∞ = M = 1000

Liczby pierwsze i ich gęstość The Prime Pages

Adam Walanus



Nieskończona ilość prostych

krzyżuje się w twoim sercu:

te proste nie mają początku - ni końca.

Ryszard Krynicki

Wszystko -
słowo bezczelne i nadęte pychą.
Powinno być pisane w cudzysłowie.
Udaje, że niczego nie pomija,
że skupia, obejmuje, zawiera i ma.
A tymczasem jest tylko
strzępkiem zawieruchy

Wisława Szymborska

infini-francuski infinito-włoski, hiszpański Unendlichkeit-niemiecki απειρο-grecki бесконечность-rosyjski nekonečno-czeski végtelenség-węgierski begalybe-litewski


Modern discussion of the infinite is now regarded as part of set theory and mathematics.


Paweł Polak, Rozwój pojęcia nieskończoności. Dialog pomiędzy filozofią a matematyką, Semina Scientiarum 2002

... nieskończoność jest w pewien sposób bliska człowiekowi, ponieważ wpisuje się w wyczuwalną przez niego potrzebę transcendencji

... Antyfon przyjął, że koło jest wielokątem o nieskończonej ilości boków ... wg Arystotelesa błąd polegający na utożsamieniu bardzo wielkiej ilości boków z nieskończoną ich ilością

Paradoks Zenona ... Stagiryta wydzielił więc:
1. nieskończoność podziałów i nieskończoność krańców;
2. nieskończoność aktualną i potencjalną.

Arystoteles odrzucił istnienie nieskończoności aktualnej. ... wszystko w świecie pozostaje skończone.

Mikołaj z Kuzy. Twierdził on, że nieskończoność nie pozostaje w proporcji do niczego, jest więc jako taka nieznana. Kuzańczyk podkreślał istotową różnicę pomiędzy tym co skończone, a tym co nieskończone.


Dopiero dzięki pracom Cantora w II poł. XIX wieku udało się wytworzyć odpowiedni grunt dla owocnej współpracy filozofii i matematyki.

... nie da się pominąć aspektu wzajemnych oddziaływań i tworzyć nawet tak abstrakcyjnej nauki jaką jest matematyka, bez uwzględnienia oddziaływań pochodzących z filozofii.

Galileo, On two New Sciences, 1638
So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.

Emmanuel Levinas, Philosophy and the Idea of Infinity Roberto Ponce Cordero
a strategy of relocating the foundation of an ethics on a concept of infinite that seems to me to be too essentialist too ahistorical

Ludwig Wittgenstein, Tractatus logico-philosophicus
Czasowo pojęta nieśmiertelność duszy ludzkiej — czyli jej wieczne życie po śmierci — nie tylko nie jest niczym zagwarantowana, lecz nade wszystko nie daje wcale tego, co zawsze chciano przez nią osiągnąć. Czy rozwiąże to jakąś zagadkę, że będę żył wiecznie? Czyż takie wieczne życie nie będzie równie zagadkowe jak obecne? Rozwiązanie zagadki życia w czasie i przestrzeni leży poza czasem i przestrzenią. (Nie chodzi tu przecież o rozwiązywanie problemów naukowych.)



Liczby naturalne Liczby wymierne Duża liczba Duża liczba zapisana Dwa miliony pierwszych























Giovanni Segantini



















To zdanie jest fałszywe.



liczenie ω



Prawdziwość tego zdania nie różni się bardzo od jego fałszywości.

p,   1-p,   (p-(1-p))2,   1-pi+1=(pi-(1-pi))2




problem stopu

bifurkacje (klikać x)

 R = 2.9 - 4 


Mandelbrot zi+1= zi2 + c


Liczby rzeczywiste

2D, kwadrat












Internet Encyclopedia of Philosophy. The Infinite.

Philosophers want to know whether there is more than one coherent concept of infinity

The density of matter at the center of a black hole is infinitely large. An electron is infinitely small. An hour is infinitely divisible. The integers are infinitely numerous. These four claims are ordered from most to least controversial

Thomas Aquinas ... God is infinitely powerful

Gauss ... scientific theories involve infinities merely as idealizations and merely in order to make for easy applications of those theories, when in fact all physically real entities are finite

Quine ... the first three sizes of Cantor’s infinities are the only ones we have reason to believe in

2,500 years ... actually infinite, potentially infinite, and transcendentally infinite

Aristotle ... “the idea of the actual infinite-of that whose infinitude presents itself all at once-was close to a contradiction in terms…,”


Dedekind in 1888

Cantor 1887 ... each potential infinite…presupposes an actual infinite.

