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First published Tue Apr 10, 2007; substantive revision Fri Jul 1, 2016 Set theory is one of the greatest achievements of modern mathematics. Basically all mathematical concepts, methods, and results admit ofrepresentation within axiomatic set theory.

Thus set theory has servedquite a unique role by systematizing modern mathematics, andapproaching in a unified form all basic questions about admissiblemathematical arguments—including the thorny question ofexistence principles.

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In 1910, Hilbert wrote that set theory is that mathematical discipline which today occupies an outstanding rolein our science, and radiates ausstr mt its powerfulinfluence into all branches of mathematics Compile a list of the very best presentation tech tools to use with upper The students can write on the screen with Teachers can set up their classes Also, if students are already familiar with PowerPoint, they can create slides on directions are inside our class' section of the website and all of their stories are then .

Hilbert 1910, 466; mytranslation This already suggests that, in order to discuss the early history, itis necessary to distinguish two aspects of set theory: its role as afundamental language and repository of the basic principles of modernmathematics; and its role as an independent branch of mathematics,classified (today) as a branch of mathematical logic.

The first section examines the origins and emergence of settheoretic mathematics around 1870; this is followed by a discussion ofthe period of expansion and consolidation of the theory up to1900.

Section 3 provides a look at the critical period in thedecade 1897 to 1908, and Section 4 deals with the time from Zermelo toG del (from theory to metatheory), with special attention to theoften overlooked, but crucial, descriptive set theory. Emergence The concept of a set appears deceivingly simple, at least to thetrained mathematician, and to such an extent that it becomes difficultto judge and appreciate correctly the contributions of the pioneers. What cost them much effort to produce, and took the mathematicalcommunity considerable time to accept, may seem to us ratherself-explanatory or even trivial.

Three historical misconceptions thatare widespread in the literature should be noted at the outset: It is not the case that actual infinity was universally rejectedbefore Cantor. Set-theoretic views did not arise exclusively from analysis, butemerged also in algebra, number theory, and geometry.

In fact, the rise of set-theoretic mathematics precededCantor’s crucial contributions. All of these points shall become clear in what follows.

The notion of a collection is as old as counting, and logical ideasabout classes have existed since at least the “tree ofPorphyry” (3rd century CE). Thus it becomes difficultto sort out the origins of the concept of set.

But sets are neithercollections in the everyday sense of this word, nor“classes” in the sense of logicians before themid-19th century. Ernst Zermelo, a crucial figure in ourstory, said that the theory had historically been “created byCantor and Dedekind” Zermelo 1908, 262 .

This suggests a goodpragmatic criterion: one should start from authors who havesignificantly influenced the conceptions of Cantor, Dedekind, andZermelo. For the most part, this is the criterion adopted here.

Nevertheless, as every rule calls for an exception, the case ofBolzano is important and instructive, even though Bolzano did notsignificantly influence later writers. In 19th century German-speaking areas, there were someintellectual tendencies that promoted the acceptance of the actualinfinite (e.

In spite ofGauss’s warning that the infinite can only be a manner ofspeaking, some minor figures and three major ones (Bolzano, Riemann,Dedekind) preceded Cantor in fully accepting the actual infinite inmathematics. Those three authors were active in promoting aset-theoretic formulation of mathematical ideas, with Dedekind’scontribution in a good number of classic writings (1871, 1872,1876/77, 1888) being of central importance.

Chronologically, Bernard Bolzano was the first, but he exerted almostno influence. The high quality of his work in logic and thefoundations of mathematics is well known.

A book entitled Paradoxien des Unendlichen was posthumously published in1851. Here Bolzano argued in detail that a host of paradoxessurrounding infinity are logically harmless, and mounted a forcefuldefence of actual infinity.

He proposed an interesting argumentattempting to prove the existence of infinite sets, which bearscomparison with Dedekind’s later argument (1888). Although heemployed complicated distinctions of different kinds of sets orclasses, Bolzano recognized clearly the possibility of putting twoinfinite sets in one-to-one correspondence, as one can easily do,e.

, with the intervals $$0, 5$$ and $$0, 12$$ by the function$$5y = 12x$$.

However, Bolzano resisted the conclusion that both setsare “equal with respect to the multiplicity of theirparts” 1851, 30–31 . In all likelihood, traditional ideas ofmeasurement were still too powerful in his way of thinking, and thushe missed the discovery of the concept of cardinality.

The case of Bolzano suggests that a liberation from metric concepts(which came with the development of theories of projective geometryand especially of topology) was to have a crucial role in makingpossible the abstract viewpoint of set theory. Bernhard Riemannproposed visionary ideas about topology, and about basing all ofmathematics on the notion of set or “manifold” in thesense of class (Mannigfaltigkeit), in his celebratedinaugural lecture “On the Hypotheses which lie at theFoundations of Geometry” (1854/1868a).

Also characteristic ofRiemann was a great emphasis on conceptual mathematics,particularly visible in his approach to complex analysis (which againwent deep into topology). To give but the simplest example, Riemannwas an enthusiastic follower of Dirichlet’s idea that a functionhas to be conceived as an arbitrary injective correspondence betweennumerical values, be it representable by a formula or not; this meantleaving behind the times when a function was defined to be an analyticexpression.

Through this new style of mathematics, and through hisvision of a new role for sets and a full program for developingtopology, Riemann was a crucial influence on both Dedekind andCantor. The five-year period 1868–1872 saw a mushrooming ofset-theoretic proposals in Germany, so much so that we could regard itas the birth of set-theoretic mathematics.

Riemann’s geometrylecture, delivered in 1854, was published by Dedekind in 1868, jointlywith Riemann’s paper on trigonometric series (1854/1868b, whichpresented the Riemann integral). The latter was the starting point fordeep work in real analysis, commencing the study of“seriously” discontinuous functions.

