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• An infinite number of photons can coexist in the same space.

• In regard to question #1: If you apply the guest checkout assumption to the hotel with an infinite number of rooms, is it possible to ever actually fill the hotel in the first place? After all, it will take an infinite amount of time to fill a hotel with an infinite number of rooms even without anyone checking out; so with guests checking out after a finite amount of time has elapsed after opening how do you ever get the hotel full?

  Since you'll never get the hotel full, you can always accommodate additional guests without moving anyone around.

  Just sayin'.

• My friend in college who was a pure math major told me about the infinity hotel, and I loved the concept! Still, as a practical person, I focused on the logistics. Since you can't move someone into a full room, but only into an empty room, you would have to bypass all the occupied rooms to find the unoccupied room to make your first move. But the first move is a mathematical impossibility, isn't it, since there is no empty room? So no "voila!" What vexes me more is the concept of "levels of infinity." If infinity means what I think it means, then an equation can't "approach" infinity. Infinity would reach right back around to the fixed point where the equation starts. It's an all or nothing concept, or else it means, as you suggest, "really big."
  New topic — is a stereo that "goes to 11" louder than one that only goes to 10? 🙂

• This is interesting, but misses the key conundrum in physics.

  If a field is quantized, there exists a discontinuity between the quanta.

  That means a Fourier transform or series representation will have an infinite number of frequencies or terms. This is a real infinity, which does not decay as is posted here.

  This infinite frequency representation is what shows up in the uncertainty principle, and also as the vacuum energy. So, the answer to the question posed in the title of this column is – indeed a YES, there exists an infinity (in frequency space). Whether it is in frequency space or real space is immaterial, as long as any quantity is truly infinite, we do have a philosophical problem.

  This is why quantum theory poses such a knotty philosophical problem that most minds trained to think classically miss.

• You cannot have a triangle of sides 100, 200, 300. Sum of any two sides must be greater than a third. If you can, then you have a triangle whose area tends to zero …

• The author's analysis in question #1 is stunningly idiotic. By adding the possibility of occupants checking out of rooms he has completely changed the hypothesis. This doesn't answer a damn thing except indicate that Pradeep has an infinite amount of oatmeal for brains squashed into a finite cranial volume.

• What if there was no space and no time. Just particles that interact with each other. And what we think of space and time is just a result of those interactions.

• Like all timelines/realities are converging and sharing the same space. Interesting you post this now at this time. 🙂 Synchronicity! Another shared space. lol

• " While you can imagine such a thing in the abstract world of mathematics, it cannot be realized at a physical level by a real-world object." – I have to disagree; anything a mind can imagine can be found as a real-world object. It is a fallacy to claim otherwise. In my view, Ron said it best….

• The infinity is measured by the poencia of the number classes,as integers,rational and irracional and transcendentes and real,when out in correspondence one-to- one,demonstrate that some infinity classes are greater that other.then the rational classes are countable,while the tal Numbers with irrational and transcendentes could not be placed one-to-one,then is the continuum,that has infinities infinitesimal,that are in theirs somatorium a finite Numbers,as the past is infinity and the future also is infinity,and onde point that behave not at the past or future is the present,that is irracional Numbers.then the complex Numbers are originated of the infinity due the left handed,positive and right handed ,negative,in theirs asymmetrics only could be conjugated in the infinity,through the Imaginary Numbers,that could be the metricmetrics of the time,that is the motions in the spacetime,it is the transformations of curves into for straight limes and vice-versa only occur by the motions,and therefore by the transformations From left handed into right handed and vice-versa,that only could occur in the infinity.the imaginara Numbers are linked for the time ,that is motions in the space with two torsions,show that the curves has infinities points as infinities holes,that is the time,measured by the relativístic time,or as the quarternions,that has the noncommutative properties as the time that encurve the space in two opposed orientations.i would like that Cantor and Dedekind,read this commentary

• The question to asked is physics really linked to mathematics. If the mathematics of the Universe magically changed, would the physical world change?

• Sigh…

  The problem of treating science as if it's true rather than just a approximation of reality described in abstract math.

  Of course such theories are going to adopt those abstractions, it's convenient for the models.

  Yes, there is evidence that time and space is discrete, but let's make sure this is due to observable fact, and not just discomfort with the concept of infinity.

• Isn't mind real? Mind. mathematics and imagination are not abstract but concrete and physical just as vacuum is real and physical. To think otherwise is both illogical and incorrect.

• You also fail to realize that if space has discrete units, then there is no such thing as a triangle. The mathematical triangle becomes an abstract approximation.

  Try to make a right triangle using only 4 pixels in Paint.exe. A 2 pixel by 2 pixel image.

  You won't get a triangle, you'll get three solid pixels and one that is a color somewhere between the background and foreground color.

• Actually, you don't have to go past all the full rooms to find an empty room.

  You order everyone to leave their room and simultaneously walk to and occupy room n + 1000.

  Actually, they don't have to move. Just renumber the rooms.

  Room 1 becomes room 1001, and so on. Now you have 1000 more rooms.

  This works because any countably infinitely set can be mapped to itself + N new items, by assigning the first 1000 to 0 – 1000, and the previous 1 to 1001 and so on.

• Slovenian mathematician Ivan Vidav has proved that we do not enter in contradiction when we say that cardinal number of real numbers is bigger than cardinal number of natural numbers. When we say that cardinal number of real numbers is equal to the cardinal number of natural numbers we also do not enter in contradiction. Vidav has resolved famous question of "infinite numbers" of Georg Cantor. Vidav achievement shows that "infinity" is not metrical term, it does not mean a given number or distance. Infinite distance and 1000 miles is still infinite distance. NASA confirms that universal space has Euclidean shape, it means is infinite. This means that amount of energy of space of the universe is infinite. "Infinity" is manifesting in the physical universe, Infinity is real, but human mind has difficulty to grasp that fully. Vidav has solved the question in details.

  1. Can a number that is finite but very large substitute for infinity? The answer is NO.

  2. What if there are physical limits to the smallest measurable amount of space? They are. Planck length is smallest possible lenght in physical universe.

• " While you can imagine such a thing in the abstract world of mathematics, it cannot be realized at a physical level by a real-world object." You can't. To imagine a geometric point with no dimension requires the physical interaction within the brain. This symbolic thought is the result of a great many physical processes. This may seem to be a trivial point but it has occupied me for many years. We err when we attempt to decouple the abstract from physical reality, an error that has distorted physics for some time

• For a long time I have wondered whether space and time are discrete or continuous or something in between. Unfortunately this discussion doesn't answer that question, and my knowledge of quantum physics is not up the the task of answering the question. It seems to me premature to "banish infinity from physics" unless a compelling theory requires us to do so. Does anyone reading this know if any current physics theory requires space to be discrete, or could it be continuous? I would love to know.

• Readers might like the discussion of the issue in the chapter on infinity An Aristotelian Realist Philosophy of Mathematics by James Franklin (2014). It says that we don't know from physics whether there is any infinity (or infinite divisibility) in nature; that it is in theory possible to do physics and the real analysis needed for physics in a discrete setting; but that it's prohibitively awkward to do it.

• Looking at the infinite hotel from a physical bent, puts a whole different spin on things.

  Imagine that the Infinite Hotel is full (let's ignore how long it took to get it full).

  Now come 1,000 new guests.

  So a new guest knocks on room #1 and instructs the occupant (on behalf of the hotel management) to move to room 1001.

  But physics says that information flow is limited by the speed of light. This means that it would take a finite amount of time for occupant of room #1 to get to room #1001 and tell that guest to move out to room 2001. Similarly with the guest in room #1001 instructing that guest to move one. And so on, ad-infinitum.

  This ripple in guests will take an INFINITE amount of time.

  So, you have not eliminated ‘infinity’. IN reality, the reason the hotel has room is that the excess guests are in the corridors between the rooms.

• You poor people need something better to do with your time and considerable intellect.

• When our math teacher told us that "at infinity parallel lines meet" I was indignant. If you are walking on a railroad track that is infinitely long, no matter how short or long your stride is, 1 micron or 1 light year, every step will land you on a track where the rails are the same distance apart (all other things being equal). So infinity may be a useful math construct for some problems, but it can be taken too far when applied to the physical world we participate with.

• …we are ants on a table called the Universe, do we know we are on a table? that the table is in a house, the house in a city, in a state, located in a country, on a planet orbiting our average star, located in a galaxy, etc etc. to infinity.

