settlers arrived from Uppland and nearby areas. Others have proposed the name as having been derived from the Swedish word helsing, an archaic form of the word hals, referring to the narrowest part of a river, the rapids.
Other Scandinavian cities at similar geographic locations were given similar names at the time, e.g. Helsingør in Denmark and Helsingborg in Sweden; when a town was founded in Forsby village in 1548, it was named Helsinge fors, "Helsinge rapids". The name refers to the Vanhankaupunginkoski rapids at the mouth of the river; the town was known as Helsinge or Helsing, from which the contemporary Finnish name arose. Official Finnish Government documents and Finnish language newspapers have used the name Helsinki since 1819, when the Senate of Finland moved itself into the city from Turku; the decrees issued in Helsinki were dated with Helsinki as the place of issue. This is; as part of the Grand Duchy of Finland in the Russian Empire, Helsinki was known as Gelsingfors in Russian. In Helsinki slang, the city is called Stadi. Hesa, is not used by natives of the city. Helsset is the Northern Sami name of Helsinki. In the Iron Age the area occupied by present day Helsinki was inhabited by Tavastians, they used the area for fishing and hunting, but due to a lack of archeological finds it is difficult to say how extensive their settlements were.
Pollen analysis has shown that there were cultivating settlements in the area in the 10th century and surviving historical records from the 14th century describe Tavastian settlements in the area. Swedes colonized the coastline of the Helsinki region in the late 13th century after the successful Second Crusade to Finland, which led to the defeat of the Tavastians. Helsinki was established as a trading town by King Gustav I of Sweden in 1550 as the town of Helsingfors, which he intended to be a rival to the Hanseatic city of Reval. In order to populate his newly founded town, the King issued an order to resettle the bourgeoisie of Porvoo, Ekenäs, Rauma and Ulvila into the town. Little came of the plans as Helsinki remained a tiny town plagued by poverty and diseases; the plague of 1710 killed the greater part of the inhabitants of Helsinki. The construction of the naval fortress Sveaborg in the 18th century helped improve Helsinki's status, but it was not until Russia defeated Sweden in the Finnish War and annexed Finland as the autonomous Grand Duchy of Finland in 1809 that the town began to develop into a substantial city.
Russians besieged the Sveaborg fortress during the war, about one quarter of the town was destroyed in an 1808 fire. Russian Emperor Alexander I of Russia moved the Finnish capital from Turku to Helsinki in 1812 to reduce Swedish influence in Finland, to bring the capital closer to Saint Petersburg. Following the Great Fire of Turku in 1827, the Royal Academy of Turku, which at the time was the country's only university, was relocated to Helsinki and became the modern University of Helsinki; the move helped set it on a path of continuous growth. This transformation is apparent in the downtown core, rebuilt in the neoclassical style to resemble Saint Petersburg to a plan by the German-born architect C. L. Engel; as elsewhere, technological advancements such as railroads and industrialization were key factors behind the city's growth. Despite the tumultuous nature of Finnish history during the first half of the 20th century, Helsinki continued its steady development. A landmark e