Boltzmannbrain

That makes sense. It is what I was thinking. Okay, that makes sense. I only meant that my intuition tells me recently that there is a smallest real. I think this all starting to sink in. Okay, this makes sense too. I see. Thanks, I forgot about that. I was only discussing it with you. I forgot/misunderstood the integral process. This seems interesting to me in that adding one real number is significant geometrically, but I don't know right how relevant it is to this discussion. Thank you very much for your help and incredible patience Sorry, I really appreciate your help, but I wasn't making the connection with what you were saying and my issue. Also, I did not have enough time to figure it out. I think I have finally understood this whole issue from my OP completely. Thank you for your help!

I read what you said about neighborhoods and infinity. Then I looked up neighborhoods because I have never heard of them before. But I still have no idea how it helps me understand anything about my issue, or even how it relates to the discussion. I am not sure if I am even ready to get into the topic of neighborhoods. It seems a little or a lot more advanced than what I have learnt so far.

I meant a space in R2. If you half the distance an infinite number of times you get 0 don't you? I only meant that my intuition has always been that there couldn't be a smallest number (as I believe everyone's intuition is), but now that has changed. Then what you said above cannot be true either about the halving process being carried out an infinite number of times, since n is always finite. Ok, I understand. That is very interesting, thanks. I never took the second semester of the advanced calculus course, so I never got a proper understanding of the more rigorous definition of the Riemann integral. I only took the typical calculus for learning integrals. Then how can we get 0 when calculating the widths of the partitions over n as n goes to infinity? What finite n value would allow this? I am interested in the integral because of what we are discussing above. It seems like that will help me understand the ideas of density, infinity vs limit to infinity, etc. from a different perspective, namely the integral (well to be more specific, the 1 dimensional aspect of the integral). I would think so too. Except when I think about that example I gave earlier about the segment not fitting because of one extra point, I am left confused and very curious.

I am interested in this geometrically. So I would like each number to represent a point/position in a space or on a line. Something interesting that keeps coming to me is when thinking about the Reimann integral. There are n partitions, and n eventually equals infinity in order for the sum to be complete. In this case there is a 1st, 2nd, 3rd etc. point/number (with a column) that "fills" in a real space. I will read more about how that is possible and how it can help me figure all of this out. I said to repeat it an infinite number of times, not a finite number of times. My intuition has always been that you can't get to a largest number or a smallest number (using typical methods). Now I am starting to rethink that intuition. I don't agree. I don't think people believe in a smallest. Yes it is definitely logical since I have yet to prove otherwise.

The reason why I am backtracking a bit on this issue is because I am not using an infinite number in place of a real number. But that is how I am using an infinite number. Since there are an infinite number of numbers/points or decreasing subsets from 0 to 1, then it would seem fair to use an infinite number where it is relevant. Yes, I fully agree. Intuition is no argument here. I briefly mentioned it earlier, but I will say it again. The "halving" proof definitely works for each individual case, but what about applying it an infinite number of times? But let's move on because I think the other proof you put below is not challengeable. So regarding proofs for this, I will accept that they exist and are unchallengeable. I did not start this thread to disprove anything. I started this thread to say that it seems illogical for the reals to have no next number or no smallest number (smallest real is what all of this really boils down to). The reductionist in me wants to define the reals to have a smallest number sort of like the naturals (and then maybe we wouldn't have to have the infinitesimal hyperreals). But this is just a side thought.

I was thinking that we could only use finite numbers to "inch" closer and closer to 1. If that is the case then ok, but I need to know the exact logic about why we can't use transfinite or infinite numbers. If we can use transfinite or infinite numbers, then it would seem that there are decreasing subsets all the way to the end of [1, 0) with it ending with a subset of 0 numbers. That final subset of 0 numbers between two numbers is what I am looking for. I agree with everything you said except for the "exactly the same" length part. This came up before in this thread. I said something like this. Imagine a line segment [0, 6] (please bare with me as I don't know how to use the proper notation for a line segment on this forum). Assume that the segment can pivot about the point 0. At the point 5 we break the segment, keeping the number 5 on the part that pivots, and leaving the part (5, 6]. We pivot the broken segment [0, 5] to some degree then reattach another 5 at the end of the part that doesn't pivot. The segment [0, 5] cannot revolve past the segment new [5, 6] anymore. The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to pass through. It would seem geometrically that there is more length on [0, 5] than [0, 5) It would just be an imaginary onion that is infinitely dense with an infinite number of layers. You peel one layer back, and you would expect a next layer, right? Yes, I have always understood that. My counter arguments that pop up in my head leave me unconvinced and also curious as to what it is that I still do not understand about this topic. And I appreciate this a lot. I have learnt a lot since starting this thread. This is just a long shot, but what if we discover something? That would be great for all of us since we have all worked on this together. Are you sure they are proofs? If so, what kind of proofs are they? Mathematical proof - Wikipedia I am quite familiar with most of them, but not all.

