Il teorema di Bolzano (1817) o di Weierstrass (1885)?

Theorem. If a function Fx is continuous from x=a to x=b inclusive, then among all the values which it takes, if we imagine that x successively takes all the values from a to b inclusive, there is always a greatest in the sense that no other is greater than it, and there is also a smallest in the sense that no other is smaller than it.

Dai manoscritti di Bernard Bolzano (Functionenlehre e Größenlehre 1817)

Paradoxes of the Infinite Dr. Bernard Bolzano 1851

Je suis tellement pour l’infini actuel, qu’au lieu d’admettre, que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte par-tout, pour mieux marquer les perfections de son Auteur.

— Leibniz, Opera omnia studio Ludov. Dutens., Tom. II, part x, p. 243 

Certainly most of the paradoxical assertions which we meet with in the area of mathematics, though not all of them as Kästner suggests, are propositions that either contain the concept of infinity directly, or depend on it in some way for their attempted proof. It is even more indisputable that precisely those mathem- atical paradoxes which deserve our greatest attention are of this kind. This is because decisions on very important questions in many another subject, such as metaphysics and physics, depend on a satisfactory resolution of their apparent contradictions.

This is the reason why in the present work I am dealing exclusively with the consideration of the paradoxes of the infinite. But it is self-evident that it would not be possible to recognize the appearance of contradiction which is attached to these mathematical paradoxes for what it is, a mere appearance, if we did not make abundantly clear what concept we actually associate with the infinite. Therefore we do this first.

Il teorema di Bolzano-Weierstrass

Esistono molte versioni equivalenti del teorema di Bolzano-Weierstrass.
Il teorema fu originalmente provato da Bernard Bolzano nel 1817 e riscoperto da Karl Weierstrass cinquanta anni dopo.

Teorema. Ogni insieme limitato ed infinito contenuto in R ammette punto di accumulazione.

 The greatest lower bound property is equivalent to what is often referred to as the Bolzano–Weierstrass theorem in a form such as ‘a bounded sequence always contains convergent subsequences’. Many introductory textbooks on analysis ask students to accept some further (equivalent) result as an axiom about the real numbers, for example,‘An increasing sequence, bounded above, must be convergent.’ 

L'infinito...ahhhh...ahhhhh

There was another deep problem stemming from the budding theory of cardinal numbers that occupied Cantor and which was not resolved during his lifetime. Because of the importance of countable sets, the symbol ℵ₀ (“aleph zero”) is frequently used for card N. The subscript “0” is appropriate when we remember that countable sets are the smallest type of infinite set. In terms of cardinal numbers, if card X < ℵ₀, then X is finite. Thus, ℵ₀ is the smallest infinite cardinal number. The cardinality of R is also significant enough to deserve the special designation c = card R = card (0, 1). The question that plagued Cantor was whether there were any cardinal numbers strictly in between these two. Put another way, does there exist a set A ⊆ R with card N < card A < card R? Cantor was of the opinion that no such set existed. In the ordering of cardinal numbers, he conjectured, c was the immediate successor of ℵ₀.
Cantor’s “continuum hypothesis,” as it came to be called, was one of the most famous mathematical challenges of the past century. Its unexpected resolution came in two parts. In 1940, the German logician and mathematician Kurt Gödel demonstrated that, using only the agreed-upon set of axioms of set theory, there was no way to disprove the continuum hypothesis. In 1963, Paul Cohen successfully showed that, under the same rules, it was also impossible to prove this conjecture. Taken together, what these two discoveries imply is that the continuum hypothesis is undecidable. It can be accepted or rejected as a statement about the nature of infinite sets, and in neither case will any logical contradictions arise.
The mention of Kurt Gödel brings to mind a final comment about the significance of Cantor’s work. 
Gödel is best known for his “Incompleteness Theorems,” which pertain to the strength of axiomatic systems in general. What Gödel showed was that any consistent axiomatic system created to study arithmetic was necessarily destined to be “incomplete” in the sense that there would always be true statements that the system of axioms would be too weak to prove.

L'infinito...ahhhh...