The young GeorgCantor entered into this area, which led him to the study ofpoint-sets. In 1872 Cantor introduced an operation upon point sets(see below) and soon he was ruminating about the possibility toiterate that operation to infinity and beyond: it was the firstglimpse of the transfinite realm.

Meanwhile, another major development had been put forward by RichardDedekind in 1871. In the context of his work on algebraic numbertheory, Dedekind introduced an essentially set-theoretic viewpoint,defining fields and ideals of algebraic numbers.

These ideas werepresented in a very mature form, making use of set operations and ofstructure-preserving mappings (see a relevant passage inFerreir s 1999: 92–93; Cantor employed Dedekind’sterminology for the operations in his own work on set theory around1880 1999: 204 ). Considering the ring of integers in a given fieldof algebraic numbers, Dedekind defined certain subsets called“ideals” and operated on these sets as new objects.

Thisprocedure was the key to his general approach to the topic. In otherworks, he dealt very clearly and precisely with equivalence relations,partition sets, homomorphisms, and automorphisms.

Thus, many of theusual set-theoretic procedures of twentieth-century mathematics goback to his work. Several years later (in 1888), Dedekind wouldpublish a presentation of the basic elements of set theory, makingonly a bit more explicit the operations on sets and mappings he hadbeen using since 1871.

The following year, Dedekind published a paper 1872 in which heprovided an axiomatic analysis of the structure of the set$$\mathbf R$$ of real numbers. He defined it as an ordered fieldthat is also complete (in the sense that all Dedekind-cuts on$$\mathbf R$$ correspond to an element in $$\mathbf R$$);completeness in that sense has the Archimedean axiom as a consequence.

Cantor too provided a definition of $$\mathbf R$$ in 1872,employing Cauchy sequences of rational numbers, which was an elegantsimplification of the definition offered by Carl Weierstrass in hislectures. The form of completeness axiom that Weierstrass preferredwas Bolzano’s principle that a sequence of nested closedintervals in $$\mathbf R$$ (a sequence such that$$a m+1 ,b m+1 \subset a m ,b m$$) “contains” atleast one real number (or, as we would say, has a non-emptyintersection).

The Cantor and Dedekind definitions of the real numbers reliedimplicitly on set theory, and can be seen in retrospect to involve theassumption of a Power Set principle. Both took as given the set ofrational numbers, and for the definition of $$\mathbf R$$ theyrelied on a certain totality of infinite sets of rational numbers(either sequences, or Dedekind cuts). With this, too, constructivisticcriticism of set theory began to emerge, as Leopold Kronecker startedto make objections to such infinitary procedures.

Simultaneously,there began a study of the topology of $$\mathbf R$$, inparticular in the work of Weierstrass, Dedekind, and Cantor.

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Cantor’s derived sets are of particular interest (for thecontext of this idea in real analysis, see e.

, Dauben 1979, Hallett1984, Lavine 1994, Kanamori 1996, Ferreir s 1999).

Cantor tookas given the “conceptual sphere” of the real numbers, andhe considered arbitrary subsets $$P$$, which he called‘point sets’. A real number $$r$$ is called a limit point of $$P$$, when all neighbourhoods of$$r$$ contain points of $$P$$.

With that concept, introduced by Weierstrass,the derived set $$P'$$ of $$P$$ can be defined, asCantor did, to be the set of all the limit points of$$P$$.

In general $$P'$$ may be infinite and have its own limitpoints. Thus one can iterate the operation and obtain further derivedsets $$P''$$, $$P'''$$… $$P (n)$$ … It is easy to giveexamples of a set $$P$$ that will give rise to non-empty derivedsets $$P (A rather trivial exampleis \(P = \mathbf Q 0,1$$, the set of rational numbers in the unitinterval; in this case $$P' = 0,1 = P''$$. This was Cantor’s first encounter with transfiniteiterations. Then, in late 1873, came a surprising discovery that fully opened therealm of the transfinite.

In correspondence with Dedekind (see Ewald1996, vol. 2), Cantor asked the question whether the infinite sets$$\mathbf N$$ of the natural numbers and $$\mathbf R$$ ofreal numbers can be placed in one-to-one correspondence.

In reply,Dedekind offered a surprising proof that the set $$A$$ of allalgebraic numbers is denumerable (i. , there is a one-to-onecorrespondence with $$\mathbf N$$). A few days later, Cantor wasable to prove that the assumption that $$\mathbf R$$ isdenumerable leads to a contradiction.

To this end, he employed theBolzano-Weierstrass principle of completeness mentioned above. Thus hehad shown that there are more elements in $$\mathbf R$$ than in$$\mathbf N$$ or $$\mathbf Q$$ or $$A$$, in the precisesense that the cardinality of $$\mathbf R$$ is strictly greaterthan that of $$\mathbf N$$.

Consolidation Set theory was beginning to become an essential ingredient of the new“modern” approach to mathematics.

But this viewpoint wascontested, and its consolidation took a rather long time. Dedekind’s algebraic style only began to find followers in the1890s; David Hilbert was among them.

The soil was better prepared forthe modern theories of real functions: Italian, German, French andBritish mathematicians contributed during the 1880s. And the newfoundational views were taken up by Peano and his followers, by Fregeto some extent, by Hilbert in the 1890s, and later by Russell.

Meanwhile, Cantor spent the years 1878 to 1885 publishing key worksthat helped turn set theory into an autonomous branch of mathematics. Let’s write $$A \equiv B$$ in order to express that the two sets$$A$$, $$B$$ can be put in one-to-one correspondence (havethe same cardinality).

After proving that the irrational numbers canbe put in one-to-one correspondence with $$\mathbf R$$, and,surprisingly, that also $$\mathbf R n \equiv \mathbf R$$, Cantorconjectured in 1878 that any subset of $$\mathbf R$$ would beeither denumerable $$(\equiv \mathbf N )$$ or $$\equiv \mathbf R$$. This is the first and weakest form of the celebrated ContinuumHypothesis.