• No… the human derived concept and construct of infinity does not and can not ever exist in physical reality. The reason is because of a basic flaw with the application of our humanly derived numbering system for the counting of things such as applied to "accounting, logistics, economics, etc. and 'misapplication' of the same numbering system to the also faulty humanly derived concepts of distance, matter, space and time." In our real physical world, our numbering system as applied to accounting, logistics and economics never breaks down. In the world of mathematics, physics, astronomy, astrophysics, cosmology, subatomic particles, light, radiation, electricity, electromagnetism, fluid mechanics and many others, our numbering system breaks down in a horrible fashion such that another freakish form of mathematics called "Quantum Mechanics" had to be humanly derived to try to approximately explain the unexplainable. There is however another approach to the so-called infinitesimal and the infinite. That is to start by defining everything that is immeasurably large and unknown as having a maximum limit of One (1) Universal Unit.
  Space will never exceed One; Time will never exceed One; Energy (both +'ve and -'ve), at the 'beginning', had a maximum level of One; Matter did not exist before Space/Time and it will never exceed One Universal Unit. In the Grand Universal scheme of things it was impossible for there to be "Nothing" or what we commonly call Zero (0). Therefore, something has always existed, and something will always exist. The Universe is made up of a multitude of wave functions which can all be described by simple Euclidian Geometry. All or most Physicists believe that the smallest unit in the "Universe of (Quantum Mechanics) Mathematics" is Plank's constant h = 6.626070040(81)×10^−34. I hypothesise that the smallest Universal Unit of Time, until now undetected or unproven, is the incremental Unit of Time t = 1×10^-64. This is the point of Time at which Energy was at its Apex and Energy was all that existed with no evidence of Matter or Space/Time. The Universe oscillates from Time t = 10^-64 to Time t = 1 and then back to Time t = 10^-64. Similarly, Space expands to its ultimate Universal Unit value of One (1) and back to a value directly related to Time. At Time and Space both having values of One, the Universe is flat, bounded and limited. Everything in the universe is nothing more than a series of wave functions which are all easily described without the use of Zero(0). The use of the words "concrete, logical and such" in these discussions is all an illusion; just as you perceive the illusion of hardness and hard facts when in fact the hardness is the meeting of two objects bounded by a multitude of internal waveforms. And, finally, that is why division by zero is not allowed and is not defined, because Zero does not exist in the smallest and largest realms of things in the Universe. Or, as some believe, it would mean something similar to infinity; which also does not exist in the "real physical world".

• 'Infinity' has always puzzled me since I was a youth decades ago. I see it as a word. Nothing more. I don't mean this in a derogatory way but using 'infinity' is a cop out. Not a lot different from using the idea of God or magic to explain what we don't understand. There are no infinities in the physical universe. Everything is measurable…we just don't have the means of doing the measuring. A number is like a word…just a symbol. There is nothing out there in existence that is a ghost, Easter Bunny or Infinity.

• to examine infinity is to examine God. To rephrase, to examine infinity is to examine the ultimate nature of existance. And yet again, the examination of infinity is the examination of the difference between life and death where the finite and the infinite intersect. And one more time, is there infinity or anything at all after death at which point the gift of life has been withdrawn by God? Is the universe itself and all it contains contingent on Life itself? It is likely that that which we observe and discern as the universe altogether ceases to be in an existential sense and it all winks out of existance as we close our eyes for the final time. When we close our eyes in death the entire notion/construct of time ceases since when dead the passage of a second and the passage of 10 to the 50th seconds are no longer perceptually different. There is no evidence one way or the other that the entirety of the universe and ourselves in it were not simply winked into existance 5 minutes ago by a higher power, said universe complete with doppler shifts and distant red shift galaxies and cosmic background microwaves and etc. So its likely that the universe and the near infinities it proclaims are contingent entirely on Life observing it and said Life concluding that the meshing quanta of info bits coupled with neuronal nets arranged into comprehensatory biointelligent algorithms sees and constructs the infinte universe from moment to moment to moment going 'forward' in time. The implication is that Life is from God. God creates Life. Life creates the percept and construct known as the universe. The universe ceases and 'unbecomes' at death. By definition infinity is therefore finite in all its possible manifestations. God is Life. The Infinity of Godhood and Godness is of a different kind and quality than the infinity-ness knowable to human understanding.

• As much as I appreciate your coverage of this interesting topic, a little more background would have benefited this article. In particular in point 2 you should note that any such minimum to space (as oppose to space-time) is inevitably in conflict with special relativity – regardless of how small that minimum is. This leads to contradictions with observations even if you push the minimal scale down to the Planck length. There is a large body of work on this, see eg the review

  http://relativity.livingreviews.org/Articles/lrr-2013-2/

• with #2, If the measuring stick cannot measure smaller than 0.001microns (mu), then I assume that 100 mu measured in this system is in fact 100.000 mu, similarly 200.000 mu and 300.000 mu.

  However, in reality, the measurements could be 100.0004 mu, 200.0008 mu and 300.0005 mu,
  in which case 100.0004 + 200.0008 = 300.0012 mu which is > 300.0005mu

• Sketchy answers: (1) an n-room hotel loses on average n/100 guests each day, which for large enough n is virtually certain to always exceed 1000; (2) for measurement resolution r=0.001 micron, and assuming rounding-down, the true dimensions could be close to 100+r, 200+r, 300 micron and this has area 1000.sqrt(6r) ~= 77.46 micron^2; (3) I have no idea what physical limit you're trying to highlight here, so I'll just say 30 degrees and wait patiently for the intended insight.

• Hilbert's Hotel is a furphy, because for countable infinities, by definition,
  infinity + any number = infinity (i.e. INF + n = INF); so rooms can be added to Hilbert's Hotel ad infinitum and no-one will have to double-up, or be excluded.

  Now, if the number of guests is N and the daily probability of a guest vacating a room is 1%, then it is probable that N/100 rooms will be vacated each day.
  If 10^n new guests arrive, and N is >>>>>>> 10^n, let N be 11 orders of magnitude bigger than 10^n, thus,
  N= 10^11n, so the number probable vacancies will be 10^11n/10^2 = 10^(11n-2),
  thus the number of probable vacancies is much greater than the number of new guests
  by 10^(11n-2)/10^n =10^ (11n-2-n) = (10^10n -2) .

  OK, so for 10^3 newbies, and 10^33 rooms, it is probable that 10^31 rooms will be vacated, which is 10^28 times greater than the number of rooms required, so in all probability the 1000 newcomers should be able to be accommodated in their own room because of rooms vacated.

• there really is only one form of infinity that can be absolutely confirmed…death…when the unitary organism dies it never ever comes back to organic life for the remainder of whatever persists…..there is no hypothesis that death/life is, like space/time, curved and turns back on itself such that by going to the end you are actually returning to the beginning

• As far as Hilbert's Hotel is concerned, I am a fan of the "solution" which says the whole construct is a non-imaginable thing in that the hotel is either full, or it is not. If it is full (which it is by definition in the puzzle), that is the end of it because there is no more empty rooms. Just because there are infinitely many rooms, does not mean that by necessity there are any unoccupied rooms. The intuition that there would be empty rooms is a bit of fallacious logic which confuses/combines infinite and finite quantities, in that is relies on the logic that for any two FINITE numbers, say V and M, V + M is always less than infinity (specifically aleph naught), which is true. This logic is then unthinkingly and fallaciously carried forth to mean that for any INFINITE number C, and any FINITE number Z, apparently Z + C is also less than infinity, a clear, plain, simple, hard to miss, contradiction. Seriously, how the heck has the thing persisted for nearly a hundred years?

• Concerning Q2: It is not clear what this discretisation would look like, but if the world was made on a grid, it would difficult to explain the apparently perfect isotropy of physical laws requiring a rotational symmetry which is by definition continuous. Actually, the best description of reality that we have: quantum field theory is entirely based on such continuous symmetries (such as the Poincaré group). Obviously, it is impossible to distinguish between a continuous symmetry and a discrete symmetry that approaches this continuous symmetry. But the simplest description favored by ocam's razor is clearly the continuous symmetry. This continuous symmetry is precisely the reason why these triangle cannot exist at the discrete level.

  Concerning Q3: the Heisenberg uncertainty principle solves this dilemma. From this uncertainty principle, one can derive the
  For an object large enough, the angular precision can be very high, but never infinite.