Cardinal numbers ... ℵ ... ℵ0, ℵ1; continuum problem

Cantor ... is not invention but rather is discovery about a mind-independent reality.

actual infinities are indispensable in mathematics and science

whether the set of all cardinal numbers has a cardinal number ... if it does, then it doesn’t

Russell’s Paradox of 1901 ... the set of all sets that are not members of themselves

Zermelo-Fraenkel’s set theory (ZF) was the best way or the least radical way

the concept of "infinite set" within ZF was claimed by many philosophers to be the paradigm example of how to provide a precise and fruitful definition of a philosophically significant concept.

we can never use the word “infinity” coherently because infinity is ineffable or inherently paradoxical

Infinity and the Mind

the infinite is beyond the grasp of the human mind

contemporary philosophers of psychology believe mental pictures are not essential to having any concept

whether we can coherently think about infinity to the extent of being said to have the concept

If we understand negation and have the concept of finite ...

might be thought of by a powerful enough mind.

Infinity in Metaphysics

person’s brain contains approximately 1027 atoms

some version of transcendental infinity that makes infinity be somehow beyond human comprehension

Levinas says the infinite is another name for the Other ... facing a practically incomprehensible and unlimited number of possibilities upon encountering another conscious being ... should say instead that there are too many possibilities to be faced

Cantor claimed his work was revealing God’s existence and that these mathematical objects were in the mind of God ... God gave humans the concept of the infinite so that they could reflect on His perfection

The connection between infinity and God exists in nearly all of the world’s religions.

The multiverse theories of cosmology in the early 21st century allow there to be an uncountable infinity of universes ... Augustine had this worry when considering infinite universes, and he responded that "Christ died once for sinners...."

Infinity in Physical Science

... examples where infinity occurs within physical science
(1) Standard cosmology based on Einstein’s GTR implies the density of the mass at the center of a simple black hole is infinitely large
(2) The Standard Model of particle physics implies the size of an electron is infinitely small.
(3) General relativity implies that every path in space is infinity divisible.
(4) Classical quantum theory implies the values of kinetic energy of an accelerating, free electron are infinitely numerous.
are not something that could be measured directly

George Berkeley and David Hume denied the physical reality of even potential infinities on the empiricist grounds that such infinities are not detectable by our sense organs. ... instrumentalists also ...

reality looks “as if” there are physical infinities ... useful mathematical fiction

theoretical terms that refer to infinities, then infinities must be accepted

Standard Model ... time is a continuum, and space is a continuum ... mass is a continuum as well as energy

space consists of discrete units called loops

Brian Greene ... the notion of being able to divide distances or durations into ever smaller units likely comes to an end at around the Planck length (10-35 m) and Planck time (10-43 s).

Roger Penrose ... The continuum still features in an essential way ... we need to take the use of the infinite seriously

Singularities ... A theory that involves singularities...carries within itself the seeds of its own destruction.

Strings have an infinite number of possible vibrational patterns each corresponding to a particle that should exist if we take the theory literally.

Big Bang ... stopped shrinking ... 10-35 meters

Gauss ... scientific theories involve infinities merely as approximations or idealizations

Penrose ... To my mind, a physical theory which depends fundamentally upon some absurdly enormous...number would be a far more complicated (and improbable) theory than one that is able to depend upon a simple notion of infinity

Erwin Schrödinger remarks, “The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is accessible to us.”

Infinity in Cosmology

Immanuel Kant (1724–1804) declared that space and time are both potentially infinite in extent because this is imposed by our own minds. ... geometry of space must be Euclidean

The volume of spacetime is finite at present if we can trust the classical Big Bang theory.

Multiverse, then both space and time are actually infinite

Infinity in Mathematics

Bertrand Russell ... thinking in an unfamiliar way

The series s1 + s2 + s3 + … converges to S if, and only if, for every positive number ε there exists a number δ such that |sn+h + sn| < ε for all integers n > δ and all integers h > 0. In this way, reference to an actual infinity has been eliminated.

infinitesimal object is as small as you please but not quite nothing ... an infinite number of infinitesimal steps

Robinson: h is infinitesimal if and only if 0 < |h| < 1/n, for every positive integer n ... the hyperreal line

A constructivist, unlike a realist, ... an unknowable mathematical object is impossible. ... potential infinites can be constructed, actual infinities cannot be

Brouwer ... intuitionist school ... Numbers are human creations.

pi is intuitionistically legitimate because we have an algorithm ... number g is not legitimate ... n consecutive 7s in the decimal expansion of pi

there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... is not a closed realm of things existing in themselves

Finitists, .. the actually infinite set of natural numbers does not exist.

ultrafinitist .. numbers such as 2100 and 21000 can never be accessed by a human mind

Quine .. some actually infinite sets are indispensable to all these scientific theories .. All this success is a good reason to believe

Quine .. only the first three alephs: ℵ0 for the integers, ℵ1 for the set of point places in space, and ℵ2 for the number of possible lines in space (including lines that are not continuous)

Zermelo-Fraenkel Set Theory

Using the axiom of choice, .. set is infinite .. for every natural number n, there is some subset whose size is n.

