This article compares actual infinity and potential infinity, using familiar examples such as 0.999... = 1, limits, Zeno's paradox, and Cantor's diagonal argument.
• An Analysis of Cantor's Diagonal Argument for Actual and Potential Infinity [2015/6/26]
Let us begin with a familiar example of actual infinity.
At first glance, it may seem that the number on the right is slightly larger than the one on the left. Nevertheless, the equality is valid, as the following calculation shows.
Let S be the repeating decimal below.
Now subtract S from ten times S:
\begin{align*} 10S &= 9.999\cdots \\ -)\quad S &= 0.999\cdots \\ 9S &= 9 \end{align*}Dividing both sides by nine gives:
Therefore,
When I first encountered this equation, I imagined an endless string of 9s already written out in full. This is the idea of actual infinity: an infinite totality treated as if it were complete. But visualizing such an infinite string is difficult. Is there a more accessible way to understand the equation? To answer that, let us introduce limits.
Consider the following sequence:
\begin{align*} a_1 & = 0.9 \\ a_2 & = 0.99 \\ a_3 & = 0.999 \\ a_n & = 0.999\cdots 9 \end{align*}This sequence gets closer and closer to 1, although each finite term is still less than 1:
The value that the sequence approaches is called its limit, and we write it using "lim":
$$ \lim_{n \to \infty} a_n = 1 $$We can interpret 0.999... not as a completed decimal written out digit by digit, but as the limit of this sequence:
$$ 0.999\cdots = \lim_{n \to \infty} a_n $$From this viewpoint, we again obtain:
In this interpretation, we do not need to picture an infinite row of 9s. Instead, we look at the finite decimals 0.9, 0.99, 0.999, and so on, and recognize that they approach 1.
One may regard this as an extension of the equals sign: it relates a convergent sequence to the number it approaches. This definition introduces no logical contradiction.
When speaking informally, we say that a sequence approaches a value indefinitely. Mathematics expresses this idea more precisely using epsilon. This is the rigorous definition of a limit for sequences.
We write the fact that the sequence an approaches 1 as:
$$ \lim_{n \to \infty} a_n = 1 $$More precisely, this means:
For every ε > 0, there exists an N such that, for all n > N, |an − 1| < ε.
Here are a few concrete examples:
For ε = 1, choose N = 0. Then for n = 1 (> N), |a1 − 1| < 1. (a1 = 0.9)
For ε = 0.1, choose N = 1. Then for n = 2 (> N), |a2 − 1| < 0.1. (a2 = 0.99)
For ε = 0.01, choose N = 2. Then for n = 3 (> N), |a3 − 1| < 0.01. (a3 = 0.999)
Reading these examples is like watching a game in which the second player always wins: no matter how small an ε you choose, we can find a suitable N that makes the condition true.
For sequences, this is often called the epsilon-N definition. For functions, the related formalism is usually called the epsilon-delta definition. The basic idea is the same: "arbitrarily close" is replaced by a precise condition involving inequalities.
The streamlined formal statement is:
For every ε > 0, there exists N such that for every n > N, |an − 1| < ε.
At first, words such as "small," "large," and "always" may make the idea easier to grasp. With practice, however, the compact formal definition becomes clearer and more useful.
Hilbert's Grand Hotel illustrates the strangeness of actual infinity. Even if every room in an infinite hotel is already occupied, the hotel can still accommodate one more guest: simply move the guest in room n to room n+1, and room 1 becomes available.
Using his diagonal argument, Cantor proved that the real numbers are uncountable. A more detailed discussion of the proof can be found in the following article.
Now consider the following sequence of decimal approximations to π:
\begin{align*} \pi_1 &= 3.1 \\ \pi_2 &= 3.14 \\ \pi_3 &= 3.141 \end{align*}We can express its limit as:
$$ \pi = \lim_{n \to \infty} \pi_n $$From the viewpoint of actual infinity, π is a fixed constant with a complete infinite decimal expansion.
From the viewpoint of potential infinity, π is understood through a process: for any required accuracy ε, we compute enough digits to meet that error bound. In this sense, the number of digits needed depends on ε.
Zeno's paradoxes of motion are classic examples of how strange infinity can be.
To move from point A to point B, we must first reach the halfway point. To reach that halfway point, we must first reach its halfway point, and so on. Therefore, reaching B appears to require passing through infinitely many points, which seems to make motion impossible.
Under potential infinity, the process of subdividing the interval never ends.
Under actual infinity, the infinitely many subdivisions are treated as a completed totality, so reaching point B is possible. Now consider the following additional thought experiment:
Each time we reach a subdivision point, we write the next decimal digit in a notebook, erasing the previous digit. At any moment, the notebook contains only one digit.
If there is a final digit, then the arrival time t at point B is finite. But if there is no final digit, what digit remains in the notebook when we reach B?
In the framework of actual infinity, the infinite process is regarded as completed. Although there is no final digit to record, reaching point B itself involves no contradiction.
The word "infinity" may refer either to actual infinity or to potential infinity. In modern mathematics, actual infinity is the dominant framework, and it is certainly free of contradiction within standard mathematical practice. Personally, however, I suspect that actual infinity does not exist in physical reality.
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