  As a matter of fact, in a finite world (for example with symmetric boundaries), quantum physics give rise to a quantization of the available quantum states even though the world is continuous. As a consequence, even in a continuous world there is a discrete number of possible configuration which translate into a finite number of possible states for the world (which is nevertheless astronomical)

• infinity is a very useful mathematical concept…but does it exist in the real world? of course not. …if there an infinite universe then there would be an infinity of almost vanishingly small nearly massless neutrinos such that there would be so many neurinos clumped everywhere that we would all be literally suffocating under their ubiquitous presence……it follows that if the universe is not infinite then nothing inside the universe is infinite

• I am from Germany – my English is not so good.
  » Is space truly infinite […] or is it in some way “pixelated” at the lowest level? «
  That sounds as if space can't be infinite if it is pixelated. Could space not be infinite (expansion-wise of course) even though it is pixelated/quantized?

• it strikes one as strangely odd that anybody could hope to characterize the ultimate nature of infinity based on a 2 sentence word puzzle…..math puzzles are great, but they remain math puzzles.

• Pradeep, Thanks very much for your article, which I thoroughly enjoyed reading.

  Is infinity real? The answer, I think, depends in which domain infinity is being used as a quantity. It also depends on whether or not we reject abstract domains from the definition of the word "real" — I think a blanket rejection would be extremely unwise in some, if not many, cases.

  If we look carefully at the focussing ring on a manual focus camera lens, we will observe that the distance scale covers the range from a short distance all the way through to infinity distance. If we take the lens apart and remove its hard stop at infinity then the lens will indeed focus beyond infinity. Modern high-performance lenses that contain one or more very-low-dispersion glass elements have to focus beyond infinity at their hard stop limit, at circa 20 °C, in order to allow for the fact that such glass changes its refractive index with temperature.

  The above is an example of infinity being an insufficiently large (limiting) quantity to adequately represent a parameter of a real-world physical device. Proposing a finite numerical upper bound, to be used in the place of infinity, would seem to be the opposite of what it required for both theoretical and applied physics.

  There are, of course, well known solutions to the above and similar practical problems that require quantities of infinity and beyond: to express the problem and its parameters in a different, and much more appropriate, domain. Before we can attempt to perform this, we have to understand *exactly* what the numerical quantities involved, and their units, really mean in practice, i.e., practical reality. The seemingly absurd notion of "focussing beyond infinity" becomes easy to understand only after switching to the correct domain, in which the unit is the dioptre (US: diopter). In this domain, infinity distance simply has a value of zero; beyond infinity has values of the opposite sign from sub-infinity distances; only a distance of zero would be an infinite number of dioptres. Easy-peasy 🙂

• Pradeep wrote: "If you had a finite number of rooms, the pigeonhole principle would apply. In this context, this common sense principle says that you cannot have n+1 pigeons in n holes if there is only room for one pigeon in each hole. But in an infinite hotel, it’s easy! We just move every resident from his or her room n to room n + 1,000. Voilà! Rooms one to 1,000 are now empty!"

  I found this part of Pradeep's article, plus the readers' comments on it, particularly fascinating because these types of [tending towards] infinity problems were more than adequately addressed decades ago by both transmission theory and queuing theory.

  A finite number of rooms is a bounded mathematical set; an infinite number of rooms is a valid unbounded mathematical set. However, there is a vast difference between an *unordered* set and an *ordered* set, on both the mathematical level and the practical level. A basic, unordered set, specifically excludes the time domain; whereas, an ordered set has either an implicit or an explicit inclusion of the time domain. A basic, unordered set problem can be solved using basic logic. An ordered set problem can be solved using only temporal [time domain] logic and its relevant mathematics, such as transmission theory and queuing theory.

  To the readers who wish to grasp the fundamentally important practical differences between non-temporal (static) logic and temporal (dynamic) logic, I recommend reading the book Water Logic by Edward de Bono.

• Cool comments from R. Morin and Pete Attkins, thanks. I think along those same lines.

  In the context of R. Morin's Universe with the addition of One Universal Unit being equivalent to a "full infinite set" in the sense of a full Hilberts Hotel, I like to rid myself metaphysically of the philosophical notion of particles in the classically thought of sense, then like Pete Atkins I grant myself use (non use? depends on how it is stated?) of the Continuum Hypothesis, and think of spacetime (at a frozen instant of time, which is a logical impossibility since that would make spacetime all one value aka (0) and thus not there due to no relative contrast, so is just a visual aid/trick/thought experiment) metaphysically as a grid of quanta or infinitesimals or Planck spheres (which we only interact with the surfaces of) which are spatially everywhere, but where the whole grid structure can move (movement in time) any amount in any direction (so the collection of Planck spheres form a four dimensional "sphere" in three dimensional space), which causes the values associated with each Planck sphere to change. This surface is discrete in it's quantized nature, but continuous in the sense that the grid/surface does not have a fixed location and so can move. Thus in my opinion it solves the whole continuous/discrete debacle.

  In this Universe, the amount of information (entropy) contained in any given Planck sphere would, in a relative sense using the Continuum Hypothesis, provide for us our idea of spacetime measure and more specifically provide the very shape of spacetime itself (and so spacetime would be an "illusion"), and therefore be the that which was gravity. Visualizing things this way, I feel like I can see how Loop Quantum Gravity and the field description of General Relativity are synonymous descriptions of the that which we call Gravity. Thus in my opinion this way of looking at things holds the potential to solve the quantum information paradox. In my opinion it also might possibly provide a hint about the nature of non-locality, entanglement, and also possibly a hint that the Axion, Graviton, and Higgs Boson might possibly all be in a way the same thing.

  So, in short summary, I think it would be a thoroughly foolish move to try and remove infinity from mathematics, specifically mathematics which we use in physics to describe reality.

• 1- if you have at least 100×1000 rooms you'll put all new people in a day. which is about 13 times higher than the hotel with the highest number of rooms currently existing on earth according to google, but of course the universe didn't impose this unfortunate restriction and I am sure we'll be able to build much bigger ones in no time.
  2- not sure…how will you measure to see if it holds? things can get crazy near h. yeah you can have like 100+0.001,200+0.001 and 300-0.001. if you have that, you have area, no? is the next one an error calculation thing? I hate those. I guess you can have 300+/-0.002 for the sum of the sides, which means it can be 300-0.002, which is smaller than 300-0.001. plus, don't you need to measure flatness? can be non-euclidean.
  3- I'd -try to- shoot along major axis. size of hole can accommodate for out of foci shoots caused by size of the ball or drunkenness of the player due to small reflection angles. any other direction may or may not work because of losses to friction and collisions. thinking of a tiny pool player as small as physically possible is too funny.

• I was wondering if the bumpers of the pool table where exactly on the ellipse and how that affects the bank shot. It might be that depending upon how hard you hit the ball it is actually effectively hitting a different concentric ellipse and that could be the cause for the different angles of reflection.

• On No Solution
  It's not obvious how discovering "infinity-free equations" or solutions to the above puzzles could ever eliminate or resolve the reality/unreality of the infinite (barring, perhaps, ambitious nominalistic attempts). For example, if our equations are constituted by mathematical objects, and if mathematical objects exist qua Platonism—as they are so often assumed to (implicitly or explicitly)—and if the infinite is constituted by these objects, it seems plausible to conclude that any equation free of the infinite is impossible, again, insofar as *there are* mathematical objects.

• My comments on the differences between infinity in mathematics and outside-the-brain reality are:

  1. I cannot conceive of a thing, like a mathematical point, existing that has any one of its dimensions at zero. Not just approaching zero, but zero. To me, something that supposedly exists with a size of zero just isn't there. It doesn't really exist.

  2. One of the thought experiments used in mathematics that's behind a lot of thinking about infinity is to imagine starting with a single set, I, of positive integers. Are there as many even integers as the total number of all integers in the set? Mathematicians suggest that if you remove the subset of even integers (E) from the original set of all integers, you can pair off each even integer in E with each integer in I and see that the subset E is the same size as the original single set I. But, it seems like what's happened is that by removing the subset of evens from the single original set, you've broken the natural relationship within that single set of each even integer marching in lockstep with the odd integers around it (and vice versa). This natural relationship in the single set suggests that there are twice as many total positive integers as even integers. In a real physical experiment (performed in outside the brain physical reality), if one removes something from its natural environment and observes it in an isolated state, you can never be sure that the situation in the isolated state is the same as in the single natural state. For instance, if you remove a cell nucleus from a cell and then study the isolated nucleus and cell remnants, you may get totally different results from what happens if the nucleus is left in the cell. This can lead to experimental artifacts. Scientists in experimental sciences, like biochemistry, always have to be careful to watch out for these artifacts. I wonder if by removing the subset of even integers from the single, original set of all positive integers, mathematicians are causing experimental artifacts in their thought experiment? It's like they're removing the nucleus (the even integers) from a single cell (the set of all positive integers) and claiming that their observations of the isolated nucleus and cell remnants are equivalent to the situation if the even integers were left inside the cell where the one-to-two relationship of evens to total integers is left intact.