I actually remember the whole context of what he said. He said that infinity is not a number, then said, well, it can be a number, but for the purposes of this course, it is a direction. We did not get to learn why he said it was also a number. Yes, I agree. This is true by definition. I don't understand what made you have to say this. I responded to WTF's comment " He noted that if p and q are real numbers then (p + q)/2 is a real number strictly between them." with this, "But what if we do this process an infinite number of times?" I thought you would have been satisfied with my answer. Wikipedia says, "Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.". I want to use standard analysis, not nonstandard analysis for topic (if we can). I have just read through them, and I definitely have not worked with them.

Yes, I definitely agree with this. And you bring up an important point: what are their abstract properties as they exist on the number line, a segment, in a space, etc? Thinking about this might help me come to a conclusion. From what I understand, a number seems to be a location on a number line or even a segment. Now do they have to "take up" a zero dimensional point or can they just exist there with no point? From what I have been reading about the linear continuum, it seems as though the point and the number are tied together. I cautiously come to that conclusion since a linear continuum is ordered and that each point represents a unique number and the converse. I actually did learn about this in university too. It was a more fundamental calculus course than what is needed for the sciences. But unfortunately I only took the one semester. Thanks though these concepts are great reminders for this topic.

Ok, this answers that issue. I was thinking that a continuum meant that the points/numbers had to be connected or attached to each other, but now after reading about continuity and the continuum I don't think that is necessary. The general idea is that the interval [0, 1] is very different after taking just one number away to make it [0, 1). Say you strip a top layer off an onion with infinite layers of infinite density, shouldn't there be another layer to take off?

I understand. This kind of argument seems a lot like the other arguments where there is always a number between two other numbers. This definitely works for a finite number of trials. But what about an infinite number trials. Don't we exhaust all divisions between the two numbers?

But what if we do this process an infinite number of times? I am trying. Except when when I think about the reals as continuous (which they are) rather than granular, the following issue pops up in my brain, "if the reals are a continuum (which they are), then the boundary must be attached to another number since there isn't anything else it could be attached to. I will explain more below. This is what I struggling to understand. The set meets 1 in a continuous and complete manner. Ordered objects in an increasing fashion connect to 1. Only one object can be larger than the rest but smaller than 1, just like the number 1 is to the rest of the set. But somehow this object that I am looking for gets lost in the fuzz. That is a perfect analogy. Removing 1 from [0, 1] is just like moving the top layer of water as well as its surface. That is why this does not make sense to me geometrically. The numbers/objects are like the H2O molecules. Remove the top layer and there should be a next layer of molecules, but there isn't. I only want to understand this using "mainstream mathematics" which I think is called Zermelo-Fraenko set theory. I know that aleph null is not in the set of naturals since it is not a natural number. I see you have posted about this, so I will read your next post.

This argument definitely makes sense. But I wonder if it is just a method that just doesn't work in finding a next number. Anyway, I have to give the counter-argument that is nagging at me. We have a set of numbers inclusively from 0 until 1, [0, 1]. The numbers increase in order to the right. There is a final number to the set, which is 1. It is furthest to the right, so logically it would be the greatest of the other numbers. We take away the number 1 from the set. Okay so far this all makes complete sense. It is all very intuitive, and there is nothing strange or even interesting happening. Now I am told that this set of increasing numbers no longer has a greatest number or an end. The number 1 seemed to have had some sort of special property that the other digits don't have. It can be at the end as well as the set's greatest number. Why can the number 1 do this and no other number? This does not seem logical to me, so I must be missing something.

Is one of these possibilities correct?

Fortunately I attended a good university. He got his PhD from Princeton. It was the advanced calculus course. I thought r was suppose to mean a real number. In any case, my point was to bring up an example to remind us of part of the scope of set theory. Ok, but where did I say that we didn't need cartesian coordinates? That makes sense. Interesting! Sure that sounds useful and interesting. I do not understand how it is logical to have no next number on a line segment in a real space. I am hoping to explore the implications of removing a number at the end of the line segment. Why does the end of the line segment no longer have an end number? The geometrical result seems illogical to me.

Because of some frustration that was expressed, I was going to bail on this thread. But I thought I would continue because I am just too interested in this subject. And I would like to explain myself better. One thing I would like to say to everyone coming into this thread, which I should've been more clear on in the OP, is that this topic is still being researched. So there probably won't be a perfect and irrefutable answer to the general concern in the OP. Having said that, I am here to learn, teach, discuss, etc. I am not here to frustrate anyone, or troll. Please give me a chance to dig into this with you all. I was told a long time ago by my math professor that aleph null (the cardinal size of the naturals/rationals or the "smallest infinity") can actually be used as a number. So in set theory, I think we can actually say that there is a number larger than any r + 1 or any natural number. It is interesting to me that when aleph null is used as a number in the case of 2^(aleph null), it equals the next largest infinity being the set of the reals. This link shows the procedure, Euclidean distance - Wikipedia I am still going through the link you gave me. There is quite a bit of information there to consider. I am also trying to think more about how the Dedikind cut comes into this.