Tuesday, June 21, 2016

Cardinality paradoxes

Some people think it is absurd to say, as Cantorian mathematics does, that there are no more real numbers from 0 to 100 than from 0 to 1.

But there is a neat argument for this:

  1. If the number of points on a line segment that is 100 cm long equals the number of points on a line segment that is 1 cm long, then the number of real numbers from 0 to 100 equals the number of real numbers from 0 to 1.
  2. The number of points on a line that is 100 cm long equals the number of distances in centimeters between 0 and 100 cm.
  3. The number of points on a line that is 1 meter long equals the number of distances in meters between 0 and 1 meter.
  4. The number of distances in centimeters between 0 and 100 cm equals the number of real numbers between 0 and 100.
  5. The number of distances in meters between 0 and 1 meters equals the number of real numbers between 0 and 1.
  6. A line is 100 cm if and only if it is 1 meter long.
  7. Equality in number is transitive.
  8. So, the number of points on a line that is 100 cm is equals the number of points on a line that is 1 meter long.
  9. So, the number of distances in centimeters between 0 and 100 cm equals the number of distances in meters between 0 and 1 meters.
  10. So, the number of real numbers between 0 and 100 equals the number of real numbers between 0 and 1.

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10 comments:

  1. Michael GonzalezJune 21, 2016 at 3:43 PM

    This seems like cheap trick, doesn't it? When dealing with real numbers simpliciter one does not have a unit of measure on which to equivocate.

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  2. Alexander R PrussJune 21, 2016 at 3:47 PM

    Which premise do you deny?

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  3. IanSJune 21, 2016 at 8:30 PM

    You could just present the bijection x ↔ 100x. From a purely mathematical point of view, this would be enough. I’m not seeing what the argument adds to it.

    The argument may seem to make the result more intuitive. But even people who feel comfortable with the result (and still more, those don’t) may well doubt that space can be modelled as a real continuum.

    A bijection between a set and a proper subset is a bit unintuitive, however it is presented. Post Cantor, most mathematicians have just got used to it.

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  4. Alexander R PrussJune 21, 2016 at 8:36 PM

    There are philosophers who reject bijection as a sufficient condition for equal number.

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  5. IanSJune 21, 2016 at 11:17 PM

    Don’t (2) and (3) implicitly depend on the assumption that bijection implies equal number (i.e. a distance for each point and vice versa implies equal numbers of points and distances)? Or are they supposed to be obvious in some other way?

    Why would philosophers who doubt that bijection implies equal number find (4) and (5) more compelling? These seem to require at very least that space (this-world, physical space) be continuous. This may indeed be true, but it does not seem obvious that it must be.

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  6. Alexander R PrussJune 22, 2016 at 12:15 AM

    Maybe the argument should thus be run with an abstraction of a line, but still with length units? Maybe that doesn't make sense.

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  7. Michael GonzalezJune 22, 2016 at 7:34 AM

    I'm saying the argument is invalid because it commits an equivocation on unit. Every centimeter within your one meter is really a fraction of the whole, and similar fractions exist between 0 and the 1st centimeter out of your 100cm. And THEN there are all the numbers that come after 1cm.

    Units are just shorthand for bundles of the smaller units. No one is suggesting that 100 small units can't be regarded as a single larger unit, bundled together. It's only by equivocating between the two that you can get your result.

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  8. Alexander R PrussJune 22, 2016 at 7:47 AM

    But I can't be equivocating on "unit" in the original argument as the argument doesn't use that word.

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  9. Michael GonzalezJune 22, 2016 at 10:09 AM

    You're equivocating on the numbers, since a given number means subunits in one premise, and a bundle of those subunits in another. I mean, doesn't it seem obvious how this is cheating, Pruss? For every fraction of a meter between 0 meters and 1 meter, there corresponds a fraction of a centimeter between 0cm and 1cm... and then you have every other fraction and whole number between 1cm and 100cm.

    This makes the argument seem unsound, even though I'm actually not sure whether I agree with the Cantorian principle in question or not. An argument might persuade me, but not one like this, which treats ones and zeros as though they are the same even though we are changing units between premises, and the unit in one premise is, by definition, a bundle of the units in another premise.

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  10. bethyadaJune 28, 2016 at 12:15 AM

    I don't think I agree with 2 and 3, though if one subscribes to 2 and 3 then I would say that they are assuming the conclusion.

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