  Just an idea. Thanks.

• Fascinating, but entirely fatuous I'm afraid. The problem is not in your mathematics, Horatio, but in your language. In brief, whether or not something akin to, or precisely equal to "infinity" exists, no amount of grappling with the concept using human language is gong to even remotely describe it, much less capture it so as to make it a practical tool. We as a species are perhaps several million generations away from understanding even a few of the bugs we have found under the rocks we have to date turned over in our garden.

• Roger,

  I totally understand your remark "I cannot conceive of a thing, like a mathematical point, existing that has any one of its dimensions at zero. Not just approaching zero, but zero. To me, something that supposedly exists with a size of zero just isn't there. It doesn't really exist."

  I present the following article just in case it helps to clarify that a "mathematical point" is sometimes a very necessary abstraction, or a very necessary limiting case, of reality:
  https://en.wikipedia.org/wiki/Point_spread_function

  Mathematics is, I think, not a science per se; it is both a formal language and an exquisite toolbox that enables branches of science to be: precise; logical; independently verifiable; and self-correcting.

  Pradeep's article has highlighted the need to always question, and to constantly peer review, the currently existing fundamental concepts that we use in both mathematics and science.

• John Baez recently wrote an excellent series of articles on struggles with continuum in physics which I highly recommend:
  https://www.physicsforums.com/insights/struggles-continuum-part-1/

• Responding to Jeff (up top), suggesting at least the potential for the physical manifestation of an infinity… While identical photons (or for that matter, any integer-spin particle, or "boson") can occupy overlapping quantum states (and consequently be in the same physical space), there is still a limit to the energy that can occupy a given space, constrained by the Planck temperature (about 1.417×10^32 kelvins). Exceeding that energy with photons, say with some unimaginably powerful laser, would create a photon black-hole (called a "kugelblitz" in astrophysics), effectively smearing the information describing the photons' energies over an event-horizon with a finite surface. At least at small scales, the universe doesn't appear to accommodate any (observable) infinities.

• I have a problem with the dimensionless point. If it has zero dimensionality, doesn't that make it a multiple of zero? In which case it doesn't mathematically exist. It is self-negating. So if you have an infinity of zeros, they would still be zero.
  The dimensionless point is presumably an ideal of location, but location is a function of space and time, so if you do eliminate all space and time from it, it has no location. Even hypothetically, if they are not all at one location, but spread around in space, this would model volume, on a plane, area and along a single dimension, a line, but since they have no extension, they don't create these configurations, only model them. It would be the dimensional forms which give definition to the non-dimensional ones, not the other way around. They just are the ones falling in the forms.

  As for planck lengths, how could one be defined, if there wasn't a smaller size level to define its length/end points? Otherwise the error bar would be no smaller than the supposed length and it would be meaningless.

  Sorry, not getting rid on infinities, without creating more problems.

  The problem with infinity is that it is a non-physical property and can't be ordered as such. Think of it as the opposite of absolute zero. We can point in the general direction, but never reach it.

• To which I would add that a dimensionless point is a very useful abstraction.
  Given thought is a process of abstracting order from nature, giving an ideal of location some infinitesimal dimensionality only confuses the premises, as it raises other problems, such a smallest possible points, how many angels can dance on the head of a pin, etc.
  So it is a tradeoff, a self negating concept, versus a fuzzy concept. Since it is an abstraction in the first place, having it self negating is the cleanest solution.

• Pete,

  Thanks for the comment. I had a hard time understanding the point spread function wikipedia page, but I totally agree that mathematical abstractions like size 0 points can be useful in both mathematics and modeling reality. But, I sometimes think that, especially, physicists forget that not all mathematical abstractions actually exist outside the mind. You're also right that mathematics may not be so much a science as it is a very useful toolbox, but it still seems like it and all areas of study could benefit from good experimental technique, which was the point of my second comment. But, I admit this is sometimes easier said than done.

  Pradeep's and all the articles at Quanta are very good, I think. This is one of the better science-related websites. Thanks, Quanta! Another good one, which isn't all science, is Nautilus magazine at http://nautil.us/

• I was always wondering if an object is moving through space, and we assume that
  between point A and B there are an infinite number of points in space, which the
  object has to pass, how much time does it take the object to move between the single
  points?

• Puzzle 3 reminds me the resolution of the so-called "ellipsoid paradox" in thermodynamics, which dates back to a letter Fallows sent to New Scientist in 1959:

  http://i.imgur.com/OYf4VUr.png

  The geometrical construction is extremely clever in case you have not seen it.

• In this post, I have attempted to answer two questions that are based on the article itself and on some of the readers' comments:
  1. Why do we need to keep infinity?
  2. Why is it crucially important to understand the domain(s) to which our numbers and our variables actually belong?

  For the purpose of this discussion, I shall answer the very interesting question posted by Christian on June 20, 2016 at 10:07 am.

  The answer to Christian's question is the number zero, but this number has *two* domains: its primary domain is the time domain, for which the SI unit is seconds; its secondary domain is a mandatory caveat that applies when the number is either zero or infinity. The caveat is: zero multiplied by infinity (and vice versa) = a precise value or a precise formula (which I have derived below).

  The velocity of an object is the rate of change of position of the object. Let s = distance in metres and t = time in seconds; now, velocity (v) = ds/dt (units are m/s). The amount of distance that the object moves during a specified time period = the integral of ds/dt over this time period.

  Let's assume for the sake of simplicity that the object is travelling in a straight line from A to B and that its velocity is constant. Let S = the distance between A and B in metres; and T = the time taken to travel from A to B in seconds. Now, it is tempting to write the equation v = S/T (units: m/s; metre·second⁻¹). This equation would be correct for the purpose of measuring the average speed of the object, but it can easily lead to a domain error [aka: a category error; a category mistake] if we rely on it to properly answer Christian's question.

  When there are N equally-spaced points between A (point 0) and B (point N), the time taken by the object to travel between two adjacent points (t) = T/N seconds. As N tends towards infinity, t tends towards zero; and when N=infinity, t=0 seconds. However, (t × N) *must always* be equal to T because the value of N cannot possibly change either the velocity of the object or the time that it takes to travel from A to B. This is the mandatory caveat that I mentioned earlier: zero multiplied by infinity = T seconds.

  This caveat is, I think, algebraically valid given the above context; however, it contains a fundamental domain error. When N = infinity, or any finite value, the time (t) it takes the object to move between adjacent points depends on its velocity (v), not on the average time (T) that it takes to travel from A to B! This domain error was caused by writing v = S/T.

  The correct equation is S/T = the integral of the velocity over time T. Likewise, the distance between adjacent N points = S/N; and the travel time (t) = the integral of the velocity (v) over time t. As N tends towards infinity, t tends towards zero, therefore the integral of ds/dt tends towards being equal to ds/dt = the instantaneous velocity at that point in time. Therefore, as N tends towards infinity, ds/dt tends towards v, and s/t tends towards v. Near to this limit:
  s = S/N (metres)
  v = s/t (metre·second⁻¹)
  Therefore t = s/v = S/N/v (metres per metre·second⁻¹), which is the numerical answer in the primary domain of time in units of seconds, as N tends towards infinity;
  and (t × N) = S/v, which is the mandatory caveat that applies when t=zero because N=infinity, and vice versa: zero multiplied by infinity = S/v (metres per metre·second⁻¹).

  As a school pupil, I was taught to always cancel out the units in derived formulae, e.g., metres per metre·second⁻¹ should always be simplified down to just seconds as the unit. This learning by rote short-circuits critical thinking skills and it prevents the proper analysis of problems; it is the main cause of committing fundamental domain errors in both mathematics and science — especially when a numerical value tends towards zero or it tends towards infinity. Removing infinity from mathematics and science would, I think, further short-circuit critical thinking and careful analysis.

  The value N appears to be a dimensionless number, but it isn't dimensionless and unit-less. It is the ratio of the distance between points A and B to the distance between each of the N points along the path from A to B, therefore its domain is distance, and its units are metres per metre.

