Working on a theory of countable cardinals

Discussion in 'Number Theory' started by Seff, May 7, 2023.

  1. Seff


    May 6, 2023
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    The natural numbers are analogous to an infinite line with a beginning and no end. We can continue that line into the integers and it becomes infinite in two directions. A line is defined by two points, we'll denote those two points as a set {x,y}. For the natural numbers x is 1 and y is infinitely large. For the integers x is infinitely small and y is infinitely large. As such I'll denote the naturals as {1,y} and the integers as {x,y}. When we expand into the rational numbers we are doubling this set in the sense that each rational number can be represented using one of infinitely many pairs of natural numbers. Thus I'll denote the rational numbers as {{1,y},{1,y}}. This new set expands in a new direction analogously connecting two infinite lines instead of two points, resulting in an infinite plane. However as the notation suggests, this plane is not infinite in all available directions. Instead it is only infinite in one direction and thus has the same cardinality as the natural numbers. When expanding this notion to the reals we find similarly that each real number can be represented as a pair of integers and natural numbers. Through replacement of like terms this means that the reals are of the form {{x,y},{1,y}} where {x,y} is the whole number component and {1,y} is the fractional component. This plane grows infinitely in three directions so it has a higher cardinality than the rationals. However, if we consider only the fractional component then we see it identical in distribution to the natural numbers. This relationship can be seen in my bijection between the reals between 0 and 1 and the natural numbers here:

    Thinking about the reals this way also shows that a bijection between the real numbers and the natural numbers should be possible utilizing a chain of reversable pairing functions such as the the Rosenbloom-Tsfasman function. Furthermore the only uncountable infinities should be those that utilize an infinite number of directions.
    Seff, May 7, 2023
  2. Seff


    May 6, 2023
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    To clarify some of the reasoning behind my representations of these sets:
    {1} A finite set, 1 can technically be any constant as it represents a static element.
    {1,1} The Cartesian product of finite sets. The notation is meant to resemble coordinates.
    {1,y} or {x,1} The Cartesian product of a finite and an infinite set, but also a line with a start and no end.
    {x,y} or {a,b} The Cartesian product of two infinite sets, also a line that grows infinitely in two directions.
    {{1,y},{1,y}} The Cartesian product of (the Cartesian product of a finite set and an infinite set) and itself.

    The reason I am representing the Cartesian products of sets in a fashion similar to coordinates is because the Cartesian product of two sets in this context represents their dimensional relationship. This notation expresses the independently infinite shared symbolic components of elements of sets in a way that allows us to see a clear relationship between them while also relating that to geometry.

    When I say a shared symbolic component of the elements of a set is independently infinite I mean that incrementing or decrementing it appropriately will never influence the other independently infinite components. Take for instance the real numbers. There are three symbolic components which may grow to infinity, the whole component, the fractional component, and the negative component. Enumerating the sequence of each of these components towards their infinities (1 to infinity, -1 to negative infinity, .0 to .999...) will never change the other components so they are independent. This relates to the Cartesian product in that this independent nature means every possible combination of these components exists in the set.

    The natural numbers have one symbolic component that grows infinitely and that is the number of digits used to represent them. The integers have two symbolic components that grow infinitely. First there are the number of digits, then there is the negative sign. We use the sign in order to represent a new infinity that mirrors the previous. This is what justifies similarly representing it as a variable in the same set.
    Seff, May 8, 2023
  3. Seff


    May 6, 2023
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    Fixing a mistake in my first post
    Using my notation as it currently stands, the appropriate representation of the rationals is {{x,y},{1,y}}. However this is of the same size as the reals, so it does not seem appropriate to relate this concept to the current assignment of ordinals.
    Seff, May 8, 2023
  4. Seff


    May 6, 2023
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    One of the thoughts that's been nagging me at the back of my mind is that the real numbers are constructable using pairs, and the integers are constructable using pairs, so perhaps the naturals are also constructable using pairs?