During the following years, Cantor explored the worldof point sets, introducing several important topological ideas (e. ,perfect set, closed set, isolated set), and arrived at results such asthe Cantor-Bendixson theorem. A point set $$P$$ is closed iff its derived set $$P'\subseteq P$$, and perfect iff $$P = P'$$.

TheCantor-Bendixson theorem then states that a closed point set can bedecomposed into two subsets $$R$$ and $$S$$, such that$$R$$ is denumerable and $$S$$ is perfect (indeed,$$S$$ is the $$a$$th derived set of $$P$$,for a countable ordinal $$a$$). Because of this, closed sets aresaid to have the perfect set property.

Furthermore, Cantor was able toprove that perfect sets have the power of the continuum (1884). Bothresults implied that the Continuum Hypothesis is valid for all closedpoint sets.

Many years later, in 1916, Pavel Aleksandrov and FelixHausdorff were able to show that the broader class of Borel sets havethe perfect set property too. His work on points sets led Cantor, in 1882, to conceive of the transfinite numbers (see Ferreir s 1999:267ff).

This was a turning point in his research, for fromthen onward he studied abstract set theory independently ofmore specific questions having to do with point sets and theirtopology (until the mid-1880s, these questions had been prominent inhis agenda). Subsequently, Cantor focused on the transfinite cardinaland ordinal numbers, and on general order types, independently of thetopological properties of $$\mathbf R$$.

The transfinite ordinals were introduced as new numbers in animportant mathematico-philosophical paper of 1883, Grundlageneiner allgemeinen Mannigfaltigkeitslehre (notice that Cantorstill uses Riemann’s term). Cantor defined them by means of two“generating principles”: the first (1) yields thesuccessor $$a+1$$ for any given number $$a$$, while the second(2) stipulates that there is a number $$b$$ which followsimmediately after any given sequence of numbers without a lastelement.

Thus, after all the finite numbers comes, by (2), the firsttransfinite number, $$\omega$$ (read: omega); and this is followed by$$\omega+1$$, $$\omega+2$$, …, $$\omega+\omega = \omega \cdot2$$, …, $$\omega \cdot n$$, $$\omega\cdot n +1$$, …,$$\omega 2$$, $$\omega 2 +1$$, …, $$\omega \omega$$,… and so on and on. Whenever a sequence without last elementappears, one can go on and, so to say, jump to a higher stage by(2).

The introduction of these new numbers seemed like idle speculation tomost of his contemporaries, but for Cantor they served two veryimportant functions. To this end, he classified the transfiniteordinals as follows: the “first number class” consisted ofthe finite ordinals, the set $$\mathbf N$$ of natural numbers;the “second number class” was formed by and allnumbers following it (including $$\omega \omega$$, and many more)that have only a denumerable set of predecessors. Thiscrucial condition was suggested by the problem of proving theCantor-Bendixson theorem (see Ferreir s 1995).

On that basis,Cantor could establish the results that the cardinality of the“second number class” is greater than that of$$\mathbf N$$; and that no intermediate cardinality exists 21 Nov 2015 - Welcome to Our Presentation. The following points are noted while writing a set. Sets are usually denoted by capital letters A, B, S, etc..

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After the second number class comes a “third number class”(all transfinite ordinals whose set of predecessors has cardinality$$\aleph 1$$); the cardinality of this new number class can beproved to be $$\aleph 2$$. The first function of thetransfinite ordinals was, thus, to establish a well-defined scale ofincreasing transfinite cardinalities. (The aleph notation used abovewas introduced by Cantor only in 1895.

) This made it possible toformulate much more precisely the problem of the continuum;Cantor’s conjecture became the hypothesis that$$\textit card (\mathbf R ) = \aleph 1$$. Furthermore, relying onthe transfinite ordinals, Cantor was able to prove theCantor-Bendixson theorem, rounding out the results on point sets thathe had been elaborating during these crucial years.

The study of the transfinite ordinals directed Cantor’sattention towards ordered sets, and in particular well-orderedsets. A set $$S$$ is well-ordered by a relation< iff< is a total order and every subset of $$S$$ has a leastelement in the<-ordering.

(The real numbers are not well-orderedin their usual order: just consider an open interval. Meanwhile,$$\mathbf N$$ is the simplest infinite well-ordered set.

) Cantorargued that the transfinite ordinals truly deserve the name of numbers, because they express the “type of order”of any possible well-ordered set. Notice also that it was easy forCantor to indicate how to reorder the natural numbers so as to makethem correspond to the order types $$\omega+1$$, $$\omega+2$$,…, $$\omega \cdot 2$$, …, $$\omega \cdot n$$, …,$$\omega (Forinstance, reordering \(\mathbf N$$ in the form: 2, 4, 6,…, 5, 15, 25, 35, …, 1, 3, 7, 9, … we obtain aset that has order type $$\omega\cdot 3$$. ) Notice too that the Continuum Hypothesis, if true, would entail thatthe set $$\mathbf R$$ of real numbers can indeed be well-ordered.

Cantor was so committed to this viewpoint, that he presented thefurther hypothesis that every set can be well-ordered as“a fundamental and momentous law of thought”. Some yearslater, Hilbert called attention to both the Continuum Hypothesis andthe well-ordering problem as Problem 1 in his celebrated list of‘Mathematische Probleme’ (1900).

Doing so was anintelligent way of emphasizing the importance of set theory for thefuture of mathematics, and the fruitfulness of its new methods andproblems. In 1895 and 1897, Cantor published his last two articles.

They were awell-organized presentation of his results on the transfinite numbers(cardinals and ordinals) and their theory, and also on order types andwell-ordered sets. However, these papers did not advance significantnew ideas.

