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I'm not intuitively convinced that this proof is sound because I feel like is ignoring the concept of "stretching" when mapping a fixed range like (0,1) to a la

I'm not intuitively convinced that this proof is sound because I feel like is ignoring the concept of "stretching" when mapping a fixed range like (0,1) to a larger or smaller set. EG, I don't care that you can map points from (0,1) to (0,2) one to one, because I know that each element in (0,2) is forced to cover twice as much space (regardless of taking the limit so that the cover goes to zero, this cover ratio is always 2:1 ). It smells fishy, like Cantor invented a way of comparing set sizes that intentionally drops units of cover so that naturally distinct elements fall into the same equivalence class. Most arguments occur downstream of this choice, so debates are imprecise and people speak past each others points. Unfortunately if you object to Cantor's ideas online, you will be flooded with people who will call you a moron and recite whatever garbage explanation they received as a child, instead of offering some thoughtful counterpoint. At the end of the day I think Cantor's ideas are a mathematical tool that the mainstream has run with, but isn't actually the most useful tool. (Monster) Curves which have continuous dimensions should be the basis for creating a infinite set system with consistent utility. (EG, there are curves which are very "spiky" that they break free of being 1 dimensional but, are not quite an area (dimension 2) -- they could be perhaps 1.6 dimensional, IE- they _sort-of_ have an area. ) With Cantor-tinted glasses a normal 1 dimensional line would have the same cardinality of points as a 1.6 dimensional one, so some richness is lost.

skh 8y ago
I don’t think anyone should call you a moron. I do think you should consider that if professional mathematicians don’t have a problem with a proof and you do then it is most likely an indication of a lack of understanding on your part. I don’t understand your objection because I’m not sure what exactly you mean by ...each element in (0, 2) is forced to cover twice as much space... I will point out that saying that (0, 2) is a larger set than (0, 1) is wrong without specifying how you are using the term size. The size of the sets, in the context of set theory, are the same. However, in terms of measure theory they are not the same. The notion of size depends on what exactly is being measured. In one context (0, 2) is larger than (0, 1) but not in another context. Unfortunately the term “size” has a connotation to most people and in set theory that term is not the connotation that most people have. A person will naturally think that an interval of length 2 is bigger than one of length 1. The word size doesn’t have any inherent meaning. It’s jut a word. It means what people agree to have it mean. In set theory the accepted meaning of size corresponds to mappings between sets. The size of the set {0, 1, 2, 3} is less than the size of the set {0, 1, 2, 3, 4} because I can map the first set injectively into the second set but not vice versa. This is an intuitive sense of size when dealing with finite collections of objects. Intuition can break down when abstracting to infinite collections of objects.
- sagebird 8y ago
  Thanks for your helpful reply, but I am still having an issue understanding part of the idea of mapping sets of points. Suppose that there was an arbitrary rule that points can only exist at integer coordinates, then (0,2] would have 2 elements and (0,1] would have one element. You would not be able to map 1 to one. Now, suppose that points are allowed to exist at coordinates m*(1/n) where n=2. (EG, 0, .5, 1, 1.5 ...) So now the first set would have 4 elements, and the second would have 2. They would still not map one to one. Now imagine that you can pick an n arbitrarily large to represent any rational number. I still don't see how you can map the sets one to one. Is there a reason that you can map them one to one, or is the set theory that we use specifically constructed/designed to afford this property?
  - skh 8y ago
    I think the problem comes from the mapping you are choosing. You can construct a map that between (0, 2) and (0, 1) that is onto but not one-to-one. But to say that (0, 2) and (0, 1) have the same size as sets (cardinality) means that there exists a bijective map between the two sets. You just haven’t found out what that mapping is. The mapping you are thinking of is not a bijection so from your perspective you are not seeing the sets as being the same size. Consider this. The set of non-negative integers is {0, 1, 2, 3, 4,....} and the set of even non-negative integers is {0, 2, 4, 6, ...}. One might think the first set is twice as large as the second set. And under a natural way of looking at it this appears to be true. But consider this mapping: 0 -> 0 1 -> 2(1) = 2 2 -> 2(2) = 4 3 -> 2(3) = 6 . . . Under this map I’ve bijectively mapped the set of all non-negative integers to the set of non-negative even integers. This is how intuition can fail when dealing with infinite sets. Under one way of looking at things one set appears to be larger than the other. But the definition of cardinality (size of the set) deals with whether or not a bijection exists and does not care that other mappings might give the appearance of one set being bigger than another set. All infinite sets have this property: They can be put in bijecitve correspondance with a proper subset. This can not happen with finite sets. Here’s a bijection from [0, 1] to [0, 2]. f(x) = 2x.