    I previously said the natural numbers have one symbolic component that grows infinitely, but it isn't independent. There is another infinite symbolic component but instead of growing infinitely it cycles infinitely, namely the numerals.
    We can represent the numerals as part of a pairing function in the following sense:
    The set infinitely repeating set of numerals m = {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}
    the infinitely growing set of digits d = {1, 11, 111, 1111, 11111, 111111, ...}

    The cartesian product of these sets looks something like:
    (0,1), (0,11), (0,111), (0,1111), (0,11111), ...
    (1,1), (1,11), (1,111), (1,1111), (1,11111), ...
    (2,1), (2,11), (2,111), (2,1111), (2,11111), ...
    (3,1), (3,11), (3,111), (3,1111), (3,11111), ...
    (4,1), (4,11), (4,111), (4,1111), (4,11111), ...
    (5,1), (5,11), (5,111), (5,1111), (5,11111), ...
    (6,1), (6,11), (6,111), (6,1111), (6,11111), ...
    (7,1), (7,11), (7,111), (7,1111), (7,11111), ...
    (8,1), (8,11), (8,111), (8,1111), (8,11111), ...
    (9,1), (9,11), (9,111), (9,1111), (9,11111), ...
    } where 1=0, 11=1, 111=2, 1111=3, 11111=4, etc
    which doesn't seem to be what I'm looking for.

    Investigating further I noticed the Cartesian product of m x m = (0-99)
    I then tried the cartesian product of m with the m x m and found the result is (0-999)
    This shows the naturals can be represented by an infinite chain of cartesian products of the set of arabic numerals.
    In other words, the number of digits is the n-fold cartesian product of m.

    As this is the second time an infinite chain of cartesian products has come up in my investigations I will denote it in my notational system and relate this concept to the natural numbers as {0^d} where d = {0,1,2,3,4,5,6,7,8,9}
    Without this notational addition the rational numbers would be represented as:
    {... {d, {d, {d, d}}}...

    In my previous investigations an infinite chain of cartesian products came up as one possible method of constructing an uncountable infinity. Namely I considered extending the reals to be uncountable in the following fashion:
    {... {{x,y}, {{x,y}, {{x,y},{1,y}}}}...

    With my new notational consideration this potentially uncountable infinite set can be represented as: {{0^{x,y}}{1,y}} where 0 represents an infinite sequence and ^ represents the an n-fold cartesian product. I've chosen to use 0 so that I can also denote a chain of cartesian products with a specific length. For example {1^d} = {d,d} and {2^d} = {d, {d,d}}.

    Finally, I can optionally evaluate other number sets to reflect this definition of naturals:
    Integers {x,y} become {{0^d}, {0^d}}
    Reals: {{x,y},{1,y}} becomes {{{0^d}, {0^d}}, {1,{0^d}}}

    I can reduce these representations by considering that the cartesian product of an infinite chain of cartesian products of some set is still just an infinite chain of cartesian products of that set. Furthermore the cartesian product of a set with one element and an infinite set returns the same infinite set. Thus I get the following representations:
    Integers {x,y} = {0^d, 0^d} = {0^d} = representable using the naturals
    Reals {{x,y}, {1,y}} = {{x,y}, {y}} = {2^y} = {2^{0^d} = {0^d} = representable using the naturals

    When I investigated the infinite chain of integers I described previously, namely {0^{0^d}}, I concluded this must be uncountable because {0^d} must be fully enumerated in order to construct its cartesian product. This notion was flawed because as we previously discussed {2^{0^d} = {0^d}, which means it's likely that {0^{0^d} = {0^d}. If true, then this seems to indicate a need for the axiom of choice.

    This relationship shows that we can define the number sets as follows:
    = {0^d} or {1^{0^d}} or {0^{0,1,2,3,4,5,6,7,8,9}
    I = {2^d} or {2^{0^d}} or {0^d, 0^d}
    R = {3^d} or {3^{0^d}} or {0^d, {0^d, 0^d}}

    This is why I said in another post that the naturals can form a bijection with the reals using a chain of pairing functions. However in order for this notion to be accurate we must consider the most specific representation of each number set and not any of the reductions I've been talking about as equivalent. This is because I was using equivalence to represent the ability for one set of dimensions to be mapped to another, as opposed to the actual mechanism by which said mapping is possible. So for instance, I just said the reals can be thought of as {3^{0^d}}, but in each link of the chain of cartesian products denoted as 3^ the dimensions are expanding from {0^d} to {{0^d}, {0^d}} to {0^d, {{0^d}, {0^d}}}.
    Last edited: May 8, 2023
    Seff, May 8, 2023
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