Unfortunately, Cantor had doubts about a third part he hadprepared, which would have discussed very important issues having todo with the problem of well-ordering and the paradoxes (see below). Surprisingly, Cantor also failed to include in the 1895/97 papers atheorem which he had published some years before which is known simplyas Cantor’s Theorem: given any set $$S$$, there existsanother set whose cardinality is greater (this is the power set$$\mathcal P (S)$$, as we now say—Cantorused instead the set of functions $$f$$: $$S \rightarrow \ 0,1\$$).

In the same short paper (1892), Cantor presented his famousproof that $$\mathbf R$$ is non-denumerable by the method ofdiagonalisation, a method which he then extended to proveCantor’s Theorem. (A related form of argument had appearedearlier in the work of P.

du Bois-Reymond 1875 , see among others Wang 1974, 570 . ) Meanwhile, other authors were exploring the possibilities opened byset theory for the foundations of mathematics.

Most important wasDedekind’s contribution (1888) with a deep presentation of thetheory of the natural numbers. He formulated some basic principles ofset (and mapping) theory; gave axioms for the natural number system;proved that mathematical induction is conclusive and recursivedefinitions are flawless; developed the basic theory of arithmetic;introduced the finite cardinals; and proved that his axiom system iscategorical.

Given a function defined on $$S$$, a set $$N \subseteq S$$, and a distinguishedelement $$1 \in N$$, they are as follows: \ \begin align \tag & \phi(N \subset N\\\tag & N = \phi o \ 1\ \\\tag & 1 \not\in \phi(N)\\\tag & \textrm the function \phi \textrm is injective.

\end align \ Condition ( ) is crucial since it ensures minimality for the setof natural numbers, which accounts for the validity of proofs bymathematical induction. $$N = \phi o \ 1\$$ is read: $$N$$ isthe chain of singleton 1 under the function , that is,the minimal closure of 1 under the function .

In general, oneconsiders the chain of a set $$A$$ under an arbitrary mapping , denoted by $$\gamma o (A)$$; in his booklet Dedekinddeveloped an interesting theory of such chains, which allowed him toprove the Cantor-Bernstein theorem. In the following years, Giuseppe Peano gave a more superficial (butalso more famous) treatment of the natural numbers, employing the newsymbolic language of logic, and Gottlob Frege elaborated his ownideas, which however fell prey to the paradoxes. An important bookinspired by the set-theoretic style of thinking was Hilbert’s Grundlagen der Geometrie (1899), which took the“mathematics of axioms” one step beyond Dedekind through arich study of geometric systems motivated by questions concerning theindependence of his axioms.

Hilbert’s book made clear the newaxiomatic methodology that had been shaping up in connection with thenovel methods of set theory, and he combined it with the axiomatictrends coming from projective geometry. Critical Period In the late nineteenth century, it was a widespread idea that puremathematics is nothing but an elaborate form of arithmetic. Thus itwas usual to talk about the “arithmetisation” ofmathematics, and how it had brought about the highest standards ofrigor.

With Dedekind and Hilbert, this viewpoint led to the idea of grounding all of pure mathematics in set theory. The mostdifficult steps in bringing forth this viewpoint had been theestablishment of a theory of the real numbers, and a set-theoreticreduction of the natural numbers.

Both problems had been solved by thework of Cantor and Dedekind. But precisely when mathematicians werecelebrating that “full rigor” had been finally attained,serious problems emerged for the foundations of set theory.

FirstCantor, and then Russell, discovered the paradoxes in set theory. Cantor was led to the paradoxes by having introduced the“conceptual sphere” of the transfinite numbers.

Eachtransfinite ordinal is the order type of the set of its predecessors;e. , is the order type of $$\ 0, 1, 2, 3, \ldots\$$, and$$\omega+2$$ is the order type of $$\ 0, 1, 2, 3, \ldots, \omega,\omega +1\$$. Thus, to each initial segment of the series ofordinals, there corresponds an immediately greater ordinal.

Now, the“whole series” of all transfinite ordinals would form awell-ordered set, and to it there would correspond a new ordinalnumber. This is unacceptable, for this ordinal $$o$$ would haveto be greater than all members of the “whole series”, andin particular $$o< o$$.

This is usually called the Burali-Forti paradox, or paradox of the ordinals (althoughBurali-Forti himself failed to formulate it clearly, see Moore &Garciadiego 1981). Although Cantor might have found that paradox as early as 1883,immediately after introducing the transfinite ordinals (for argumentsin favour of this idea see Purkert & Ilgauds 1987 and Tait 2000),the evidence indicates clearly that it was not until 1896/97 that hefound this paradoxical argument and realized its implications.

By thistime, he was also able to employ Cantor’s Theorem to yield the Cantor paradox, or paradox of the alephs: if there existed a“set of all” cardinal numbers (alephs), Cantor’sTheorem applied to it would give a new aleph $$\aleph$$, such that$$\aleph< \aleph$$. The great set theorist realized perfectly wellthat these paradoxes were a fatal blow to the “logical”approaches to sets favoured by Frege and Dedekind.

Cantor emphasizedthat his views were “in diametrical opposition”to Dedekind’s, and in particular to his “na veassumption that all well-defined collections, or systems, arealso ‘consistent systems’ ” (see the letterto Hilbert, Nov An Introduction To Sets, Set Operations and Venn Diagrams, basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, The concept of sets is an essential foundation for various other topics in mathematics. Back to Top | Interactive Zone | Home..

### Sets and set theory | math goodies

) Cantor thought he could solve the problem of the paradoxes bydistinguishing between “consistent multiplicities” orsets, and “inconsistent multiplicities”. But, in theabsence of explicit criteria for the distinction, this was simply averbal answer to the problem.