  I hope that the above has made it abundantly clear that, just because an object can theoretically traverse two adjacent points (in an infinite set of points) in zero seconds, this most definitely does *not* imply that the object can therefore traverse an almost infinite number of points, or infinity − 1 points, in zero seconds!

  I also hope that my usage of UTF-8 and Unicode characters in this post display correctly on the website. Apologies in advance if they don't.

• I would like to congratulate Pradeep on eliciting such a smörgåsbord of aperçus in the comments thread. (I hope my usage of extended ASCII diacritics displays properly …)

• There are infinite infinities. The foundation of reality is smooth.

  Every universe, in the multiverse, has a unique quantization unit.

• In question 1, about a hypothetical hotel with a countably infinite number of rooms, all of which are occupied, room n+1000 is already occupied for any n. The guests in room n can't be moved into room n+1000 without increasing the number of guests in room n+1000. Still no vacancies.

• Imagine our hotel have n occupied room. To accommodate 1000 more guests, just increase size of hotel by 1000 empty rooms.

• @Paul, Rooms n+1000 (for any value of n) were indeed previously occupied, but, if every room occupant currently staying in this hotel is reassigned to their current room number + 1000 then the rooms numbered from 1 to 1000 become available to new guests. The total number of rooms in this hotel was specified as being infinity therefore it will retain its infinite number of rooms after the reassignment of its current occupants.

• Interesting that somebody thought my comments on "Is Infinity Real?" were "cool". Somebody else understands my concept of "near zero" to one; or, as he views it: zero to One being a full "Infinite Set" which can include, measure and count all things – while at the same time I suggest that Zero does not exist – thus precluding the existence of infinity and the seemingly paradox of his "Infinite Set" . It is not an infinite set at all, because as I stated at the beginning, Zero does not exist. One can keep adding the numeric "0" between the "." and the "1" in ".1" such that the "number" t = Time in the denominator of the "Special Equation for The Big Bang or of perhaps Everything" 'approaches' zero; but it can never become or reach Zero. Why? Because in the Grand Scheme of all things and reality, "Zero = 0" is absolutely meaningless. And, invisible Energy is at its maximum of 1.

  "J E says:

  June 18, 2016 at 3:47 pm

  Cool comments from R. Morin and Pete Attkins, thanks. I think along those same lines.

  In the context of R. Morin's Universe with the addition of One Universal Unit being equivalent to a "full infinite set" in the sense of a full Hilberts Hotel, I like to rid myself metaphysically of the philosophical notion of particles in the classically thought of sense, then like Pete Atkins I grant myself use (non use? depends on how it is stated?) of the Continuum Hypothesis, and think of spacetime (at a frozen instant of time, which is a logical impossibility since that would make spacetime all one value aka (0) and thus not there due to no relative contrast, so is just a visual aid/trick/thought experiment) metaphysically as a grid of quanta or infinitesimals or Planck spheres (which we only interact with the surfaces of) which are spatially everywhere, but where the whole grid structure can move (movement in time) any amount in any direction (so the collection of Planck spheres form a four dimensional "sphere" in three dimensional space), which causes the values associated with each Planck sphere to change. This surface is discrete in it's quantized nature, but continuous in the sense that the grid/surface does not have a fixed location and so can move. Thus in my opinion it solves the whole continuous/discrete debacle." More at:

  https://www.quantamagazine.org/20160616-infinity-puzzle/

  In addition: While it is 'true' that for applied physics 'to work' in a defined 'domain', it depends again on humanly defined applied mathematics with all of the approximated humanly defined SI Units of Time, Mass and Dimensions. Given that Space/Time is expanding at an accelerating rate; and, given that "Time" slows down in a given domain as we spin and circle through this universe; e.g., "The orbital speed of Earth averages about 29.8 km/s (107,000 km/h)" while the rotational velocity of the Milky way is estimated that "The Milky Way as a whole is moving at a velocity of approximately 600 km per second ( = 2 160 000 kilometer/hour) with respect to extragalactic frames of reference." In addition, the Andromeda Galaxy and the Milky Way are converging at about 110 kilometres per second (68 mi/s or = 2 160 000 kilometer/hour). "Absolute measurement" in any domain is meaningful at that location (domain) only in terms of humanly defined units. In "relative terms", all of the combined velocities will have a 'relative' impact on both increase of mass, and shortening of Time and distance/measurement of length.

  I reiterate that in "Universal Mathematics", where the maximum value of Energies can never exceed +'ve 1 and -'ve 1 "Unit less" entities, Space/Time are also "Unit less" entities as is all Matter (or congealed energy). One may 'say' that Matter was Zero at one point in Time; but, that Zero is meaningless because all matter existed in its precursor – Energy. In addition, Space/Time was so insignificantly small that the conversion of Energy to plasma > subatomic particles > building blocks of the first hydrogen atoms > coalescence of hydrogen clouds > the furnaces of atomic fusing suns fabricating helium and all of the heavier elements had "yet to start".

  Unit less entities are required, out of necessity, to avoid tainting the laws of the Universe – at the subatomic, galactic and universal scales – by erroneously (and even arbitrarily) defined, approximating and limiting humanly defined units of measure, SI or otherwise.

  Finally, there is "no infinite set" within the Unit less set of Energy, Matter and Space/Time. Space/Time starts at a Time approximately equal to 1 X 10^-63. It 'seems' that this incredibly small number approaches Zero; but, at Time = 1 X 10^-64 "everything is meaningless" because only Energy existed. There is no infinity. There is the infinitesimally small at the point of the Big Bang. And, there are and exceedingly large number (10^64) of steps until the fullness of Space/Time; but they are all finite.

• @Pete Attkins, the trouble I see with that logic (which is just a re-statement of the puzzle of Hilbert's Hotel, so this is the trouble with the classic logic behind Hilbert's Hotel) is that in re-naming room n to be room n + m for any finite integer m, you did not change the occupancy in any room of the hotel. They are all still occupied. As to the claim that doing so, re-naming room n as room n+1000, somehow creates 1000 additional rooms in the hotel, I disagree. To see this you have to formally introduce the idea of the time it takes to count all the rooms into the problem, as opposed to letting time ruminate in the background of the puzzle, imparting it's unspoken assumptions into the situation. I will try to explain.

  Think about the finite situation: if the hotel had 100 rooms, would re-numbering the rooms from 1-100 to 101-200 somehow change the number of rooms? What then changes if we had 1-"infinity" rooms? For what reason does substituting the total number of rooms with "infinity" (as defined as counting "forever", and thus I would argue that if one had "forever", one could count all the rooms. Notice here "forever" is just our familiar notion of infinity, just applied to specifically time instead of specifically space.) imbue the hotel with magical properties?

  It seems to me that taking rooms 1-infinity and then using the rule that for any room n, re-name the room to n + m for any integer m, we just end up with a hotel full of occupied rooms, just numbered m – (infinity + m), and which would take exactly forever to count. Thus it seems to me that in the original formulation of Hilbert's Hotel we have let some kind of notion of a lack of countability slip into the most basic of infinities in terms of our space and time proportioning. Specifically we let space be infinite but time be either finite or at least less infinite than space, and thus created for ourselves some logical conundrums. Specifically we have said something to the effect that even if we could count forever (so had infinite time), we could not count all the rooms (so space is also infinite and what's more, it is more infinite than time).

  Thus my claim is effectively that if one imagines space and time as being equivalently infinite, one cannot consistently logically construct the paradox which is Hilberts Hotel. To construct the classic version of Hilbert's Hotel, one must make one of the two, space or time, be more infinite than the other. I hope this helps explain my reasoning :).

• @ "Pete Attkins says:
  June 22, 2016 at 11:37 am
  @Paul, "Rooms n+1000 (for any value of n) were indeed previously occupied, but, if every room occupant currently staying in this hotel is reassigned to their current room number + 1000 then the rooms numbered from 1 to 1000 become available to new guests."

  I am very sorry to state that your apparent 'logic' is circular in nature, uses slight of hand, breaks the premise of the definition of ∞ and attempts to not only pull itself up by its own boot straps; but, also attempts to stand on 'nothing' or that which equals zero. In other words Paul, your illogical story is a non sequitur and thus fallacious and illogical.