The power set axiom (which says every set has a power set, namely a set of all its subsets) then generates many more infinite sets of larger cardinality, a surprising result that Cantor first discovered in 1874

The Axiom of Choice and the Continuum Hypothesis

Platonists tend to like the axiom

mathematics’ most unintuitive theorem, the Banach-Tarski Theorem, requires the axiom of choice

continuum hypothesis and the axiom of choice are independent of ZF

the concept of infinite set is so vague that we simply do not have any intuitions that .. ZFC

finite axiomatizability ... infinitary logic

logic should reflect the finitude of the human mind

Tarski also suggested allowing formulas to have a sequence of quantifiers of any transfinite length.

Infinitely Long Proofs

infinite number of truth values .. truth is a matter of degree .. from 0 to 1 .. Lofti Zadeh

true sentences of languages lower in the hierarchy ... Tarski's hierarchy of metalanguages

Renormalization in quantum field theory. ... When describing space and time as a continuum, certain statistical and quantum mechanical constructions are ill-defined.

Ellis, Meissner, Nicolai: pdf Fala prostokątna (js)

Real analysis, from Wikipedia

n(1)=9 n(i)=n(i-1)n(i-1) n(n(9)) n(n(n(n(n(n(n(n(n(9))))))))) m(9)=n(n(n(n(n(n(n(n(n(9))))))))) m(m(9))

Big Number Calculator

Infinity: New Research Frontiers Editors: Michael Heller, W. Hugh Woodin

Introduction Rudy Rucker
Part I. Perspectives on Infinity from History:
1. Infinity as a transformative concept in science and theology, Wolfgang Achtner
Part II. Perspectives on Infinity from Mathematics:
2. The mathematical infinity, Enrico Bombieri
3. Warning signs of a possible collapse of contemporary mathematics, Edward Nelson
Part III. Technical Perspectives on Infinity from Advanced Mathematics:
4. The realm of the infinite, W. Hugh Woodin
5. A potential subtlety concerning the distinction between determinism and nondeterminism, W. Hugh Woodin
6. Concept calculus: much better than Harvey, M. Friedman
Part IV. Perspectives on Infinity from Physics and Cosmology:
7. Some considerations on infinity in physics, Carlo Rovelli
8. Cosmological intimations of infinity, Anthony Aguirre
9. Infinity and the nostalgia of the stars, Marco Bersanelli
10. Infinities in cosmology, Michael Heller
Part V. Perspectives on Infinity from Philosophy and Theology:
11. God and infinity: directions for future research, Graham Oppy
12. Notes on the concept of the infinite in the history of Western metaphysics, David Bentley Hart
13. God and infinity: theological insights from Cantor's mathematics, Robert J. Russell
14. A partially skeptical response to Hart and Russell, Denys A. Turner.

Introduction, Rudy Rucker:
Hart feels that the starting point of the metaphysical notion of infinity is the notion of absolute indeterminacy . He also remarks that the metaphysical infinite is a domain in which the principle of noncontradiction fails, and both A and the negation of A can be true. Waxing a bit territorial, he says that things such as numbers, matter, space, or time can never really attain to a truly metaphysical infinitude. As he puts it, “We see here, then, that between the mathematical and the metaphysical senses of ‘infinite’ there exists not merely a distinction, but very nearly an opposition…any possible analogy is at best pictorial, affective, and immeasurably remote.”
I was fortunate enough to meet with the great logician Kurt Gödel at his office in Princeton. I was intrigued by his remarks about infinite sets being objectively existing objects that he could in some sense perceive. I asked him how to see the infinite sets. I’ll quote his answer from my book Infinity and the Mind : He said three things. (i) First one must close off the other senses, for instance, lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind. (ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects – the mind is capable of directly perceiving infinite sets. (iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute . 9 Many of us have some immediate feeling for the infinite. We may express it in mathematics, in philosophy – or in art.

1. Infinity as a Transformative Concept in Science and Theology, Wolfgang Achtner:
ת = {ℵ0,...,ℵ1,...,ℵn}        (ת - Tav)
This ת, Georg Cantor claimed, is God, the creative source of all quantities existing in the world, and an intuitive insight of God is possible.

