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Drocket i copied text from cardinal numbers because you suggested cardinal numbers as a proof and can not be questioned, in the text if you fully read,it gives many paradoxes in the theory it self. besides cardinal numbers has nothing to do with infinity or finity subject that i am giving. but thank you very much for the hints, i do believe every information and idea is important and has the potential value to be read and researched. but i wished you could have understood the conflict in my mind that i am questioning ,how can you intend to solve physical related conflicts with the axioms of infinity,where infinity doesnt exist in the universe itself. so how can you explain the density of blackholes? citation ''Gazing into the sublime immensity of the starry night sky, pondering the awesome depths of the past and future, many hold that the Universe must be “infinite” in space and time. Now infinity is certainly a profound notion, holding a powerful emotional appeal across cultures. But while infinity is widely believed in, it has simply been taken for granted that infinity is a logically coherent concept. Such an assumption is mistaken. 1 I will argue that infinity is in fact a logical absurdity (that is, a self-contradictory notion) like a square circle or a four-sided triangle. 2 And since logical absurdities cannot refer to anything that actually exists, it follows that there is nothing infinite. I will show why we must conclude that the Universe cannot, therefore, be infinite in either space or time. 3 My case against infinity as a coherent notion will begin by defining infinity and its related terms (such as infinitude, “the infinite,” finitude, finite, indefinite, indefiniteness, etc.). Elucidating these terms will enable us to grasp the traditional understanding of infinity with enough detail to critically assess it. Once all of our terms are defined and understood, I’ll then show why the traditional understanding of inf init y, based on these definitions, is a self-contradictory notion. 4 Now if my case against infinity is sound, some radical implications follow for certain fields of inquiry. For instance, even if mathematics is operationally consistent in terms of its use of rules for calculation, it cannot be logically consistent when it makes use of infinity—at least, not as infinity has been traditionally construed. Consequently, mathematicians should abandon the use of infinity in making calculations in favor of a more logically consistent alternative. Since I’ll show that infinity is not coherent and ought to be dropped as a mathematical value, the question naturally arises as to how mathematics would then conceptualize unfathomably large quantities in calculation. An alternative concept to the traditional notion of infinity must be proposed. Fortunately, such a concept is available to us—a concept called indefiniteness. After debunking the traditional notion of infinity, I will explain how indefiniteness serves as a rational alternative. Indefiniteness is not simply conceptual sleight-of-hand; it really is different from infinity in that it avoids the logical contradictions inherent in the notion of infinity but while still retaining the mathematical utility that infinity has. Indeed, to throw out the traditional notion of infinity in favor of indefiniteness would not lose anything significant to the field of mathematics; to the contrary, it would actually make mathematics even more logically consistent than it is now with the notion of infinity still in use. And so I will offer indefiniteness as the new mathematical concept that ought to be used in place of infinity. In addition, the logical failure of the traditional notion of infinity, and the necessity of replacing infinity as a mathematical value with indefiniteness, carries even more serious implications for physics and cosmology: If infinity as a mathematical value is logically absurd, and if logical absurdities refer to nothing that can actually exist, then infinity as a mathematical value refers to nothing that really exists—at least, not according to the traditional definition of infinity. As a result, measures of space and time INTRODUCTIONThe Case Against Infinity 2 cannot really be “infinite” in the usual sense of the term. Thus in showing the selfcontradictions involved with the traditional notion of infinity, I will also be presenting reasons why cosmology and physics must hold that neither space nor time can be infinite—no matter how indefinite the vastness of space or time may be, the Universe as further reading: http://philpapers.org/archive/SEWTCA.2.pdf

at least you accept the universe might be finite. finally,there is an approach, please also read that the boundless universe is just an ilusion as it is explained in the nature article, also please explain to me how can you solve such physics questions with the explanation of infinity if mathematics doesnt obey the rules of universe, that makes it metephysics, it can be a language that consits of axioms but if it doesnt obey the rules of the universe, it is metaphysics, i believe in the near future,with the new discoveries in cosmology, the idea of infinity would be only applied to only time related problems in mathematics, because it will be known that time only can be infinite nothing else. i am accepting the concept of infinity and infinite series in mathematics ,but what i am trying to explain it can only be applied to time related issues in theories and in related to the laws of universe.otherwise it would turn it self to metaphysics. this is article taken from wikipedia ''the practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations,but rather from more methodological and pragmatic motivations.[citation needed]One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result,which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any objectof finite mass would be attracted with infinite force (and hence acceleration)by the infinite mass object, which is not what we can observe in reality.Sometimes infinite result of a physical quantity may mean that the theory bein gused to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory. This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations,theories and approximations often use infinite series, unbounded functions,etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physicall ymeaningful result, a process called renormalization. However, there are some theoretical circumstanceswhere the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thusinfinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does no tnecessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.'' Statements and proofs In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one can speak about one being greater or less than another. Then Cantor's paradox is: Theorem: There is no greatest cardinal number.This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set. Proof: Assume the contrary, and let C be the largest cardinal number. Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2C which, by Cantor's theorem, has cardinality strictly larger than that of C. But the cardinality of C is C itself, by definition, and therefore we have exhibited a cardinality (namely that of 2C) larger than C, which was assumed to be the greatest cardinal number. This contradiction establishes that such a cardinal cannot exist.Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result. Theorem: If S is any set then S cannot contain elements of all cardinalities. In fact, there is a strict upper bound on the cardinalities of the elements of S.Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and hence is of cardinality less than or equal to the cardinality of T. Cantor's theoremthen implies that every element of S is of cardinality strictly less than the cardinality of 2T. [edit]Discussion and consequences Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least inZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

