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Premise 1: All straight line segments have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero.

Premise 2: Lengths of certain value exist if they can be constructed by increasing the points to a certain extent.

Premise 3: Lengths of non-terminating decimal form value can't exist because they can't be constructed by increasing the number of points to a particular extent.
Ex: 1.9999...; Length of this value can't exist because, length of this value can't be constructed by increasing the number of points to a certain extent. In order to reach that value of length, first we need to achieve 1.99, then 1.99999, then 1.9999999, and so on. We don't know where to stop. If we don't know where to stop, we can't construct that line of that value.

Conclusion: From 3, we can't construct line segments of length equal to non-terminating decimal form value. So, there is no question of end point, and no question of non-terminating decimal form value.

From this, it seems lines can't have non-terminating decimal form value. But I don't think it is what we know. What is going wrong in the argument? Is it that length of line segment can't have non-terminating decimal form value?

Sensebe
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  • I asked a related question which didn't have much clarity: http://math.stackexchange.com/questions/1196989/irrational-diagonal-length-problem – Sensebe Apr 02 '15 at 04:59
  • Premise 3 sounds more like its own sub-conclusion; it sounds like a conclusion from the first two premeses. You're also not really asking a question here, are you? – pjs36 Apr 02 '15 at 05:10
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    Lines do not have start and end points. Line segments have start and end points. – Thomas Andrews Apr 02 '15 at 05:10
  • What does this have to do with mathematical physics? – Thomas Andrews Apr 02 '15 at 05:12
  • @ThomasAndrews: I have edited the question. – Sensebe Apr 02 '15 at 05:13
  • @pjs36: Sorry, I am not really understanding where I am going wrong in this argument. I feel, lengths of non-terminating decimal form value can't exist. – Sensebe Apr 02 '15 at 05:24
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    What does "increasing the points" mean to you? Also, why should the decimal system of representing numbers be relevant to which lines can & can't be constructed? Under your premises some lines that are impossible in base 10 become possible in base 12, and vice versa. And that would be weird! – PM 2Ring Apr 02 '15 at 05:47
  • Your 3rd premise reminds me of the Achilles and the tortoise "paradox". – Martin R Apr 02 '15 at 05:56

2 Answers2

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Your faulty premise is most certainly the 3rd, that line segments of certain lengths cannot be constructed.

The representation of a number as a decimal has no bearing on the line itself. Let's say we're interested in the length $L$, that has a non-terminating decimal. We could use a different number system, where our base is $L$, and the length $L$ would be represented as $10$, which certainly has a terminating base-$L$ representation, and consequently be exempt from premise 3.

Here's another reason why your premises aren't appropriate. If you look at Euclid's Elements, the very first proposition is the construction an equilateral triangle, constructed like so (image from this site):

enter image description here

We could draw a line segment connecting the intersection of the two circles, and then use that to mark off an altitude of our triangle:

enter image description here

If the triangle has base length $1$, the line segment connecting our two blue points would have length $\frac{\sqrt3}{2}$, which is certainly irrational. (Of course it takes a while for Euclid to get to perpendicular bisectors and so forth, but that's beside the point).

My point is that plane geometry is incredibly old, and it might be helpful for you to pick up Euclid's Elements to see what axioms geometers have used for millenia. You'll see how reasonable yet powerful those axioms are.

pjs36
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There are few things more constructible than $\sqrt 2,$ being the length of the diagonal of a square with sides of length $1.$ Yet this is an irrational number whose decimal expansion is nonterminating and non repeating.

zhw.
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