Potential vs. Actual Infinity

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Potential vs. Actual Infinity

Postby JeffJo » Thu Jan 28, 2016 11:34 am

Draw a horizontal line across the center of a piece of paper. Make a tick mark on it near the left edge, and another about an inch to the right of it. Call the length between them "1 unit." Make more marks on the line, moving toward the right, of the same length between each. Call them "2", "3", "4", etc. These marks determine a numeric value for every point on this line, even the ones between the tick marks.

Now imagine that the line could be continued endlessly to the right. Tick marks can continue to be made, with values getting ever bigger with no end. This is the modern concept called "potential infinity." You can never reach an end, but the values keep getting bigger. Potential Infinity is not a numeric value, per se, but it is similar. Call it a value-like concept. It is the abstract goal that the actual values are trying to reach, but never can.

Some fields of mathematics, in particular the Calculus of Limits, use this value-like potential infinity. Even though some notation makes it look like it is treated as an actual value, it is not. The expression "at the limit" is not used for this value, either. It represents only what happens to a function as its argument becomes arbitrarily large.

For example, the line I described is obviously supposed to be the x-axis of a Cartesian coordinate system, with a y-axis at the first tick mark. Plot the function y=1/(1+x) on this system, and imagine you could keep plotting it endlessly. Even though you can't reach the end of the x-axis, you can show that y gets arbitrarily close to 0 as you keep trying. This is called the limit of y. There is no limit of x, and any attempt to label one is a misuse of the potential infinity.

Now draw the quarter part of a circle, of radius 1 with its center at the point (0,2), between (0,1) and (1,2). You can draw a line (maybe an imaginary one) from the center of the circle to any point on the (positive) x-axis. The line will intersect the quarter circle at a different point for every x. So each of point on the quarter circle, except(1,2), can be associated with either a point on the x-axis, or its value.

An archaic interpretation of "actual infinity" is the value-like concept you might assign this way to the point on the quarter circle at (1,2). That is, an actual point corresponding to the potential infinity. While this is a unique point "at the limit" (by the correct use of the phrase), there is no actual line from the center of the circle that goes through it and intersects the x-axis. There is no such value as this archaic interpretation of "actual infinity."

The set-theory concept of infinity, which some people call "actual infinity" or "completed infinity," is not a value-like concept in any way. All it means is that this quarter circle exists, can clearly be constructed, and represents an endless set of points and/or values. This correct interpretation of "Actual Infinity" has nothing to do with the values assigned to the points, which get endlessly bigger. It only means that this curve, with points representing any possible value approaching potential infinity, has the same existence as the line segment and quarter circle you drew.

Which it clearly does.
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Re: Potential vs. Actual Infinity

Postby bill » Mon Feb 22, 2016 11:47 am

I think of infinity as a number - not all infinities - . Take the square root of two for example, its decimal expansion has to end or repeat somewhere. If it didn't then there would be no length exactly equal to the square root of two. Pi = 3.14159.....etc. has to terminate or repeat somewhere because if it didn't a circle would grow ever so slightly larger forever.
Last edited by bill on Tue Feb 23, 2016 9:54 pm, edited 1 time in total.
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Re: Potential vs. Actual Infinity

Postby JeffJo » Tue Feb 23, 2016 9:23 am

Another example of the Potential Infinite is way we represent irrational numbers in decimal format. Weak minds sometimes believe that this representation somehow defines the number. It does not.

All it really is, is a simplification of an infinite formula. That is, if X=0.d1d2d3d4..., where d1, d2, d3, d4, etc. represent individual digits, all it means is
    X = d1/10 + d2/100 + d3/1000 + d4/10000 + ...
This is no different than the proven expansion (which is greater than 3 for every partial sum after those listed):
    pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 ...
Or the similar Bailey–Borwein–Plouffe formula, which finds the nth hexadecimal digit of pi for any n. Belief that such this hexadecimal expansion must end, or repeat, is a version of the (fallacious) archaic interpretation of "actual infinity" I described above. That they go on endlessly without repeating is an example of the potential infinity. That the digits can be calculated for that endless set of integers n is an example of the modern Axiom of Infinity.
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Re: Potential vs. Actual Infinity

Postby Edward Marcus » Tue Feb 23, 2016 9:50 am

JeffJo wrote:Another example of the Potential Infinite is way we represent irrational numbers in decimal format. Weak minds sometimes believe that this representation somehow defines the number. It does not.


On behalf of weak minds everywhere, I think you are not only expanding a series for the purpose of making a speech, you are also expanding upon the axiomatic assumption of infinity, which I believe (the experts be damned) is simply this- the axiom of infinity is the fundamental "belief" (religious or otherwise) that for any number n there exist other numbers which are larger than 'n'.

The concept of convergence by a series to the exact value of an irrational number is simply the byproduct of this fundamental assumption, because if the axiom of n always having the potential for larger values could not be assumed- then the construction of a series as the representation of an irrational number would be silly- as would the divvying up of a line segment etc... Hence the axiom at work is the basic assumption n is unbounded- and all the rest is frosting on the cake.
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Re: Potential vs. Actual Infinity

Postby JeffJo » Tue Feb 23, 2016 12:48 pm

Edward Marcus wrote:... the axiomatic assumption of infinity, which I believe (the experts be damned) is simply this- the axiom of infinity is the fundamental "belief" (religious or otherwise) that for any number n there exist other numbers which are larger than 'n'.
No, it is the assumption that you can call all such numbers a "set."

Wikipedia wrote:In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I.
What is "weak" about some minds, is that they think it has to be constructed to be called a set.

So I constructed one above.

The weak minds then keep trying to find a contradiction in it, but every one they claim to find starts with denying the possibility that such a set exists.

The concept of convergence by a series to the exact value of an irrational number is simply the byproduct of this fundamental assumption,
It is an unrelated point. The Axiom of Infinity is part of set theory. The convergence you describe is used by fields like the Calculus of Limits, which use the Potential Infinity only.
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