skh 8y ago

I don’t think anyone should call you a moron. I do think you should consider that if professional mathematicians don’t have a problem with a proof and you do then it is most likely an indication of a lack of understanding on your part. I don’t understand your objection because I’m not sure what exactly you mean by ...each element in (0, 2) is forced to cover twice as much space... I will point out that saying that (0, 2) is a larger set than (0, 1) is wrong without specifying how you are using the term size. The size of the sets, in the context of set theory, are the same. However, in terms of measure theory they are not the same. The notion of size depends on what exactly is being measured. In one context (0, 2) is larger than (0, 1) but not in another context. Unfortunately the term “size” has a connotation to most people and in set theory that term is not the connotation that most people have. A person will naturally think that an interval of length 2 is bigger than one of length 1. The word size doesn’t have any inherent meaning. It’s jut a word. It means what people agree to have it mean. In set theory the accepted meaning of size corresponds to mappings between sets. The size of the set {0, 1, 2, 3} is less than the size of the set {0, 1, 2, 3, 4} because I can map the first set injectively into the second set but not vice versa. This is an intuitive sense of size when dealing with finite collections of objects. Intuition can break down when abstracting to infinite collections of objects.

sagebird 8y ago
Thanks for your helpful reply, but I am still having an issue understanding part of the idea of mapping sets of points. Suppose that there was an arbitrary rule that points can only exist at integer coordinates, then (0,2] would have 2 elements and (0,1] would have one element. You would not be able to map 1 to one. Now, suppose that points are allowed to exist at coordinates m*(1/n) where n=2. (EG, 0, .5, 1, 1.5 ...) So now the first set would have 4 elements, and the second would have 2. They would still not map one to one. Now imagine that you can pick an n arbitrarily large to represent any rational number. I still don't see how you can map the sets one to one. Is there a reason that you can map them one to one, or is the set theory that we use specifically constructed/designed to afford this property?
- skh 8y ago
  I think the problem comes from the mapping you are choosing. You can construct a map that between (0, 2) and (0, 1) that is onto but not one-to-one. But to say that (0, 2) and (0, 1) have the same size as sets (cardinality) means that there exists a bijective map between the two sets. You just haven’t found out what that mapping is. The mapping you are thinking of is not a bijection so from your perspective you are not seeing the sets as being the same size. Consider this. The set of non-negative integers is {0, 1, 2, 3, 4,....} and the set of even non-negative integers is {0, 2, 4, 6, ...}. One might think the first set is twice as large as the second set. And under a natural way of looking at it this appears to be true. But consider this mapping: 0 -> 0 1 -> 2(1) = 2 2 -> 2(2) = 4 3 -> 2(3) = 6 . . . Under this map I’ve bijectively mapped the set of all non-negative integers to the set of non-negative even integers. This is how intuition can fail when dealing with infinite sets. Under one way of looking at things one set appears to be larger than the other. But the definition of cardinality (size of the set) deals with whether or not a bijection exists and does not care that other mappings might give the appearance of one set being bigger than another set. All infinite sets have this property: They can be put in bijecitve correspondance with a proper subset. This can not happen with finite sets. Here’s a bijection from [0, 1] to [0, 2]. f(x) = 2x.