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Physics, Astronomy & Cosmology / Re: What difference does this make to our understanding of infinity?
« on: 14/04/2020 15:36:45 »
A nice way of showing this 1-1 correspondence of the uncountably infinite reals between 0 and 1 (or any finite distance) and all the reals is to imagine an arc (semicircle) connecting the two endpoints. Every point on that arc corresponds to a point on the line beneath it (we can define every point on the semicircle uniquely using only the x coordinate, so if the diameter of the circle is 1, this is the continuum from 0 to 1. Every point on this arc also has a slope (slope of the tangent line), and it contains all real numbers, from arbitrarily large negative slopes to arbitrarily large positive slopes. Therefore there is a 1:1 correspondence between the continuum from –∞ to +∞ and the continuum between 0 and 1.
Screen Shot 2020-04-14 at 10.42.13 AM.png (15.17 kB . 394x306 - viewed 4888 times)
In other words: if you think there are more slopes than x values in the setup described above, choose any real slope, and I will find you the only x value that gives it. Likewise, if for some bizarre reason you think that there should be more x values than slopes, I challenge you to find an x value in the domain that does not uniquely correspond to a slope.
Screen Shot 2020-04-14 at 10.42.13 AM.png (15.17 kB . 394x306 - viewed 4888 times)
In other words: if you think there are more slopes than x values in the setup described above, choose any real slope, and I will find you the only x value that gives it. Likewise, if for some bizarre reason you think that there should be more x values than slopes, I challenge you to find an x value in the domain that does not uniquely correspond to a slope.