Having divided the universe of sets into disjoint groups, it would be con- venient to attach a “number” to each collection which could be used the way natural numbers are used to refer to the sizes of finite sets. Given a set X, there exists something called the cardinal number of X, denoted cardX, which behaves very much in this fashion. For instance, two sets X and Y satisfy cardX = cardY if and only if X ∼ Y. (Rigorously defining cardX requires some significant set theory. One way this is done is to define cardX to be a very particular set that can always be uniquely found in the same equivalence class as X.)

Looking back at Cantor’s Theorem, we get the strong sense that there is an order on the sizes of infinite sets that should be reflected in our new car- dinal number system. Specifically, if it is possible to map a set X into Y in a 1–1 fashion, then we want card X ≤ card Y . Writing the strict inequality card X < card Y should indicate that it is possible to map X into Y but that it is impossible to show X ∼ Y . Restated in this notation, Cantor’s Theorem states that for every set A, card A < card P (A).

There are some significant details to work out. A kind of metaphysical problem arises when we realize that an implication of Cantor’s Theorem is that there can be no “largest” set. A declaration such as, “Let U be the set of all possible things,” is paradoxical because we immediately get that cardU < cardP(U) and thus the set U does not contain everything it was advertised to hold. Is- sues such as this one are ultimately resolved by imposing some restrictions on what can qualify as a set. As set theory was formalized, the axioms had to be crafted so that objects such as U are simply not allowed. A more down-to- earth problem in need of attention is demonstrating that our definition of “≤” between cardinal numbers really is an ordering. This involves showing that cardinal numbers possess a property analogous to real numbers, which states that if card X ≤ card Y and card Y ≤ card X, then card X = card Y . In the end, this boils down to proving that if there exists f : X → Y that is 1–1, and if there existsg:Y →X that is 1–1, then it is possible to find a function h:X→Y that is both 1–1 and onto. 

A proof of this fact eluded Cantor but was eventually supplied independently by Ernst Schr ̈oder (in 1896) and Felix Bernstein (in 1898). 

Il teorema degli zeri - Bernard Bolzano (1817)

Purely Analytic

Proof of the Theorem

that

between any two Values, which give Results of Opposite Sign,

there lies at least one real Root of the Equation

by

Bernard Bolzano

Priest, Doctor of Philosophy, Professor of Theology and Ordinary Member of the Royal Society of Sciences at Prague

For the Proceedings of the Royal Society of Sciences

Prague, 1817 Printed by Gottlieb Haase

Theorem. If two functions of x, fx and φx, vary according to the law of continuity either for all values of x or for all those lying between α and β, and furthermore if fα < φα and fβ > φβ, then there is always a certain value of x between α and β for which fx = φx.

What is a number?

Euler’s death in 1783 was followed by a period of stagnation in mathematics. He had indeed solved everything: an unsurpassed treatment of infinite and differential calculus (Euler 1748, 1755), solvable integrals solved, solvable differential equations solved (Euler 1768, 1769), the secrets of liquids (Euler 1755b), of mechanics (Euler 1736b, Lagrange 1788), of variational calculus (Euler 1744), of algebra (Euler 1770), unveiled. It seemed that no other task remained than to study about 30,000 pages of Euler’s work.
The “Theorie des fonctions analytiques” by Lagrange (1797), “freed from all considerations of infinitely small quantities, vanishing quantities, limits and fluxions”, the thesis of Gauss (1799) on the “Fundamental Theorem of Algebra” and the study of the convergence of the hypergeometric series (Gauss 1812) mark the beginning of a new era.
Bolzano points out that Gauss’s first proof is lacking in rigor; he then gives in 1817 a “purely analytic proof of the theorem, that between two values which produce opposite signs, there exists at least one root of the equation”. In 1821, Cauchy establishes new requirements of rigor in his famous “Cours d’Analyse”. The questions are the following:

– What is a derivative really? Answer: a limit.
– What is an integral really? Answer: a limit.
– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.

This leads to:

– What is a limit? Answer: a number.

And, finally, the last question:

                                        What is a number?

Weierstrass and his collaborators (Heine, Cantor), as well as Me'ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence, uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series.