  Let us assume for a moment that infinity does exist in the physical world. The basic problem is that infinity is so poorly defined and little understood by anybody. However let us accept the usual definition: "Infinity (symbol: ∞) is an abstract concept describing something without any bound or larger than any number." Ref.: Wikipedia: https://en.wikipedia.org/wiki/Infinity

  The illogical slight of hand in adding 1000 rooms to "a number 'already' without bound" is meaningless. It breaks the definition of infinity which is already "larger than any number". It is thus a non sequitur. If infinity exists in reality – it cannot be added to, since it is already boundless – otherwise we could arbitrarily make these additional arbitrary mathematical functions, which while appearing to be true give meaningless results to ∞ using the same "illogical" tricks and slight of hand such as:

  1. ∞ + 1 = ∞ (is really ≠ ∞)
  2. ∞ + 10X10 = ∞ (is really ≠ ∞)
  3. ∞ X 10^3 = ∞ (is really ≠ ∞)
  4. ∞ X 10^10 = ∞ (is really ≠ ∞)
  5. ∞ X 10^10^10 = ∞ (is really ≠ ∞)
  4. ∞ X ∞ = ∞ (is really ≠ ∞)

  The above are all "Non sequitur in nature and in formal logic, they are all arguments with conclusions that do not follow from its premises. In a non sequitur, the conclusion could be either true or false (because there is a disconnection between the premise and the conclusion), but the argument nonetheless asserts the conclusion to be true, and is thus fallacious – illogical.

• We can also see infinite sets as a good approximation for finite sets, since things are often simpler with infinite objects; here are two examples :
  -Statistician often use the Gaussian distribution to approximate the binomial distribution (by the central limit theorem). The advantage is that there is a unique Gaussian distribution, but one distinct binomial distribution for each N.
  -For any infinite sequence of integers, there exists an infinite subsequence that is monotonous. A discrete version of this theorem may be : for any sequence of n^2+1 integers, there exists a monotonous subsequence of size at least n+1. Fun exercise, but tricky !

  There is a large but finite number of other examples where people use infinity because it's simpler.

• @J E, Many thanks for your comment. I worded my reply to Paul such that it conveyed the problem as it is presented in the article, without giving away the fallacies that it contains.

  One of the fallacies is the reassigning the the occupant(s) of room n to room n+1000 (for all values of n in the infinite set of rooms) because this turns the set problem into both a queue problem and an ordered set problem. As you pointed out, queue problems and ordered set problems do include the time domain; whereas unordered set problems do not.

  Suppose there's a very wide vehicle W occupying all three lanes of a three-lane motorway and it is travelling at 30 mph. As time passes, the queue of traffic behind it will increase in length. Now suppose that the motorway is infinitely long and there is an infinite number of vehicles in front of W that are all travelling at 30 mph or faster: this makes no difference whatsoever to the queue of traffic behind W. The hypothetical infinitely-long motorway that's already full is just a red herring, as is the infinite number of rooms in the hotel paradox. What matters is not the things that are already ahead of the queue bottleneck, it is the flow rate of the queue *at* its bottleneck. The speed of light is the upper bound to the speed at which objects can travel through a queue.

  There is a finite maximum rate at which people can be checked in to a hotel, and a finite rate at which existing occupants can be reassigned to different rooms. Even if people could move around at the speed of light, a hotel having an infinite number of rooms can never be filled; but if, for the sake of argument, it is full then the room reassignments would take an infinite amount of time to complete, so the room reassignment is also a red herring. If the hotel had, say, only 10,000 occupied rooms, I think it would take a long time to reassign all of the guests to different rooms! And think of the nightmare of serving and billing the guests according to their room number.

  I think the most glaring red herring in Hilbert's hotel thought experiment is that, as the room number increases, it takes longer to speak the number and to write or print the number; there reaches a point at which it would take a whole day just to print the room number. But of course, a red herring:

  "may be either a logical fallacy or a literary device that leads readers or audiences towards a false conclusion. A red herring might be intentionally used, such as in mystery fiction or as part of rhetorical strategies (e.g. in politics), or it could be inadvertently used during argumentation." — Wikipedia.

• @R. Morin wrote:
  "1. ∞ + 1 = ∞ (is really ≠ ∞)
  2. ∞ + 10X10 = ∞ (is really ≠ ∞)
  3. ∞ X 10^3 = ∞ (is really ≠ ∞)
  4. ∞ X 10^10 = ∞ (is really ≠ ∞)
  5. ∞ X 10^10^10 = ∞ (is really ≠ ∞)
  4. ∞ X ∞ = ∞ (is really ≠ ∞)

  The above are all "Non sequitur in nature and in formal logic, they are all arguments with conclusions that do not follow from its premises. In a non sequitur, the conclusion could be either true or false (because there is a disconnection between the premise and the conclusion), but the argument nonetheless asserts the conclusion to be true, and is thus fallacious – illogical."

  Thank you for formalizing that. That is exactly what I meant when I wrote:

  "As far as Hilbert's Hotel is concerned, I am a fan of the "solution" which says the whole construct is a non-imaginable thing in that the hotel is either full, or it is not. If it is full (which it is by definition in the puzzle), that is the end of it because there is no more empty rooms. Just because there are infinitely many rooms, does not mean that by necessity there are any unoccupied rooms. The intuition that there would be empty rooms is a bit of fallacious logic which confuses/combines infinite and finite quantities, in that is relies on the logic that for any two FINITE numbers, say V and M, V + M is always less than infinity (specifically aleph naught), which is true. This logic is then unthinkingly and fallaciously carried forth to mean that for any INFINITE number C, and any FINITE number Z, apparently Z + C is also less than infinity"

  Although I would have been more clear if I had written the last bit of the last line as "Z + C is always less than the version of infinity the proponent of fallacious logic is mentally referencing", as I claim that someone who mistakes Z + C as being equivalent in terms of cardinality with the counting forever infinity* has a priorily granted themselves the logical paradox that infinity is always bigger than itself as their baseline definition of infinity. This is the mistake you have run clear of in your above scenario.

  * Note that the wiki definition "without bound", potentially if one is not careful and specific in what they mean, runs the risk of making the mistake of letting the time and/or space infinities get mis-proportioned, and hence I prefer the logical cleanliness of "counting forever", which I also argue is just our a priori notion of infinity, and also fits in perfectly with what R. Morin wrote above. I talk about "counting forever" as it pertains to Hilbert's Hotel in the context of time and space in the comment which is right above the one I quoted from R. Morin. Sorry for the awkwardness of having to reference such a complicated topic across 60+ individual comments. I nonetheless enjoy this hodgepodge conversation.

• @R. Morin,

  You misread my comment, which was just my attempt at stating the paradox as it is presented in the article. The final part was: "The total number of rooms in this hotel was specified as being infinity therefore it will retain its infinite number of rooms after the reassignment of its current occupants."

  The number of rooms will indeed stay the same after the reassignment. I did not claim that all of the current occupants would be assigned to new rooms by the reassignment. 1,000 of them probably won't be assigned to a room and they will have to join the back of the queue for booking a room in the hotel.

  In a previous comment (June 18, 2016 at 11:10 am), I used the example of a lens that focusses beyond infinity and I provided the solution to this apparent paradox/impossibility.

• @ Sand:
  "-For any infinite sequence of integers, there exists an infinite subsequence that is monotonous. A discrete version of this theorem may be : for any sequence of n^2+1 integers, there exists a monotonous subsequence of size at least n+1. Fun exercise, but tricky !"

  I think you are playing fast and loose with "infinity" in terms of the concept of cardinality, and so are playing fast and loose with what you are referring to as the "integers" as well. As long as you grant that the n^2+1 "infinity" (even though in my opinion cardinality does not exactly permit the addition of a single unit to an already whole set, so presents yet another consistency issue for you, as in what exactly you mean by "infinity +1".) is larger than the n+1 infinity, and that the n+1 infinity is at least as large as Aleph Naught, you may have a consistent statement for the first sentence i.e. "For any infinite sequence of integers, there exists an infinite subsequence that is monotonous."

  I have no qualms with this part: "for any sequence of n^2+1 integers, there exists a monotonous subsequence of size at least n+1." Although without the re-working of the first sentence I would re-state it: for any integer n (so a finite number), then for any sequence 1,2,3,…,n^2+1, there exists a monotonous subsequence of size at least n+1.

• @Sand: "-For any infinite sequence of integers, there exists an infinite subsequence that is monotonous."

  This is reminiscent of the Bolzano–Weierstrass theorem, which in turn suggests a construction yielding some very large (but non-infinite!) numbers: see theorem 3.5 of the wonderful https://u.osu.edu/friedman.8/files/2014/01/EnormousInt.12pt.6_1_00-23kmig3.pdf .