Drocket, i do understand why do you question my knowlodge and give the general old answer, you do not have the proper knowlodge to question these topics.First of all , i expect you to make my questions and answers rubbish my answering with detailed scientific proof ,citation and good explanation, not by questioning my background with a look of scientific elitist.Science doesnt care about someone's academic background ,knowlodge, science only cares the questions with a strong scientific data and proof ,so you could be a professor in Mit and i can be a simple undergraduate but still question your work, your seat at Mit or anywhere doesnt give you the right to rubbish any scientific question. Well you are right ,i am not a graduate in physics or mathematics. i did my undergraduate in industrial engineering.but i still have the passion for science and physics,eventhough i am working in industrial fields. i do accept to challenge the mainstream science by no background or lack of knowlodge would look silly and wont work. But please understand me, i am not challenging anything, i am just asking questions, but if you tell me only graduates from math and physics can do that, or tell me i do not have the right to ask questions, you are definately a scientific elitist. i also believe even if i was not an engineer or a simple cash register in a shopping mall , i would still have the right to question many subject in physics. Please read the article from nature,it suggest that universe might be finite. ''The doughnut is making a comeback – at least as a possible shape for our Universe.The idea that the universe is finite and relatively small, rather than infinitely large, first became popular in 2003, when cosmologists noticed unexpected patterns in the cosmic microwave background (CMB) – the relic radiation left behind by the Big Bang. The CMB is made up of hot and cold spots that represent ripples in the density of the infant Universe, like waves in the sea. An infinite Universe should contain waves of all sizes, but cosmologists were surprised to find that longer wavelengths were missing from measurements of the CMB made by NASA's Wilkinson Microwave Anisotropy Probe. One explanation for the missing waves was that the universe is finite (see 'Universe could be football-shaped'). A mirror ball "You can think of the Universe as a musical instrument - it cannot sustain vibrations that have a wavelength that is bigger than the length of the instrument itself," explains Frank Steiner, a physicist at Ulm University in Germany. Cosmologists have suggested various 'wrap-around' shapes for the Universe: it might be shaped like a football or even a weird 'doughnut'. In each case, the Universe would appear to be infinite, because you would never physically reach its edge - if you travelled far enough in any direction you would end up back where you started, just as if you were circumnavigating the globe. But the notion soon suffered a setback. Cosmologists predicted that a wrap-around Universe would act like a hall of mirrors, with images from distant objects being repeated multiple times across the sky. Glenn Starkman at Case Western Reserve University in Cleveland, Ohio, and his colleagues searched for the predicted patterns, but found nothing. Undeterred, Steiner and his colleagues have re-analysed the 2003 data from NASA's Wilkinson Microwave Anisotropy Probe, looking for different shapes, including the so-called '3-torus', also dubbed the 'doughnut universe'. Despite its catchy nickname, this shape is tough to visualize, says Steiner. The 3-torus is an extension of the familiar doughnut shape and can be formed from a rectangular piece of paper. You can imagine gluing together first one set of opposite edges to make a cylinder, and then the second set of opposing edges to make a doughnut shape, explains Steiner. The 3-torus is formed in a similar way, but you begin with a cube and glue together each of the opposite faces. So if you were to attempt to exit one of the cube's faces, you would immediately find yourself entering again through the opposite one. Other shapes are possible Steiner's team used three separate techniques to compare predictions of how the temperature fluctuations in different areas of the sky should match up in both an infinite Universe and a doughnut one. In each case, the doughnut gave the best match to the Wilkinson Microwave Anisotropy Probe data. The team has even been able to pin point the probable size of the Universe, which would take around 56 billion light years to cross. Starkman, however, is not convinced that Steiner's team has done enough to win people over. "It could be true that the Universe is small," he says, "but this doesn't provide an answer one way or the other."Jean-Pierre Luminet at the Paris Observatory in France, who proposed the football-shaped universe in 2003, likes Steiner's work. He agrees that the analysis shows that the doughnut is still a likely candidate, but adds that other shapes are also possible. "One must remember that the (football universe) is still alive and well," says Luminet. Steiner believes that new and more precise measurements of the cosmic microwave background to be made by Europe's Planck satellite, which is due to be launched later this year, will help answer the question. "Philosophically, I like the idea that the Universe is finite and one day we could fully explore it and find out everything about it," Steiner says. "But since physics cannot be decided by philosophy, I hope it will be answered by Planck."

