Gabriel's Horn and the painter's paradox 

Cavalieri's principle i guess

Yes, that's true. The shape would simply have a wider mouth if the domain were extended to some value less than 1 (but greater than 0).

No, it is not true that an infinite surface must contain an infinite volume "by definition." You have clearly never heard of fractal shapes before.

​@@angelmendez-rivera351 Yes I have. You dont understand my critique.....if you claim that the surface area of Gabriel’s horn is infinite and the volume is finite then how do you explain that IF you remove the theoretical horn, what is left is the volume, the surface area exposed of which is precisely the same in all respects to that of the horn itself, is not also infinite and that is by definition, the case. These paradoxes are nonsense, nothing more than conceptual contradictions, word games. Some talk about a 1" square for example as being infinitely subdividable, which it is not if you think within the context of the logic which governs unsympathetically, the material realm. Or do you think that that context should be consequent of illusion only by which infinity, an abstraction , can be defined? Consider...if a 1” box is subdivided by any fraction, let’s say by half, each of those segments, as would be for a line which designated the subdivision, would be quantifiable and thus would require a quantifiable point or means of demarcating that point of division, IF we are to work from the logical context respecting materiality. You don’t get to have it both ways. To clarify, one cannot claim infinite points along a line segment 1” long because we know that somewhere in the entirety of space the line begins and ends a quantifiable 1” away. Since all claim it contains infinite points, we know that it is of them that it is composed. This means, by the logic by which the line segment was initially defined as such and which must apply, that each end of that line must be designated by a point beyond which there is no other (obviously or the line segment would be greater than 1” or be able to have no end). I will avoid the page and a half validation of this for now and just insist that this being the case, each of these points must be quantifiable in an aspect that the end of the line can be defined at all. Additionally, if we subdivide this line segment by half, let’s say, the above issue remains and is itself validated by another understanding that at that point of subdivision, each segment also must be designated by a point beyond which there is no other, by which it too must be quantifiable in some aspect by which it can be understood as adjacent to that point of the other end of the divided line where the segmentation was made. You cannot explain this away without admitting compromise of the very logic by which the problem was defined to begin with. Piffle, nothing more. So, the infinitely subdividable box to which the above if applied, reveals the same problem. A final point is that one cannot in any quantifiable instant of time claim that this or that box or shape of any kind is infinitely subdivided because in that context (of time) it would be a process of subdivision which could never, ever end. It would continue on forever (there can be no “complete infinity”). A final point would be that to claim infinite points along a line segment or infinite subdivisions of a finite shape would be necessarily to define these points as “infinitely small” which is another conceptual contradiction. Infinitely small would mean non-existent. Small is a concept which can only exist within the context of understanding of a quantifiable measure. Infinity cannot be quantified and is devoid of limits. The two concepts cannot be paired if logic and truth are to be respected. A quantifiable line segment could not have infinite parts by virtue of the very means by which it is quantifiable. It cannot be both. A quantifiable box cannot be infinitely subdivided and Gabrial’s horn cannot have an infinite surface area which when removed did not leave behind and infinite “surface area” of the once contained volume. Sorry, but logic and reason must not be dispensed with in these discussions or the conclusions drawn are very suspect.

