1

Every time I’ve read about the staircase paradox it refers to dividing the horizontal and vertical legs of the square by repeated integer divisions. Their lengths are of the form a/b, with a usually equal to 1. I think the idea is that the smallest rational number in a sequence cannot ever exactly equal zero.

I read the post <https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4 > My post is about a reason why the statement below is true:

“The limiting curve is approached, but never reached.”

Hence the step function can never be the same as the line y=x.

Let f(x) be a mathematical function. Its input is a real number and its output is a large real number, R. by evaluating f(x) repeatedly, then taking the reciprocal of R, a sequence of small real numbers is generated. I’m sure this can be accomplished without the use of a computer. These would be the length of each step in successive constructions of the square.

This is my logic:

  1. On the real number line, there are infinitely many points on a line segment that is a subset of 0 to 1.
  2. For any two successive outputs of the function f(x) there exists a real number in between them.
  3. Between zero and the output of f(x) from the millionth execution of the function there is a real number.
  4. Between zero and any repeated evaluation of f(x) there is another real number.
  5. There is no smallest real number greater than zero.
  6. ---> steps cannot be eliminated no matter how small each step becomes. This is still true when the square legs are divided into segments, the lengths of which are of the form 1/R, as R->infinity. The only way to not have steps is to consider the entirely different curve: the actual diagonal.

I made the following assumptions:

  1. I’m speaking in terms of math only, not physics (and quantum mechanics).
  2. The fact that lines do not behave intuitively in the smallest infinitesimal (smaller than a “neighborhood”) means one can’t assign an accurate length to the smallest steps. It does not mean steps representing the application of the output of f(x) indefinitely don’t exist. An output of f(x), when executing indefinitely, called R^n and its reciprocal 1/R^n would not be actual numbers with a value. Therefore, it makes sense to say the length is not accurate, but the step still exists. The infinitesimal never assumes an actual number value. I have read the vague statement that there are an infinite number of points between two adjacent points. Even if adjacent points are difficult to distinguish geometrically, it can still be true in R2 every point is a distinct point, to be differentiated from others, no matter how small the neighborhood is and the space being non-discrete.
GaryW
  • 31
  • What did switching from rationals to reals achieve? Which property of real numbers did you use? – Karl Apr 07 '23 at 03:17
  • 2
    I am confused by the whole posting. It is true that as smaller and smaller steps are taken, the area of the enclosed region approaches the area of the circle. However, despite this, the perimeter of the enclosed region is constant at $~4.~$ Since the foundation of the assertion that $~\pi = 4,~$ is based on comparing the circumference of the circle to the (unchanging) perimeter of the enclosed region, rather than comparing the area of the circle to the area of the enclosed region, all of the analysis is clearly specious. What am I missing? – user2661923 Apr 07 '23 at 03:56
  • Using rational numbers seemed like apples to oranges. The diagonal is composed of reals. So have the steps composed of reals too. The thrust was to show at the hypothetical point to point level, steps still don't vanish. Even though one can't easily conceive of connecting 2 points with an infinitesimal line. Rationals have the problem that reals are always stuck between them. I thought one would find fewer rationals, conversely, stuck between two reals. So the property I used would be density. – GaryW Apr 07 '23 at 12:26
  • I've read about non-standard analysis. I agree, combining infinitely many infinitely small things requires great care. That is what prompted my question - seeking a very basic model (excluding limits) to explain why the step function will always differ from the diagonal. I attempted a (obviously too much so) simple explanation. If one hypothesizes two adjacent points, one could say they form a smallest step. – GaryW Apr 08 '23 at 02:28
  • It seemed logical to think of the division of the legs to be by real numbers because the transfinite number beth (infinity of reals) is larger than aleph (infinity of natural numbers). The inverse of beth would therefore be smaller than the inverse of aleph. This would represent the two points more accurately (or so I thought). it probably doesn't make sense to envision an individual point or points. The concept of a point, lacking any tangible representation, requires the tools of calculus and analysis. – GaryW Apr 08 '23 at 02:28
  • Derivatives and integrals are used even for the basics like the metric on the real number line. So, I started out with an impractical approach. Part of my motivation is to understand how an infinite number of real numbers can exist just from 0 - 1. I think the only way is to work through dozens of examples of limits using epsilon and delta. – GaryW Apr 08 '23 at 02:28

1 Answers1

0

No.

The distinction between rational and real numbers has no impact on the paradox. The fact that every step will still have a path length which is larger than the piece of diagonal it covers remains the same whether the steps are rational or not.

Reading your reasoning, you mainly appear to go for “there is no smallest real number.” Well, that's true for rationals, too: there is no smallest rational number either. That doesn't change the fact that for any step, no matter how small it is, the step is longer than the diagonal by the factor of $\sqrt2$.

I can't say I follow all your reasoning, but some of what you write sounds like nonstandard analysis to me. That's a way to consistently deal with infinitesimal quantities that are smaller than any real value.

I'm not an expert in that field, but I assume that if you were to natively assume all your steps to be of such infinitesimal size, then any finite sum of such steps would still cover only an infinitesimal part of the diagonal. As usual, combining infinitely many infinitely small things requires great care.

I wouldn't be surprised if nonstandard analysis didn't change the staircase paradox in any significant way. As I understand it, the formalism should allow comparing infinitesimal quantities, so you might find that even for infinitesimal steps, the stair step is larger than the diagonal by a factor of $\sqrt2$. But as I said, I have no real experience in this area.

MvG
  • 42,596