4.2.1 Woodin

The sum total of human experience in mathematics to date (i.e., the number of manuscript pages written to date) is certainly less than 1012 pages. The shortest proof from ZFC0 that no such sequence exists must have length greater than 1024. This is arguably beyond the reach of our current experience, but there is an important issue that concerns the compression achieved by the informal style in which mathematical arguments are actually written. This is explored a little bit further in Woodin (1998).

With proper inputs and global determination, one could verify with current technology that a given sequence of length at most 1024 is a proof of (¬Ξ) from ZFC0. However, we obviously do not expect to be able to find a sequence of length less than 1024 that is a proof of (¬Ξ) from ZFC0. This actually gives a prediction about the physical universe because one can code any candidate for such a sequence by a binary sequence of length at most 1026. The point is that, assuming the validity of the quantum view of the world, it is possible to build an actual physical device that must have a nonzero chance of finding such a sequence if such a sequence can exist. The device simply contains (a suitably large number of independent) modules, each of which performs an independent series of measurements that in effect flips a quantum coin. The point, of course, is that by quantum law any outcome is possible. The prediction is simply that any such device must fail to find a sequence of length less than 1024 that is a proof of (¬Ξ) from ZFC0. One may object that the belief that any binary sequence of length 1026 is really a possible outcome of such a device requires an extraordinary faith in quantum law; but any attempt to build a quantum computer that is useful (for factoring) requires the analogous claim where 1026 is replaced by numbers at least as large as 105.

This, of course, requires something like quantum theory. In the universe as described by Newtonian laws, the argument just described does not apply because truly random processes would not exist. One could imagine proving that for a large class of chaotic (but deterministic) processes (“mechanical coin flippers”), no binary sequence of length 1024 that actually codes a formal proof can possibly be generated. In other words, for the nonquantum world, the prediction that no such sequence (as just presented) can be generated may not require that the conception of Vn is meaningful where n = | V1000|.

Granting quantum law, and based only on our collective experience in mathematics to date, how can one account for the prediction (that one cannot find a sequence of length less than 1024 that is a proof of (¬Ξ) from ZFC0) unless one believes that the conception of Vn is meaningful where n = | V1000|?

Arguably (given current physical theory) this is already a conception of a nonphysical realm.

Skeptic's Attack: The mathematical conception of infinity is meaningless and without consequence because the entire conception of the universe of sets is a complete fiction. Further, all the theorems of set theory are merely finitistic truths, a reflection of the mathematician and not of any genuine mathematical “reality.” Throughout this section, the “Skeptic” simply refers to the metamathematical position that denies any genuine meaning to a conception of uncountable sets. The counterview is that of the “Set Theorist.” Set Theorist's Response: The development of set theory, after Cohen, has led to the realization that there is a robust hierarchy of strong axioms of infinity. Elaborating further, it has been discovered that, in many cases, very different lines of investigation have led to problems whose degree of unsolvability is exactly calibrated by a notion of infinity. Thus, the hierarchy of large cardinal axioms emerges as an intrinsic, fundamental conception within set theory. To illustrate this, I discuss an example from modern set theory that concerns infinite games.

8. Cosmological intimations of infinity, Anthony Aguirre
8.6 Conclusions
It seems inescapable that, as finite beings, we can never prove that the universe is physically infinite: we cannot travel through infinite spaces or times or experience an infinite number of states. Nonetheless, I have argued that in modern cosmology, we may face the fascinating situation that the theories (particularly inflation) devised to explain the finite observed region of the universe also naturally produce an infinite universe , through a process called everlasting inflation. This can be the case even if the universe is initially finite, because the dynamics of inflation, played out over an infinite available duration, allow the creation of an infinite universe – even, in a sense, many such universes. Thus, although we cannot prove that the universe is infinite, strong evidence for inflation, along with the (strong but imperfect) theoretical link between inflation and “everlasting” inflation, leads to a strong inference of an infinite universe.
On even more speculative ground, I have discussed the possibility that in cosmology, finity might be problematic in certain ways, so that the very coherence and comprehensibility of our physical world is pointing to an infinite duration, or infinite number of states of the universe. Even if this conclusion is overreaching, however, t he analysis of the paradoxes or solutions to paradoxes that infinity can generate in cosmology can bring novel perspectives on some ancient riddles.