@@jamestagge3429 *if you claim that the surface area of Gabriel’s horn is infinite and the volume is finite then how do you explain that IF you remove the theoretical horn, what is left is the volume, the surface area exposed of which is precisely the same in all respects to that of the horn itself, is not also infinite and that is by definition, the case.* The surface area of the exposed previously enclosed volume is equal to the surface area of the Gabriel's horn itself, and is thus infinite. This much was made clear by the video. It is clear you do not understand what the properties of Gabriel's Horn are. No one ever claimed that the surface area stops being infinite, except you. *These paradoxes are nonsense, nothing more than conceptual contradictions, word games.* The reason this is nonsense is because you have presented it inaccurately in your comments, because you do not understand it. *Some talk about a 1" square for example as being infinitely subdividable, which it is not if you think within the context of the logic which governs unsympathetically, the material realm.* Logic does not govern the material realm. Logic is entirely conceptual. The material realm is studied not through logic, but through the scientific method. Also, calling it the material realm is very ignorant, since there are physical systems not comprised of matter, such as boson fields. There is nothing in the material realm that prevents a 1 meter line segment in spacetime to be infinitely subdivisible. Find me a scientific law that says otherwise. I have a bachelor's degree in physics, and I know for a fact no such law exists. *Or do you think that that context should be consequent of illusion only by which infinity, an abstraction , can be defined?* Context is important, even within mathematics. If you are working with a finite field of characteriatic 2, then 1 + 1 = 0. If you are working in a Boolean lattice, then 1 + 1 = 1. If you are working with any other recognizable algebraic structure, then 1 + 1 = 2. *Consider...if a 1” box is subdivided by any fraction, let’s say by half, each of those segments, as would be for a line which designated the subdivision, would be quantifiable and thus would require a quantifiable point or means of demarcating that point of division, IF we are to work from the logical context respecting materiality.* There is no such a thing as "logical context respecting materiality." You are pulling nonsense out of your rectum. There are only three laws of classical logic. The law of identity, the law of noncontradiction, and the law of excluded middle. None of these say anything about the material world whatsoever. Logic is the study of propositions, not of the material realm. Besides, for how much you think you understand the material realm and you respect it, I know for a fact there is so little you know about it, as far as science is concerned. *To clarify, one cannot claim infinite points along a line segment 1” long because we know that somewhere in the entirety of space the line begins and ends a quantifiable 1” away.* No, you absolutely can claim that there are infinitely many points in the line, and we can prove it. Your line of reasoning is fallacious. *I will avoid the page and a half validation of this for now and just insist that this being the case,...* Thank you for admitting that you are not willing to accept proofs or disproofs, and that you are going to insist that you are right, regardless of what anyone else tells you. This saves me a lot of work. Knowing that I am having a discussion with a big baby, rather than with an adult, is helpful. *.You cannot explain this away without admitting compromise of the very logic by which the problem was defined to begin with.* No, I absolutely can explain it. However, you already admitted you are not willing to listen to what anyone else has to say on the matter, and that you are going to keep insisting without changing your mind. In other words, you have admitted you are a brick wall. So, I will not waste my time giving said explanations. If one day you become a mature, grown adult, and you decide you are willing to listen to others who disagree with you, and are willing to learn from the experts, rather than being arrogant, egotistical, and proud of your ignorance, then that will be day when I can give you a chance and provide you with such explanations. *A final point is that one cannot in any quantifiable instant of time claim that this or that box or shape of any kind is infinitely subdivided because in that context (of time) it would be a process of subdivision which could never, ever end. It would continue on forever (there can be no “complete infinity”).* Ah, so you are an ultrafinitist. Well, thank you for clarifying that, now I know talking to you is a waste of time. *A final point would be that to claim infinite points along a line segment or infinite subdivisions of a finite shape would be necessarily to define these points as “infinitely small” which is another conceptual contradiction. Infinitely small would mean non-existent.* This is false, but I am not going to waste my time explaining this to someone else who only wishes to continue being wrong. *Small is a concept which can only exist within the context of understanding of a quantifiable measure. Infinity cannot be quantified and is devoid of limits. The two concepts cannot be paired if logic and truth are to be respected.* You must have failed calculus when you were in high school. Although, fine, calculus is taught veeeeeeeeery poorly anyway in high school. *A quantifiable box cannot be infinitely subdivided and Gabrial’s horn cannot have an infinite surface area which when removed did not leave behind and infinite “surface area” of the once contained volume.* I already addressed this. *Sorry, but logic and reason must not be dispensed with in these discussions or the conclusions drawn are very suspect.* If you were to take a course in logic, you would fail, miserably. It is clear your understanding of what logic and reason are is just as lacking as your understanding of scientific concepts and mathematical concepts. Your ignorance is so large, and your condescension is even larger, to the extent that it is not actually worthwhile trying to convince you that you are wrong.