Being aware of deficiencies in his newideas, Cantor never published a last paper he had been preparing, inwhich he planned to discuss the paradoxes and the problem ofwell-ordering (we know quite well the contents of this unpublishedpaper, as Cantor discussed it in correspondence with Dedekind andHilbert; see the 1899 letters to Dedekind in Cantor 1932, or Ewald1996: vol. Cantor presented an argument that relied on the“Burali-Forti” paradox of the ordinals, and aimed to provethat every set can be well-ordered. This argument was laterrediscovered by the British mathematician P.

Jourdain, but it isopen to criticism because it works with “inconsistentmultiplicities” (Cantor’s term in the above-mentionedletters). Cantor’s paradoxes convinced Hilbert and Dedekind that therewere important doubts concerning the foundations of set theory.

Hilbert formulated a paradox of his own (Peckhaus & Kahle 2002),and discussed the problem with mathematicians in his G ttingencircle. Ernst Zermelo was thus led to discover the paradox of the“set” of all sets that are not members of themselves (Rang& Thomas 1981).

This was independently discovered by BertrandRussell, who was led to it by a careful study of Cantor’sTheorem, which conflicted deeply with Russell’s belief in auniversal set. Some time later, in June 1902, he communicated the“contradiction” to Gottlob Frege, who was completing hisown logical foundation of arithmetic, in a well-known letter vanHeijenoort 1967, 124 .

Frege’s reaction made very clear theprofound impact of this contradiction upon the logicist program. “Can I always speak of a class, of the extension of a concept?And if not, how can I know the exceptions?” Faced with this,“I cannot see how arithmetic could be given a scientificfoundation, how numbers could be conceived as logical objects”(Frege 1903: 253).

The publication of Volume II of Frege’s Grundgesetze(1903), and above all Russell’s work The Principles ofMathematics (1903), made the mathematical community fully awareof the existence of the set-theoretic paradoxes, of their impact andimportance. There is evidence that, up to then, even Hilbert andZermelo had not fully appreciated the damage.

Notice that theRussell-Zermelo paradox operates with very basicnotions—negation and set membership—concepts that hadwidely been regarded as purely logical. The “set” $$R =\ x: x \not\in x\$$ exists according to the principle ofcomprehension (which allows any open sentence to determine a class),but if so, $$R \in R \textit iff R \not\in R$$.

It is a directcontradiction to the principle favoured by Frege and Russell. It was obviously necessary to clarify the foundations of set theory,but the overall situation did not make this an easy task.

Thedifferent competing viewpoints were widely divergent. Cantor had ametaphysical understanding of set theory and, although he had one ofthe sharpest views of the field, he could not offer a precisefoundation.

It was clear to him (as it had been, somewhatmysteriously, to Ernst Schr der in his Vorlesungen berdie Algebra der Logik, 1891) that one has to reject the idea of aUniversal Set, favoured by Frege and Dedekind. Frege and Russell basedtheir approach on the principle of comprehension, which was showncontradictory.

Dedekind avoided that principle, but he postulated thatthe Absolute Universe was a set, a “thing” in histechnical sense of Gedankending; and he coupled thatassumption with full acceptance of arbitrary subsets. This idea of admitting arbitrary subsets had been one of the deepinspirations of both Cantor and Dedekind, but none of them hadthematized it.

(Here, their modern understanding of analysis played acrucial but implicit background role, since they worked within theDirichlet-Riemann tradition of “arbitrary” functions. ) Asfor the now famous iterative conception there were some elements of it(particularly in Dedekind’s work, with his iterative developmentof the number system, and his views on “systems” and“things”), but it was conspicuously absent from many ofthe relevant authors.

, Cantor did not iterate theprocess of set formation: he tended to consider sets of homogeneous elements, elements which were taken to belong“in some conceptual sphere” (either numbers, or points, orfunctions, or even physical particles—but not intermingled). Theiterative conception was first suggested by Kurt G del in 1933 ,in connection with technical work by von Neumann and Zermelo a fewyears earlier; G del would insist on the idea in his well-knownpaper on Cantor’s continuum problem.

It came only postfacto, after very substantial amounts of set theory had beendeveloped and fully systematized. This variety of conflicting viewpoints contributed much to the overallconfusion, but there was more.

In addition to the paradoxes discussedabove (set-theoretic paradoxes, as we say), the list of“logical” paradoxes included a whole array of further ones(later called “semantic”). Among these are paradoxes dueto Russell, Richard, K nig, Berry, Grelling, etc.

, as well as theancient liar paradox due to Epimenides. And the diagnoses and proposedcures for the damage were tremendously varied.

Some authors, likeRussell, thought it was essential to find a new logical system thatcould solve all the paradoxes at once. This led him into the ramifiedtype theory that formed the basis of Principia Mathematica (3volumes, Whitehead and Russell 1910–1913), his joint work with AlfredWhitehead.

Other authors, like Zermelo, believed that most of thoseparadoxes dissolved as soon as one worked within a restrictedaxiomatic system. They concentrated on the “set-theoretic”paradoxes (as we have done above), and were led to search foraxiomatic systems of set theory.

Even more importantly, the questions left open by Cantor andemphasized by Hilbert in his first problem of 1900 caused heateddebate. At the International Congress of Mathematicians at Heidelberg,1904, Gyula (Julius) K nig proposed a very detailed proof thatthe cardinality of the continuum cannot be any ofCantor’s alephs.

His proof was only flawed because he had reliedon a result previously “proven” by Felix Bernstein, astudent of Cantor and Hilbert. It took some months for Felix Hausdorffto identify the flaw and correct it by properly stating the specialconditions under which Bernstein’s result was valid (seeHausdorff 2001, vol.

Once thus corrected, K nig’stheorem became one of the very few results restricting the possiblesolutions of the continuum problem, implying, e.

, that$$\textit card (\mathbf R )$$ is not equal to $$\aleph \omega$$.