Hello Mike Thank you for your reply,, if BB has happened everywhere how can we explain singularity? i also consider the m theory and believe there is millions of universes beyond our universe,as you also know einstein suggested that time was present before BB. this is the research i got from nature website '' The doughnut is making a comeback – at least as a possible shape for our Universe.The idea that the universe is finite and relatively small, rather than infinitely large, first became popular in 2003, when cosmologists noticed unexpected patterns in the cosmic microwave background (CMB) – the relic radiation left behind by the Big Bang. The CMB is made up of hot and cold spots that represent ripples in the density of the infant Universe, like waves in the sea. An infinite Universe should contain waves of all sizes, but cosmologists were surprised to find that longer wavelengths were missing from measurements of the CMB made by NASA’s Wilkinson Microwave Anisotropy Probe. One explanation for the missing waves was that the universe is finite (see 'Universe could be football-shaped').'' for more details the link is :http://www.nature.com/news/2008/080523/full/news.2008.854.html

well i think you should understand the mainstream science, the first part should be to learn the universe is not infinite. taken from the nasa website ''The WMAP spacecraft can measure the basic parameters of the Big Bang theory including the geometry of the universe. If the universe were flat, the brightest microwave background fluctuations (or "spots") would be about one degree across. If the universe were open, the spots would be less than one degree across. If the universe were closed, the brightest spots would be greater than one degree across.Recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know that the universe is flat with only a 0.5% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe.''

infinity (symbol: ∞) is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end infinity is not a number, even though it is anumber , it is not metaphysics, it has to obey the rules of physics, you can define the universe as a cone or lets define like this, when the big bang happened, the light started to travel with the speed of C. i am trying to explain , the universe is not infinite at any given time ,will never be infinite, the universe can only be infinite at a given time t(x) with the function of time ,when t reaches infinity, so what i am trying to suggest is that time can only be finite , of course if there is smth beyond bing bang as it is mentioned in m theory ,it proves that there was time before big bang, but as you may know the law of physics in our universe started with big bang ,so the physics ,mathematics can only work in our universe not beyond big bang,because the law of our universes wont work there. So the term of infinity can only be explained by the function of time. Thank you for your shallow answer, my idea is wrong because you just said so. so for you science is not reasoning questioning and discussing,i have never seen a scientific research which quotes the whole book or says read this book for quotation. there is room for all kinds of debates in physics ,mathematic or science where everything can be questioned and discuss even it is silly. Pls give me specific quotes from each book that you mentioned on your reply. And pls answer my questions. you are telling me that universe has no boundaries,how can the universe has no boundaries at any given time,when it has a begining ,with start ,many years ago it was proved that universe was expanding and not stable ,how come a universe with no boundaries can expand pls explain. you told me that mathematics already defined infinity ,and the laws of universe has nothing to do with mathematics,so how can you explain physical problems such electro magnetic equations by using the term infinity or how certain electrons behave. i am not saying what i am suggesting is an absolute truth, it can be also or totaly rubbish but i belive it is worth to debate on ,but not like as the way you suggest which is plain and absolutely wrong because you just said so.

First of all thank you for response, i value every response eventhough you think and define these ideas as gibberish or rubbish. i do accept infinity is explained well in mathematics.Also i do accept for the base mathematics is the assumed axioms and logic.But also please note that everything has to obey the laws of universe. you can assume ,i dont know anything about cosmology or science or you can define me a moron.But it wont change the facts of physics and science. if you want i can make many quotes from many scientific sources that universe is not infinite,it is expanding. if smth has a begining, like big bang, it can never be infinite. i am not saying the term of infinity is not real , i accept the term of infinity,but i am trying to explain ,that only time can be infinite nothing else,not numbers not universe or not anything only time.

but isnt the topic both the subject of math and quantum

As everybody knows, the infinity is a term that can never be reached and defined in our current mathematics system. But as you may know,all our mathematic theories must obey the laws of the universe.As it is accepted by many physicians that the begining of our universe started with the big bang ,where the universe started to have a volume and a mass.As it is also known the fastest moving thing in the universe is the speed of light. Even though the universe is expanding, at the current time, it has a certain volume with certain borders. So at the current time ,lets say the given time is T(n) Lets define the time where the big bang happened and everything started T(f) Speed of light is C So lets define the border of universe in 3 dimensions in a given time is X(n-f) the volume of the universe or the border of the universe in the far end in a given time equals to : X(n-f)= C x (T(n)-T(f)) So the infinity at a given time can be explained in 3 dimensions which has an end at a given time, So how can we accept the infinity as unexplained and unreachable in our mathematical system? i believe time only can be infinite cos we already know there was time before big bang as it is asserted in the m theory,but in our mathematical modeling we use the term infinity and explain many things by it. it is true that we will never be able to reach the end of the universe at any given time, because we will never be faster than the speed of light,but this doesnt mean the term of infinity is true for such mathematical modellings.