@@angelmendez-rivera351 First of all, you are rude and arrogant and so deserve little better treatment. The problem here is that you know your material but you really do not understand it. The material realm is not studied through logic? Really? So I can appeal to truths to define a position which denies the existence of truth? I can say “I think I am not thinking” and expect that it can ever be true? Can YOU show me motion without an object? These are presented and understood by what? Not logic? With regard to materiality, we know that a composite entity such as a cup is a material construct, as are the molecules of which it is composed, then it of the atoms of which it is composed and it of the sub-atomic particles of which it is composed. These are objective, deterministic and cannot be posited, scientifically or otherwise by other than logic. So too with the theories you guys propose and the means of their validation. Logic is entirely conceptual? That is the most ridiculous remark I have ever heard. If 2 plus 2 is four, by the logic of that formulation, is not 4 plus 2, 6? Are you telling me that is not bound to a truth of it by logic? Is any truth not just that by means of logic? Conceptual only? Logic is the architecture of materiality, removed for examination of what we think that architecture is. Logic governs the material realm. And by the way, saying “you’re wrong” is not an argument. Explain how and why logic is only conceptual. That would be an argument for your position. Well your degree notwithstanding is not of consequence. You should know that one should never argue from authority. I know you are wrong because what you say cannot be defended. If you know anything about the scientific method I would not have to so advise you. As for materiality, as for the infinite subdivision of a line segment, “in materiality” there is the Plank length beneath which there is no distance so in a material context, no line can be infinitely sub-divided. As for the theoretical subdivision, you made no effort to refute my points. Why? Are you unable? That is all I can conclude from your extensive response indicating that you are willing to spend the time, but leaving out any point for point refutation. As for space/time, is it not a material construct? If the cup mentioned above for example is, then by the logic for which you have little understanding, so too must the molecule, atom and sub-atomic particle be. Extend that understanding to the energy fields to which the sub-atomic particles of different kinds are peculiar. What are these particles? As I understand it, they are in actuality, according to the standard model, wave forms or discreet packets of energy (quantifiable) vibrating as a local phenomenon in an energy field, these the very foundations of all material entities which exist. These fields then in concert are or fill empty space which itself can be distorted or warped by the presence of a mass Are you trying to tell me that that is not also a material construct? If the Higgs field is of like kind and the particle is of like kind that it is of materiality and you really don’t know your own discipline. If I am wrong, prove it with an actual argument, not insults. If I am, I didn’t have the luxury of your education but I do see what is logical and what not, I think so far, more than you. Trying to bowl me over with math you know I have never studied is a cheap maneuver. Regardless, nothing you have presented is possible absent a system of logic, necessary for the conceptualization, the expression, etc. It is easy to see that you have no capacity for understanding anything you have studied beyond that which was presented to you in the exact manner. You obviously are incapable of abstraction. You resort to indicting my position only for my lack of familiarity with the terminology you use, instead of trying to see the point being made, which I now realize is quite beyond your capacity. So, I will make it simple enough for you. Can you have a cup without molecules? No. Logic of materiality. Can you have a molecule of the cup without atoms? No. Logic of materiality. Can you have atoms without protons? No? Logic of materiality…and so on. To claim that this or that is prerequisite to another thing, by definition, is the logic of the material context of that thing. Wow, that you have to have this explained to you from what I posted… Logic is the a reflection of relationships of abstractions and those concepts by which we define materiality and the relationships of its component parts. Then prove it smart guy. I gave you the reasoning which refuted the conventional wisdom. If I am wrong, demonstrate. The fact is, and you should be smart enough to see this, that there is no infinity in materiality and by that, there is not way to pair concepts born of the quantifiable with that of infinity a mere abstraction which cannot be quantified. That I said I would not burden the reader with a page and a half of validation, did not mean I would not listen to anyone’s response. How is it you got that from what I posted? Refute point for point what I posted or be quiet about it, a sure admission that you are not equipped for this discussion. What you are is typical. You are a coward who clings to what you have studied for comfort and reject anyone who politely offers what is minimally an interesting take on these things. My condifence in my view is not born of arrogance of the kind you have displayed for all to see, but as a consequence of my success in a debate with no less than five graduate students in physics, all of whom were willing participants in the discussion but not one of whom could refute what I said. All that tells me is you are a coward. And you are the rudest and most arrogant jerk I have encountered to date on RU-vid. Good riddance.