Meanwhile, Zermelo was able to present a proof that every set can bewell-ordered, using the Axiom of Choice 1904 . During the followingyear, prominent mathematicians in Germany, France, Italy and Englanddiscussed the Axiom of Choice and its acceptability.

This started a whole era during which the Axiom of Choice was treatedmost carefully as a dubious hypothesis (see the monumental study byMoore 1982). And that is ironic, for, among all of the usualprinciples of set theory, the Axiom of Choice is the only one thatexplicitly enforces the existence of some arbitrary subsets.

But,important as this idea had been in motivating Cantor and Dedekind, andhowever entangled it is with classical analysis, infinite arbitrarysubsets were rejected by many other authors. Among the mostinfluential ones in the following period, one ought to emphasize thenames of Russell, Hermann Weyl, and of course Brouwer. The impressive polemics which surrounded his Well-Ordering Theorem,and the most interesting and difficult problem posed by thefoundations of mathematics, led Zermelo to concentrate on axiomaticset theory.

As a result of his incisive analysis, in 1908 he publishedhis axiom system, showing how it blocked the known paradoxes and yetallowed for a masterful development of the theory of cardinals (andordinals) 23 Jun 2009 - Sets notation: Elements Notation: Note For example, A = {a, b, c, d} b A ' b is an element of set A' or 'b is in A' f A ' f is not an element of .

From Zermelo to G del In the period 1900–1930, the rubric “set theory” wasstill understood to include topics in topology and the theory offunctions.

Although Cantor, Dedekind, and Zermelo had left that stagebehind to concentrate on pure set theory, for mathematicians at largethis would still take a long time. Thus, at the first InternationalCongress of Mathematicians, 1897, keynote speeches given by Hadamardand Hurwitz defended set theory on the basis of its importance foranalysis.

Around 1900, motivated by topics in analysis, important workwas done by three French experts: Borel 1898 , Baire 1899 andLebesgue 1902 1905 . Their work inaugurated the development ofdescriptive set theory by extending Cantor’s studies ondefinable sets of real numbers (in which he had established that theContinuum Hypothesis is valid for closed sets).

They introduced thehierarchy of Borel sets, the Baire hierarchy of functions, and theconcept of Lebesgue measure—a crucial concept of modernanalysis. Descriptive set theory (DST) is the study of certain kinds ofdefinable sets of real numbers, which are obtained from simple kinds(like the open sets and the closed sets) by well-understood operationslike complementation or projection.

The Borel sets were thefirst hierarchy of definable sets, introduced in the 1898 book of mile Borel; they are obtained from the open sets by iteratedapplication of the operations of countable union and complementation. In 1905 Lebesgue studied the Borel sets in an epochal memoir, showingthat their hierarchy has levels for all countable ordinals, andanalyzing the Baire functions as counterparts of the Borel sets.

Themain aim of descriptive set theory is to find structural propertiescommon to all such definable sets: for instance, the Borel sets wereshown to have the perfect set property (if uncountable, they have aperfect subset) and thus to comply with the continuum hypothesis (CH). This result was established in 1916 by Hausdorff and by Alexandroff,working independently.

Other important “regularityproperties” studied in DST are the property of being Lebesguemeasurable, and the so-called property of Baire (to differ from anopen set by a so-called meager set, or set of first category). Also crucial at the time was the study of the analytic sets,namely the continuous images of Borel sets, or equivalently, theprojections of Borel sets.

The young Russian mathematician MikhailSuslin found a mistake in Lebesgue’s 1905 memoir when herealized that the projection of a Borel set is not Borel in general Suslin 1917 . However, he was able to establish that the analyticsets, too, possess the perfect set property and thus verify CH.

By1923 Nikolai Lusin and Wac aw Sierpi ski were studying the co-analytic sets, and this was to lead them to a newhierarchy of projective sets, which starts with the analyticsets $$(\sum 1 1$$ sets), the projections of these last($$\sum During the 1920s much work was done on these new types ofsets, mainly by Polish mathematicians around Sierpi ski and bythe Russian school of Lusin and his students. A crucial resultobtained by Sierpi ski was that every \(\sum 1 2$$ set is theunion of $$\aleph 1$$ Borel sets (the same holds for$$\sum 1 1$$ sets), but this kind of traditional research on thetopic would stagnate after around 1940 (see Kanamori 1995 ). Soon Lusin, Sierpi ski and their colleagues were finding extremedifficulties in their work.

Lusin was so much in despair that, in apaper of 1925, he came tothe “totally unexpected” conclusion that “one doesnot know and one will never know” whether the projective setshave the desired regularity properties (quoted in Kanamori 1995: 250). Such comments are highly interesting in the light of laterdevelopments, which have led to hypotheses that solve all the relevantquestions (Projective Determinacy).

They underscore the difficultmethodological and philosophical issues raised by these more recenthypotheses, namely the problem concerning the kind of evidence thatbacks them. Lusin summarized the state of the art in his 1930 book Le ons sur les ensembles analytiques (Paris,Gauthier-Villars), which was to be a key reference for years to come.

Since this work, it has become customary to present results in DST forthe Baire space $$\omega \omega$$ of infinite sequences of naturalnumbers, which in effect had been introduced by Ren Baire in apaper published in 1909.

Baire space is endowed with a certaintopology that makes it homeomorphic to the set of the irrationalnumbers, and it is regarded by experts to be “perhaps the mostfundamental object of study of set theory” next to the set ofnatural numbers Moschovakis 1994, 135 . This stream of work on DST must be counted among the most importantcontributions made by set theory to analysis and topology.

But whathad begun as an attempt to prove the Continuum Hypothesis could notreach this goal. Soon it was shown using the Axiom of Choice thatthere are non-Lebesgue measurable sets of reals (Vitali 1905), andalso uncountable sets of reals with no perfect subset (Bernstein1908).