9. Infinity and the nostalgia of the stars, Marco Bersanelli
the cosmological principle was widely accepted well before observations could effectively verify its validity. Remarkably, recent data confirm the gradual tendency toward uniformity at large dimensions. 3 If we look at portions of the universe of sizes >100 Mpc or so, 4 we find that the general statistical distribution is maintained, whereas the details change from region to region, suggesting that the cosmological principle is indeed a good approximation of the real universe – at least within the limits of our current data. But how far can we verify its validity with observation?

some 13.7 billion years ago the temperature and energy density reached fantastic values everywhere in space.

Eventually, however, we approach the ultimate barrier of our cosmic horizon: we cannot get information from regions farther away than the distance traveled by light in the entire lifetime of the universe. It is not a matter of improving our instruments and observing strategy; rather, it is a fundamental limit set by the finite speed of light and the finite cosmic age. As a consequence, we can observationally test the validity of the cosmological principle only within a limited region of space. The extension of isotropy and uniformity to the entire universe – something we may call a “strong cosmological principle ” – is ultimately unverifiable. That would be the case for the extrapolation to the universe as a whole of any other physical property observed within our cosmic horizon. The expanding hot Big Bang universe brings with it the notion of a horizon that delimits the part of cosmic space we can probe: the observable universe is definitely finite.

It is clear that strong claims on the infinity of space based on the apparent spatial flatness of the observable universe might be as naive as those ancient ideas on the flatness of the earth based on local and inaccurate observation, but with an important difference: in the case of cosmic space, the ambiguity would remain even with infinitely accurate data. This is because our cosmic horizon, unlike the visibility horizon on the earth surface, is a fundamental boundary (Barrow 1999 ; Ellis 2006 ) not surmountable with improved instruments or more refined theory.

Part V. Perspectives on Infinity from Philosophy and Theology:
11. God and infinity: directions for future research, Graham Oppy

... God's possessing a given property to a certain maximal extent forces us to say that God possesses that property to a given infinite cardinal degree. (If, for example, there are 15 true propositions ... (AW: less similar)

13.1 Robert John Russell, Introduction
Western monotheism begins with the fundamental assertion that God is Absolute Mystery

explore ways in which modern mathematics bears important implications for our theological conversation about God. The key mathematical piece will be Georg Cantor's work

Georg Ferdinand Ludwig Philip Cantor ... born in St. Petersberg, 1845. His mother was a devout Roman Catholic, his father a Jew who had converted to Lutheranism. Raised in a musical family that relocated to Germany and converted to Lutheranism, Cantor was an accomplished violinist.

Cantor's views were supported by Richard Dedekind (1831–1916), but they were opposed by many others, notably Leopold Kronecker (1823–91). This opposition may have contributed to Cantor's declining health ... Cantor died 1918 in a mental institution.

antor actually distinguished between an unending but finite sequence of elements, such as the sequence 1, 2, 3, …, a sequence that is potentially infinite but always, in fact, finite, and the complete infinite sequence of these numbers thought of as a whole, that is, the set {1, 2, 3, …}. He called the potential infinite a “variable finite” and symbolized it as ; the actual infinite he symbolized by 0 , as we saw earlier.

Wolfhart Pannenberg (1928 – 2014) God of the Philosophers

In modern philosophical theology after Descartes and especially after Kant, the concept of being lost the fundamental importance it had in medieval philosophy. The traditional philosophical “demonstrations” of the existence of God as first cause of the universe and therefore as first being were replaced over time by the idea of God as the presupposition of human subjectivity and of its intellectual and ethical functions. The idea of God as first cause of the universe was not abandoned, but it was approached by another line of argument. In Hegel’s rehabilitation of the traditional demonstrations of God’s existence, in response to Kant’s powerful critique, the arguments were recast in terms of the rise of the human mind beyond finite reality to the idea of the infinite.

In this understanding, the idea of the infinite is the prior condition for perceiving the finite. Finite beings are conceived”as Descartes had already argued in Meditations (1641)”by being delimited by the infinite, which therefore is prior to anything finite, including even the human subject itself. Rising beyond the finite to the idea of the infinite belongs to the very nature of the human intellect. This, it was said, is evident historically in the fact of religion and is expressed theoretically in the rational arguments for the existence of God.

Friedrich Schleiermacher’s speeches of 1799 developed this line of argument with respect to the fact of religion. Religion expresses the human sense of the infinite as the prior condition for conceiving anything finite. In Hegel, too, the idea of the infinite replaces the concept of being or of the highest being as that concept functioned in medieval philosophical theology. This new approach to philosophical theology, however, did not abandon the conception of God as first cause of the universe, the creator of everything. What was changed was the way of reaching that conception.

Opozycja do Chaitin'a, dużo tekstów źródłowych www

Conway's życie Collatz conjecture Cayley graph Fraktal Infinity v.0







Przykład nieskończoności