• @J E : Thank you for the comment, I could be more accurate indeed : in the second example my infinity is Aleph 0 (implied by the term "sequence", but maybe not clear on the context), and it is not only valid for sequences of integers, but also sequences of real number or elements of any totally ordered sets.

  Thus,
  -in my first example the infinity is Aleph 1 and there is the additional structure of a measurable vector space,
  -in the second example the infinity is Aleph 0 with the additional structure given by ordering. And I think of n^2+1 as the same infinity as n+1, since A^2 has the same cardinal as A for A infinite.

  By the way, the second example is a theorem of Erdös and Szekeres.

  @eJ : Thank you for the reference, I didn't know it. Theorem 3.5 also illustrate that it is often easier to deal with infinity than with finite but enormous numbers.

  To elaborate on your comment about Bolzano-Weierstrass : what would be a non-trivial BW theorem for finite sequences ? First we need a non-trivial version of the notion of limit of a finite sequence, for example : x is the "(k, e)-limit" of (x_1, …, x_n) if the k last terms of (x_i) are at distance less that e of x. The BW theorem would be something like : for any sequence (x_1, …, x_n) in [0,1] , there exists a subsequence with a (k, e) limit, for e=k /m with m^2+1< n.

  It may be possible to transpose many theorems to finite sets, but the number of parameters like n, k, e, m… will grow exponentially with the complexity of the theorem

• @Sand: No comment from me. You interact like more of a troll than anything. In case you are not though, I disagree about the infinity present in your second example, because I consider the statement of the theorem of Erdös and Szekeres to itself misunderstand infinity and thus be illogical nonsense. This is because I disagree about the definition of Aleph Naught that Cantor, Hilbert, yourself, Erdös, and Szekeres, as well as pretty much every other mathematician uses, and the conversation we would need to have about it is way beyond the scope of this awkward comments section, especially given the tone deaf reply you just gave.

• @Sand : Keep in mind this is all under my own possibly whacky version of infinite set theory, which I have not fully shared and feel is beyond the scope of this comments section, and which is not in any text books as far as I have found. Nonetheless, I will try to explain :). You wrote:
  "in the second example the infinity is Aleph 0 with the additional structure given by ordering."

  Under my view this is then some infinity greater than Aleph0, except I am willing to bet you and I agree the actual cardinality of the set is not greater than Aleph0 (though I contend for different reasons), and under my view this has implications that do not permit your conclusion in connection to the theorem statement i.e. there is a major inconsistency. For clarity, I refer to cardinality as expressing the total count of the members of the set. See below.

  "And I think of n^2+1 as the same infinity as n+1, since A^2 has the same cardinal as A for A infinite."

  In the n + 1 example, under my view, the cardinality of the set of all n is Aleph0, as defined as "counting forever" (as I have discussed several comments up in this conversation, sorry for the awkwardness), and so the cardinality of n+1 just does not really exist as the +1 is implied in the definition of Aleph0, so you can always add one more infinitesimal (as defined as a single member of the set, and has cardinality 1/theCardinalityOfTheSet) to the set without changing the sets cardinality. Thus we agree that the cardinality of the set created by the sequence of n + 1 is Aleph0. We could also get here by just saying n + 1 is just a re-numeration of the integers, but I wanted to add the above to the conversation.

  In the n^2 + 1 you have real problems though, as under my view the infinite set which represents the sequence n^2 + 1 is just a re-numeration of the set of integers, so is like giving different names to every member of the set of integers ala Hilbert's Hotel, and so has cardinality Aleph0, and so is 1-1 with the sequence of all n, n + 1, or any other integer sequence. The way you are thinking about it, however, I believe, is that when you take an integer and apply the rule n^2 +1 you define a new bound to the set of integers.

  To examine this idea, let us first drop that pesky + 1 as per the above discussion, leaving n^2. Your reasoning seems to me to be as follows: take any finite n and square it, and the result is less than infinity, thus if I took the integer notion of infinity and squared it, the result will be less than the integer notion of infinity (i.e. 'since A^2 has the same cardinal as A for A infinite'), which is an obvious logical contradiction.

  Think about it: If I had a count of Aleph0 blocks, and I divided every block into a subcount of Aleph0, the set would have cardinality, via induction, of Aleph0^2 (so it would have Aleph0 infinitesimals of size 1/Aleph0 and Aleph0^2 infinitesimals of size Aleph0^2, so is countable as Aleph0 if you have one forever as you have Aleph0 infinitesimals of size 1/Aleph0^2 for each infinitesimal you have of size 1/Aleph0, and "countable/orderable" at Aleph0^2 if you have forever^2 worth of time.), and would represent the entire set of rationals.

  Thus, under my view, you can either say your sequence of n^2 +1 (never minding the + 1 for now, especially in light of infinitesimal cardinality i.e. how many infinitesimals of what size is the 1 comprised of) has a cardinality in the rationlas, thus permitting your subsequence to be made. Or you say the sequences n^2 +1 and n + 1 have the same cardinality, and thus dis-allow your subsequence. What can't be done is to make the two sequences, make them infinite but different sizes, have them both be in the integers, and furthermore make one an infinite subsequence of the other. That whole ship does not hold water under my view.

  I also want to note that due to my definition of Aleph0 the whole way I think about cardinal numbers is very non-standard, and so likely represent a different way of thinking about cardinality than people are accustomed. I am sorry if I that, or anything else, utterly confused anyone, and I hope this helps.

  (editors note: since this comment is much nicer than the other one from me sent today which you have not posted yet, please do not put that one up, please take it down if you did, and please view this as the reply I meant to send. This is a really technically tough thing to talk about it and it has become a sometimes touchy subject for me over the years so I apologize for my bad manners 🙂 )

• this universe is in itself not infinite but what it is expanding into might well be.
  there is one infinity within this universe. the progression of numbers one by one.
  what is the last number? is there one? or is this progression in itself infinite?

• @E J : I had not read your previous discussion with R. Morin and misunderstood your comment, so my answer probably made things even less clear, I'm sorry. As you said in your first comment, Erdös and Szekeres 's theorem is about finite sequences (n is finite). My sentence about "A^2 and A has the same cardinal" was essentially there to convince you that the first statement (any infinite sequence has a monotone infinite subsequence) could indeed be seen as a version of Erdös-Szekeres dealing with infinite sequences.

  My whole point was that the notion of infinity is "an extremely convenient approximation" as is quoted Mark Tegmark in the article. And the notion of countable set as defined by Kantor and explained by Hilbert in the infinite Hotel metaphor is extremely important in measure theory, probability and topology, so I'm not ready to dump it like that 😉 For Kantor and Hilbert, in an infinite hotel you can fit an infinity of groups with an infinite number of people in each, and this is an important fact in measure theory. Their ship holds water since 1874 (Kantor's article) , and gets mathematician across the sea since 1902 (Lebesgue article on measure theory).

  However, Einstein, Poincaré and Lorenz got free of the extremely solid notion of "Galilean referential" to develop relativity, so maybe at some point you need to get free of this notion of infinity to have a better understanding of the universe. But for this you need to show how your new paradigm solves existing problems, as did Einstein.

• Unrelated to the previous discussion, let me attempt to talk about the elliptic billard. The link with "does infinity exists" is very loose, but it's still an interesting problem.

  Here is my point of view : consider that the effect of spin is neglected, and that the collision with the border is elastic; they are clearly not negligible in the video (the angle after the bouncing depends on the speed !) but the question is not about that.

  Now, the fact that the ball is not a point makes that it will collide the border a little earlier than in the "ideal trajectory" where it is a point. Assuming it is small enough, we can neglect the curvature of the border near the point of collision. The actual trajectory after the collision will then be parallel to the ideal trajectory, but at distance r/sin a where r is the radius of the ball and sin a is the angle of the collision.

  Thus, to hit a target of radius 1.5x r , we need that sin a > 1/1.5, ie a > 41°. The best way is thus to shoot in the opposite direction, where a=90°. With axis of lenght 1 and 2, I computed that you can shoot in an angle of at most 110° to the large axis. And this, even is r is very small !

  In the video, Alex Bello is able to hit the target with any angle. The axis of his billard are not of ratio 2, but more like 1.06. In this case you can shoot in any direction. It would be interesting to see whether you can hit the target with a real billard of axis with ration 2 and shooting with angle more than 110°.

• @Sand: you wrote:
  " For Kantor and Hilbert, in an infinite hotel you can fit an infinity of groups with an infinite number of people in each, and this is an important fact in measure theory."