Such results made clear the impossibility of reaching the goalof CH by concentrating on definable and “well-behaved”sets of reals. Also, with G del’s work around 1940 (and also with forcingin the 1960s) it became clear why the research of the 1920s and 30shad stagnated: the fundamental new independence results showed thatthe theorems established by Suslin (perfect set property for analyticsets), Sierpinski ($$\sum 1 2$$ sets as unions of $$\aleph 1$$Borel sets) and a few others were the best possible results on thebasis of axiom system ZFC. This is important philosophically: alreadyan exploration of the world of sets definable from the open (orclosed) sets by complement, countable union, and projection hadsufficed to reach the limits of the ZFC system.

Hence the need for newaxioms, that G del emphasized after World War II G del1947 . Let us now turn to Cantor’s other main legacy, the study oftransfinite numbers.

By 1908 Hausdorff was working on uncountableorder types and introduced the Generalized Continuum Hypothesis$$(2 He was also the first to considerthe possibility of an “exorbitant” cardinal, namely aweakly inaccessible, i. , a regular cardinal that is not a successor(a cardinal \(\alpha$$ is called regular if decomposing $$\alpha$$into a sum of smaller cardinals requires $$\alpha$$-many suchnumbers). Few years later, in the early 1910s, Paul Mahlo was studyinghierarchies of such large cardinals in work that pioneered what was tobecome a central area of set theory; he obtained a succession ofinaccessible cardinals by employing a certain operation that involvesthe notion of a stationary subset; they are called Mahlo cardinals.

But the study of large cardinals developed slowly. Meanwhile,Hausdorff’s textbook Grundz ge der Mengenlehre(1914) introduced two generations of mathematicians into set theoryand general topology.

The next crucial steps into the “very high” infinite weredone in 1930. The notion of strongly inaccessible cardinals was thenisolated by Sierpi ski & Tarski, and by Zermelo 1930 .

Astrong inaccessible is a regular cardinal $$\alpha$$ such that $$2 x$$is less than $$\alpha$$ whenever $$x< \alpha$$.

While weakinaccessibles merely involve closure under the successor operation,strong inaccessibles involve a much stronger notion of closure underthe powerset operation. That same year, in a path-breaking paper onmodels of ZFC, Zermelo 1930 established a link between theuncountable (strongly) inaccessible cardinals and certain“natural” models of ZFC (in which work he assumed, so tosay, that the powerset operation is fully determined).

In that same year, Stanislaw Ulam was led by considerations coming outof analysis (measure theory) to a concept that was to become central:measurable cardinals. It turned out that such cardinals, defined by ameasure-theoretic property, had to be (strongly) inaccessible. Indeed,many years later it would be established (by Hanf, working uponTarski’s earlier work) that the first inaccessible cardinal isnot measurable, showing that these new cardinals were even more“exorbitant”.

As one can see, the Polish school led bySierpi ski had a very central role in the development of settheory between the Wars.

### Set theory - online math learning

This againvindicated G del’s convictions, expressed in what issometimes called “G del’s program” for newaxioms A solid foundation on sets is provided for students of all ages. is introduced as a shorthand for writing sets, including formulas, notation and restrictions..

Set-theoretic mathematics continued its development into the powerfulaxiomatic and structural approach that was to dominate much of the20th century.

To give just a couple of examples,Hilbert’s early axiomatic work (e. , in his arch-famous Foundations of Geometry) was deeply set-theoretic; ErnstSteinitz published in 1910 his research on abstract field theory,making essential use of the Axiom of Choice; and around the same timethe study of function spaces began with work by Hilbert, MauriceFr chet, and others. During the 1920s and 30s, the firstspecialized mathematics journal, Fundamenta Mathematicae, wasdevoted to set theory as then understood (centrally including topologyand function theory).

In those decades structural algebra came of age,abstract topology was gradually becoming an independent branch ofstudy, and the study of set theory initiated its metatheoretic turn. Ever since, “set theory” has generally been identifiedwith the branch of mathematical logic that studies transfinite sets,originating in Cantor’s result that $$\mathbf R$$ has agreater cardinality than $$\mathbf N$$.

But, as the foregoingdiscussion shows, set theory was both effect and cause of the rise ofmodern mathematics: the traces of this origin are indelibly stamped onits axiomatic structure. Baire, Ren , 1899, “Sur les fonctions de variablesreelles”, Annali di Matematica Pura ed Applicata, SerieIIIa, vol. –––, 1909, “Sur la repr sentationdes fonctions discontinues”, Acta Mathematica, 32:97–176.

Bernstein, Felix, 1908, “Zur Theorie der trigonometrischenReihen”, Sitzungsberichte der K niglich SachsischenGesellschaft der Wissenschaften zu Leipzig, Math. du Bois–Reymond, Paul, 1875, “Ueber asymptotischeWerthe, infinit re Approximationen und infinit reAufl sung von Gleichungen”, Mathematische Annalen,8: 363–414.

Bolzano, Bernard, 1851, Borel, mile, 1898, th edn 1950with numerous additions. Cantor, Georg, 1872, “ ber die Ausdehnung eines Satzesaus der Theorie der trigonometrischen Reihen”, MathematischeAnnalen, 5: 123–132.

–––, 1883, Grundlagen einer allgemeinenMannigfaltigkeitslehre, Leipzig: B.

–––, 1884, “ ber unendliche, linearePunktmannichfaltigkeiten, 6”, Mathematische Annalen,23: 453–88.

–––, 1892, “ ber eine elementareFrage der Mannigfaltigkeitslehre”, Jahresbericht derDeutschen Mathematiker Vereinigung, 1: 75–78.

–––, 1895/97, “Beitr ge zurBegr ndung der transfiniten Mengenlehre”, in Cantor 1932:282–351.

