Comment by veltas
3 days 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.
3 days 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.
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.
Sorry I apologise, I didn't realise I wasn't allowed to care about countability.
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.
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.
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.
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