@@jamestagge3429 *you are rude and arrogant and so deserve little better treatment.* You are projecting. I have been nowehere near as rude and arrogant as you have been since you posted your original comment. I understand that you need to project, though, because you have no valid arguments to present to support your claims. *The material realm is not studied through logic? Really?* Yes, really. *So I can appeal to truths to define a position which denies the existence of truth?* Who said anything about denying the existence of truth? I did not. You are strawmanning me. *I can say “I think I am not thinking” and expect that it can ever be true? Can YOU show me motion without an object?* How does this have anything to do with whether logic studies the material world or not? In your first question, the question runs into the problem of solipsism. The second question is asking me if I can study the physical without the physical, to which I am saying "No, because logic is not about studying the material world." *These are presented and understood by what? Not logic?* Correct. They are presented and understood by the scientific method. I already explained this to you. *With regard to materiality, we know that a composite entity such as a cup is a material construct, as are the molecules of which it is composed, then it of the atoms of which it is composed and it of the sub-atomic particles of which it is composed. These are objective, deterministic and cannot be posited, scientifically or otherwise by other than logic.* They absolutely can be posited scientifically. These discoveries were made by scientists using science, not logicians using logic. *So too with the theories you guys propose and the means of their validation.* Theories are validated by experiment, not by logic. *Logic is entirely conceptual? That is the most ridiculous remark I have ever heard.* It sounds ridiculous to you because of your utter ignornace. *If 2 plus 2 is four, by the logic of that formulation, is not 4 plus 2, 6?* No. Because in a finite ring of characteristic 5 (which is a field, since 5 is a prime integer), 4 + 2 = 1, and in a finite ring of characteristic 6, (which is not a field, as it has zero divisors), 4 + 2 = 0. This is basic abstract algebra, and you use this whenever you do modular arithmetic without realizing it. Instead of using the notation 4 + 2 = 1, you are more likely to encounter 4 + 1 == 1 (mod 5), but both notations means the exact same thing in the context of finite rings of characteristic 5. *Are you telling me that is not bound to a truth of it by logic? Is any truth not just that by means of logic?* I just demonstrated an example of a case where your "truth" is not actually a truth. *Logic is the architecture of materiality, removed for examination of what we think that architecture is.* Citation needed. *And by the way, saying “you’re wrong” is not an argument.* You are right, which is why you should stop doing that. After all, that is what your original comment boils down to. *Explain how and why logic is only conceptual.* I did explain it. Logic is the study of propositions. If you knew how to read (which evidently, you do not), you would have understood it the first time around, and I would not have had to repeat myself. This is why I said talking to you is a waste of time. The only reason I am still replying is because you already made it clear you are not going to continue replying yourself. *Well your degree notwithstanding is not of consequence. You should know that one should never argue from authority.* I am not arguing from authority. I am merely pointing out that you are arrogant by believing that all the experts of the world in logic, science, and mathematics, are wrong about how these things work, yet you yourself are not wrong. *I know you are wrong because what you say cannot be defended.* It can easily be defended, but as I said, you already admitted you are not willing to listen to explanations, so I am not going to waste my time providing them. *As for materiality, as for the infinite subdivision of a line segment, “in materiality” there is the Plank length beneath which there is no distance so in a material context, no line can be infinitely sub-divided.* This is false. String theory posits that the Planck length is a fundamental length that cannot be further subdivided. However, there is no evidence for string theory. Until string theory is experimentally verified, the fact remains that spacetime can be infinitely subdivided. *As for the theoretical subdivision, you made no effort to refute my points. Why? Are you unable?* No. I explained why I made no effort: because you are not willing to listen. *That is all I can conclude from your extensive response indicating that you are willing to spend the time, but leaving out any point for point refutation.* If that is all you can conclude, then your reading comprehension skills are extremely lacking. *As for space/time, is it not a material construct? If the cup mentioned above for example is, then by the logic for which you have little understanding, so too must the molecule, atom and sub-atomic particle be. Extend that understanding to the energy fields to which the sub-atomic particles of different kinds are peculiar. What are these particles? As I understand it, they are in actuality, according to the standard model, wave forms or discreet packets of energy (quantifiable) vibrating as a local phenomenon in an energy field, these the very foundations of all material entities which exist.* I can see that Google is your best friend, lol. You totally looked this up, and copy-pasted without having any idea of what any of the words mean. But fair enough, this is a decent-ish description of quantum field theory's premise, suitable enough for a layperson. *Are you trying to tell me that that is not also a material construct?* I never said spacetime is nonphysical. Get your head out of the gutter, and learn how to read. You are making shit up that I never said. *If I am wrong, prove it with an actual argument, not insults.* I have not insulted you at all in this conversation. I have given you arguments, but as confirmed, you are not willing to even acknowledge them. But, with that being said, I will now insult you, because of the sentence you wrote next. *I didn’t have the luxury of your education...* Luxury? You think you know me personally, huh? Allow me to tell you that I came from a homeless family, and that my education is something I had to work extremely hard for, spending many days without eating and many nights without sleeping. You're a shithead if you think you know my history. Yes, now I am insulting you, because you totally deserve it. *...but I do see what is logical and what not, I think so far, more than you.* In your dreams.