English translation in Cantor, Contributions to thefounding of the theory of transfinite numbers, New York: Dover,1955. His Mathematics andPhilosophy of the Infinite, Cambridge, MA: Harvard UniversityPress. Dedekind, Richard, 1871, “ ber die Komposition derbin ren quadratischen Formen”, Supplement X to G.

Dedekind, Vorlesungen berZahlentheorie, Braunschweig: Vieweg. Later editions asSupplement XI, of which the fourth is reprinted in New York: Chelsea,1968.

–––, 1872, Stetigkeit und irrationaleZahlen, Braunschweig: Vieweg. –––, 1876/77, “Sur la th orie desnombres entiers alg briques”, Bulletin des Sciencesmath matiques et astronomiques, 1st series, XI(1876): 278–293; 2nd series, I (1877): 17–41,69–92, 144–164, 207–248.

Stillwell: Theory of algebraic integers, Cambridge: Cambridge UniversityPress, 2004. –––, 1888, Was sind und was sollen dieZahlen?, Braunschweig: Vieweg.

& Heinrich Weber, 1882, “Theorie deralgebraischen Funktionen einer Ver nderlichen”, Journal f r reine und angew.

by JohnStillwell, Theory of Algebraic Functions of One Variable, AMS/ LMS, 2012.

, 1996, From Kant to Hilbert: A source bookin the foundations of mathematics, 2 vols.

Ferreir s, Jos , 1995, “‘What Fermentedin Me for Years’: Cantor’s Discovery of TransfiniteNumbers”, Historia Mathematica, 22: 33–42.

–––, 1999, Frege, Gottlob, 1903, Grundgesetze der Arithmetik, vol. G del, Kurt, 1933, “The present situation in thefoundations of mathematics”, in S.

–––, 1947, “What is Cantor’scontinuum problem?”, American Mathematical Monthly, 54. Hallett, Michael, 1984, Hausdorff, Felix, 1914, Grundz ge der Mengenlehre,Leipzig: Viet. Reprinted New York: AMS Chelsea Publishing, 1949.

The third edition(1937) was translated into English, 1957, –––, 1916, “Die M chtigkeit derBorelschen Mengen”, Mathematische Annalen, 77(3):430–437.

1007/BF01475871 –––, 2001–, Gesammelte Werke, 9volumes, E.

van Heijenoort, Jean, 1967, From Frege to G del: A sourcebook in mathematical logic, 1879–1931, Cambridge, MA:Harvard University Press. Kanamori, Akihiro, 1995, “The emergence of descriptive settheory”, Synthese, 251: 241–262. Lavine, Shaughan, 1994, Lebesgue, Henri, 1902, “Int grale, longueur,aire”, Annali di Matematica Pura ed Applicata, 7 (1):231–359.

–––, 1905, “Sur les fonctionsrepres ntables analytiquement”, Journal deMath matiques, (6e serie), 1: 139–216. Lusin, Nikolai, 1925, “Sur les ensembles projectifs de M.

–––, 1930, Le ons sur les EnsemblesAnalytiques et leurs Applications, with a preface by Lebesgue anda note by Sierpinski, Paris: Gauthier-Villars. Garciadiego, 1981,“Burali-Forti’s Paradox: A reappraisal of itsorigins”, Historia Mathematica, 8: 319–50. Kahle, 2002, “Hilbert’sParadox”, Historia Mathematica, 29 (2):157–175.

Ilgauds, 1987, Georg Cantor1845–1918, Basel: Birkh user. Thomas, 1981, “Zermelo’sDiscovery of the ‘Russell Paradox’”, HistoriaMathematica, 8: 15–22. Riemann, Bernhard, 1854/1868a, “ ber die Hypothesen,welche der Geometrie zu Grunde liegen” (Habilitationsvotrag), Abhandlungen der K niglichen Gesellschaft der Wissenschaftenzu G ttingen, 13 (1868): 133–152.

English translation by Clifford, reprinted in Ewald1996: vol.

–––, 1854/1868b, “ ber dieDarstellbarkeit einer Function durch eine trigonometrischeReihe”, (Habilitationsschrift), Abhandlungen derK niglichen Gesellschaft der Wissenschaften zuG ttingen, 13 (1868): 87–132.

–––, 1892, Gesammelte mathematische Werkeund wissenschaftlicher Nachlass, H.

Sierpi ski, Waclav & Alfred Tarski, 1930, “Sur unepropri t caract ristique des nombresinaccessibles”, Fundamenta Mathematicae, 15:292–300.

Steinitz, Ernst, 1910, “Algebraische Theorie derK rper”, Journal f r die reine und angewandteMathematik, 137: 167–309. , 1917, “Sur une d finition desensembles measurables B sans nombres transfinis”, ComptesRendues Acad. & Bertrand Russell, 1910–1913, Principia Mathematica, 3 volumes, Cambridge: CambridgeUniversity Press.

, 2000, “Cantor’s Grundlagenand the Paradoxes of Set Theory”, Between Logic andIntuition: Essays in Honor of Charles Parsons, G.

Tieszen (eds), Cambridge: Cambridge University Press, pp.

Reprinted in his The Provenance of PureReason, Oxford: Oxford University Press, 2005, pp.

Wang, Hao, 1974, “The concept of set”, in FromMathematics to Philosophy, London, Routledge; reprinted in P.

Putnam, Philosophy of Mathematics: selectedreadings, Cambridge Univ.

Zermelo, Ernst, 1904, “Beweis, dass jede Menge wohlgeordnetwerden kann”, Mathematische Annalen, 59: 514–516;in Zermelo 2010 , vol.

English translation in vanHeijenoort 1967 (“Proof that every set can bewell-ordered”).

–––, 1908, “Untersuchungen ber dieGrundlagen der Mengenlehre”, Mathematische Annalen, 65:261–281; ; in Zermelo 2010 , vol. Englishtranslation in van Heijenoort 1967 (“Investigations in thefoundations of set theory I”).