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The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom

by veltas 11mo ago

The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom in for any fraction. I can't do that for real numbers.

zkmon 11mo ago
That doesn't make one set "larger" than the other. You need to define "larger". And you need to make that definition as weird as needed to justify that comparison.
- veltas 11mo ago
  The fact that I can't even fit the real numbers between 0 and 1 on a single page, but I can fit every possible fraction in existence, doesn't mean anything? I don't think this definition is that weird, for example by 'larger' I might say I can easily 'fit' all the rational numbers in the real numbers, but cannot fit the real numbers in the rational numbers.
  - zkmon 11mo ago
    It doesn't mean anything because, with arbitrary zooming for precision, every real number is a fraction. You can't ask for infinite zooming. There is no such thing. So, let's inspect pi. It's a fraction, precision of which depends on how much you zoom in on it. You can take it as a constant just for having a name for it.
    - veltas 11mo ago
      Hi if you noticed, I never said anything about 'infinite zooming', instead I said I can write them all on the paper, which I can. The zooming is finite for every fraction.
      - zkmon 11mo ago
        If infinite zooming/precision is ruled out, all real numbers can be written as fractions. Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision.
        veltas 11mo ago
        > If infinite zooming/precision is ruled out, all real numbers can be written as fractions. How? > Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision. That's the least of your problems, I couldn't even draw different sized splodges per real number.
gloftus 11mo ago
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line! The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
- veltas 11mo ago
  Sorry I apologise, I didn't realise I wasn't allowed to care about countability.
  - gloftus 11mo ago
    Countability is the whole point, there's no need to apologize. I was merely offering the perspective that "towers of infinity" is possibly the least useful consequence that comes from defining the notion of countability. To my mind, what we really reap from Cantor's work is a better understanding of the topology of the real numbers. But you have to define countability first in order to understand what uncountability really implies.

by veltas 11mo ago

zkmon 11mo ago

That doesn't make one set "larger" than the other. You need to define "larger". And you need to make that definition as weird as needed to justify that comparison.

veltas 11mo ago
The fact that I can't even fit the real numbers between 0 and 1 on a single page, but I can fit every possible fraction in existence, doesn't mean anything? I don't think this definition is that weird, for example by 'larger' I might say I can easily 'fit' all the rational numbers in the real numbers, but cannot fit the real numbers in the rational numbers.
- zkmon 11mo ago
  It doesn't mean anything because, with arbitrary zooming for precision, every real number is a fraction. You can't ask for infinite zooming. There is no such thing. So, let's inspect pi. It's a fraction, precision of which depends on how much you zoom in on it. You can take it as a constant just for having a name for it.
  - veltas 11mo ago
    Hi if you noticed, I never said anything about 'infinite zooming', instead I said I can write them all on the paper, which I can. The zooming is finite for every fraction.
    - zkmon 11mo ago
      If infinite zooming/precision is ruled out, all real numbers can be written as fractions. Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision.
      - veltas 11mo ago
        > If infinite zooming/precision is ruled out, all real numbers can be written as fractions. How? > Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision. That's the least of your problems, I couldn't even draw different sized splodges per real number.

gloftus 11mo ago

You can't enumerate the real numbers, but you can grab them all in one go - just draw a line! The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.

veltas 11mo ago
Sorry I apologise, I didn't realise I wasn't allowed to care about countability.
- gloftus 11mo ago
  Countability is the whole point, there's no need to apologize. I was merely offering the perspective that "towers of infinity" is possibly the least useful consequence that comes from defining the notion of countability. To my mind, what we really reap from Cantor's work is a better understanding of the topology of the real numbers. But you have to define countability first in order to understand what uncountability really implies.