  Hence the problems with infinite set theory.

  Any actual intellectual critique of my ideas? Do you know how the general structure of arguments work pertaining to the philosophy of science with regards to consistence and closeness to the Truth?

  Speaking of which, it is Cantor and Hilbert's idea of infinity which leads to the necessary introduction of Lebeusge Measure Theory in order to make the Real Line back into a measure, as that property goes away when the continuum, which the real line is intended to model, (the continuum is the Truth and our real line is a model of it, and thus is the truth: it may or may not be the model which is closest to the Truth) is declared an uncountable (and thus unmeasurable and unproportionable) infinite set by Cantor via the diagonal argument (which btw ironically does not use his version of Aleph Naught to establish the 1-1 relationship, otherwise he could not establish said 1-1 relationship, which is yet another inconsistency of Cantors). Sooo….

• ^^I meant "which btw ironically does not use his version of Aleph Naught to establish the lack of a 1-1 relationship, otherwise he could not establish said 1-1 relationship, which is yet another inconsistency of Cantors"

• Doh! Third time..
  I meant "which btw ironically does not use his version of Aleph Naught to establish the lack of a 1-1 relationship, otherwise he could not establish the lack of a 1-1 relationship that he is apparently able to show, which is yet another inconsistency of Cantors"

  I also wanted to add that there was a LOT of pushback on Cantors work in his time due to the fact that it is all non-constructive (math up to that point was 100% constructive out of The Cogito), as well as full of "paradoxes", which in reality are just contradictions built into the logical structure of the definitions used for the paradoxical theory, at least in this case.

  Additionally, years were spent by guys like Bertrand Russel trying to fix those paradoxes to no avail. It is not that anyone EVER thought our version of infinite set theory was the Truth, it was that math was more fruitful with than without it, and mathematicians assumed in time the problems would get addressed.

  Somehow contemporary mathematicians are blissfully unaware of these historical tidbits when they defend infinite set theory, as they defend it both in spirit as well as in strategy as if it is the Truth. Sand, you are no exception to this.

• @J E :
  1) I do not agree with your critics of infinite set theory. I insist that Hilbert's hotel is essentially a metaphor, what he shows is essentially that there exist a map from the set of integers into itself which is injective but not bijective; the injection is n -> n+1. We could work without assuming the existence of the map n->n+1 (and without induction), but it would double the lenght of any mathematical argument, without significant improvement. I also think that the diagonal argument of Cantor (sorry for the K!) shows indeed that there is no 1-1 map between R and N, within the set of axioms of infinite set theory. And finally, Lebesgue measure theory is not here to "make the Real line back into a measure" but to define the measure of possibly very complicated subsets of R^k (for simple sets Riemann integration is sufficient).
  2) I did not make any actual critics of your theory because I don't understand it (which is a bad reason indeed). What I have understood is that you think of Aleph0 as "counting forever", but that's obviously not a real definition. So, maybe you want to consider a theory like : Zermelo-Fraenkel set theory, but without the axiom of infinity ? It's a possibility, but then what is a circle ? Is it connected ? Is Pi defined ?
  A more specific critic on your way to see the real line : you pretend it may be countable, and this should solve the problem of measure. But then, what is the measure of a point ? If it's zero, then the measure is alwas zero. If it's not, then are the interval made of a finite number of points ? If yes, your real line does not seem to be continuous for me. If no, then the measure of an interval is infinite, and so your measure is useless. The point of this is to make clear what I don't understand in your ideas.

• Aren't the possible frequencies (at both "ends" of the spectrum) of light infinite?

• JP asked the very interesting and important question: "Aren't the possible frequencies (at both 'ends' of the spectrum) of light infinite?"

  The frequency, f (Hz), of each individual photon is proportional to its energy, E (joules). E=hf, which is known as the Planck–Einstein relation, therefore f=E/h, where h is the Planck constant: approx. 6.626E-34 J·s.

  The emission of photons from a surface that has a temperature above absolute zero is a stochastic process that has a well established wavelength versus temperature spectral density function, where each individual wavelength in the spectrum = c/f metres.

  If we collect, say, one thousand light photons, it is extremely unlikely that any two of them will have exactly the same energy (exactly the same frequency in Hz). If we collect, say, 1E50 photons, most if not all of them will have different levels of energy. If we could accurately measure the frequency of all of the N electromagnetic photons that have ever existed in the whole universe, the number of different photon frequencies must be less than or equal to N. This will hold true until the end of the universe, which will occur long before N reaches infinity 🙂

  The above has, I sincerely hope, demonstrated the vast difference between *epistemic* possibility/probability, which lies on a continuum from zero to unity; and the *ontic* reality of that which has already manifested in our universe, and that which *can possibly actually manifest during the future* of our universe.

  I also hope that the readers will have spotted the flaw in my reasoning thus far. What about the increasing level of redshift that all photons undergo as they journey through the circa 13.8 billion years of our current universe? Does the energy level of each photon (therefore its frequency) decrease along a continuum as it travels through space-time; or does its energy level jump down in discrete steps, perhaps with each of its traversals through a Planck length of approx. 1.616E-35 metres? If the photon energy jumps in discrete steps then, by definition, its highest frequency component is octaves above its nominal frequency — unless time itself is discretized rather than existing on a continuum.

  Frequency is not actually a fundamental property of an object. Frequency is a derivative parameter of a rotating phase vector. Frequency can be expressed as either the number of cycles per second of the phase vector, or the inverse of the seconds per cycle of the phase vector. Throughout each individual cycle of the rotating phase vector, the vector may undergo second and higher order phase variations (phase perturbations) and/or magnitude variations (amplitude perturbations). These perturbations increase the spectral bandwidth occupied by the rotating phase vector. NB: A perfectly pure, single frequency, has a spectral bandwidth that reduces towards zero only after its elapsed time tends towards infinity.

  The Dirac delta function is a normalized singularity: it has an infinite peak magnitude that exists for the duration of zero seconds; its mandatory secondary-domain caveat stipulates that infinity multiplied by zero must equal unity. This function has a frequency spectrum that extends from minus infinity to plus infinity. The Dirac delta function is a hypothetical idealized function that is used for the purposes of system design and system analysis via its conjugate mathematical transforms. It seems obvious to me that it is highly applicable to the analysis of the proposed singularity that instigated our universe.

• Should we build a hotel with an infinite number of rooms at infinite expense, hoping for infinite returns on our investment, or is it better to go with a very large, finite number of rooms to cap our costs? Would we need more rooms than a very large number? Suppose we build in Shanghai, with low costs and good prices, a really good breakfast buffet, and all the accoutrements that tele-commuters need for work. We include cloning labs (I think that's legal in China), with the rule that any guest may use the labs, provided that the clones that result continue to stay at our hotel. Sure a few people check out, but with the cloning, the numbers are increasing pretty fast. Then whoosh, the curve takes off: the clones want the good life, so they clone themselves, lodging their clones in the hotel, and with better and better cloning techniques there will be more and more clones, who will each make more, and more, and more clones. Bottom line: there isn't a number of rooms big enough to house our guests–there just isn't. And by the way, you have all forgotten more than I know about Quantum mechanics. I am a classical musician who read some Cantor in grad school because we have sets in 20th c. music and I was interested. If memory serves, some of those Cantor articles were entitled not "infinity" but "transfinity", at least in the original German, which seemed to me to capture the concept well. Thank you for the puzzle !

• This is a PS because I forget to mention the resource questions. It might be objected that there is not enough space in Shanghai to build such a hotel. We will go up and up. In fact we will pass the zero gravity point, which will make it easy to reach our hotel from outer space, so we will have off-planet transport, which will attract telecommuters from other planets, and we can order in from off planet, so they will always have enough to eat. Perhaps they will gain a reputation for working on tedious tasks like tax returns (they must have those throughout the universe), accounting in general, or statistics (!). In short we are not limited by space for building, resources, or job opportunities for our clones, who will work throughout the infinity of the universe. In short, no big number will do the job. Thanks for reading.

• "And by the way, you have all forgotten more than I know about Quantum mechanics."

  You appear to have forgotten that those who claim to properly understand quantum mechanics, don't. Secondly, those who know even a smidgeon of quantum mechanics know full well that capitalising the "q" and/or the "m" is a bright red warning flag of BS.

• @Jeff – An infinite number of photons cannot occupy the same space, for at least 2 reasons…

  1. an infinite number of photons represents an infinite amount of energy.
  2. At some point the number of photons would cause the space to collapse into a black hole.

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