We consider the Cantor's diagonal argument for actual infinity and potential infinity.
• Consideration of Cantor's diagonal argument for actual infinity and potential infinity [2015/6/26]
We consider the following equation, in order to explain the actual infinity.
Some person might seem that the right-hand side is greater than the left-hand side. However, this is the correct equation. We calculate it as follows.
We suppose that the following value is the variable S.
We subtract the variable S from ten times of the variable S.
\begin{align*} 10S&=9.999\cdots \\ -)\qquad S&=0.999\cdots \\ 9S&=9 \end{align*}We divide both sides of the above formula by nine.
Therefore, we have the following equation.
When I saw the expression mentioned above for the first time, I imagined the state that the number 9 formed a line endlessly. We call the state that the number 9 forms a line endlessly "actual infinity." It put a high load on my imagination. It is very difficult to imagine the state that the number 9 formed a line endlessly. Isn't there a way to understand this equation more easily? Then, we introduce the concept of a limit.
Here, we consider the following series.
\begin{align*} a_1 & = 0.9 \\ a_2 & = 0.99 \\ a_3 & = 0.999 \\ a_n & = 0.999\cdots 9 \end{align*}The series approachs the value 1 endlessly. However, it never arrives at the value 1.
We call the value 1 that the series an never arrives at a "limiting value." A limiting value is the value that the series never arrives at. We get a limiting value by taking the limit. We express the limiting value by using the following symbol, lim.
$$ \lim_{n \to \infty} a_n = 1 $$We interpret the string 0.999... as not a number but a pure symbol, and we define it by the following symbol lim.
$$ 0.999\cdots = \lim_{n \to \infty} a_n $$Then, we notice that we have the following equation.
It is not necessary for us to imagine the state that the number 9 forms a line endlessly if I think in this way. First, we imagine that the finite series, 0.9, 0.99, and 0.999. Next, we should write the value 1 that the series never arrives at on the right-side of the equation.
We can also interpret this as the expansion of an equal sign concept. It means that we define a new equal sign between a value and a series approaching the value endlessly. Mathematical contradiction does not occur even if we define such a new equal sign.
In the explanation of a limit, we expressed that a series approaches a value endlessly. We used the expression that the series approaches the value endlessly in the explanation about the limit. We can define the limit by the combination of simpler concepts. We call the definition the epsilon-delta definition of limit.
We expressed the series an approaching a value 1 endlessly by the symbol lim as follows.
$$ \lim_{n \to \infty} a_n = 1 $$We define the lim symbol by the epsilon-delta definition of limit as follows.
For all small ε > 0, there exists a big N such that for all n > N, the inequality |an − 1| < ε always holds.
This is a definition of the symbol lim. We express it as the concrete numbers as follows.
For a $ \varepsilon=1\ \ \ \ \ $, there exists a N = 0 such that for a $n=1>N$, the inequality $|a_1-1|<1\ \ \ \ \ $ holds. (a1 = 0.9)
For a $ \varepsilon=0.1\ \ $, there exists a N = 1 such that for a $n=2>N$, the inequality $|a_2-1|<0.1\ \ $ holds. (a2 = 0.99)
For a $ \varepsilon=0.01 $, there exists a N = 2 such that for a $n=3>N$, the inequality $|a_3-1|<0.01 $ holds. (a3 = 0.999)
It seems to play the game that the second player wins, when we read the above expressions. Even if a person takes out any epsilon, we can prepare N which we can win.
Though we use epsilon and N in the above expressions, we usually use epsilon and delta. Therefore, the definition is called the epsilon-delta definition of limit.
If we have the expression for a small error ε (epsilon), we have the expression for a bigger epsilon. Therefore, we do not actually need the adjective "small." We also do not need the adjective "big." We also do not need the adverb "always." According to the thought of mathematics, we need to describe the definition as concisely as possible. Therefore, we define the limit generally as follows.
For all ε > 0, there exists N such that for all n > N, the inequality |an − 1| < ε holds.
I think that it is easy to understand the expression with the adjective "small" and "big" and the adverb "always", until we used to the above expression.
"Hilbert's paradox of the Grand Hotel" is known as the paradox that shows the mystery of the actual infinity. Even if the all rooms of the hotel are occupied, it is possible to make a room for a new guest, because we can make a room 1 empty by moving the guest in room 1 to room 2, and moving the guest in room 2 to room 3 simultaneously. A new guest is able to stay in the room 1.
Cantor proved the real number is uncountable by Cantor's diagonal argument. I show the detail of the proof in the following article.
Here we consider the following series.
\begin{align*} \pi_1&=3.1 \\ \pi_2&=3.14 \\ \pi_3&=3.141 \end{align*}We show the limit of this series as follows.
$$ \pi = \lim_{n \to \infty} \pi_n $$According to the actual infinity, the circle ratio is the constant π. The numbers of all digits are fixed.
According to the potential infinity, the circle ratio is the function π (ε ). The digit number varies depending on the variable epsilon of the epsilon-delta definition of limit.
"Zeno's paradoxes of motion are known as the paradox that shows the mystery of the actual infinity.
In order to move from the point A to the point B, we need to arrive at the halfway point. In order to move to the point, we need to arrive at the halfway point additionally. Therefore, in order to arrive at the point B, we need to pass through points of the infinite number. However, because it is impossible, we cannot move from the point A to the point B.
According to the potential infinity, we do not arrive at the point B, because we cannot finish dividing the space endlessly.
According to the actual infinity, we arrive at the point B, because we can divide the space endlessly. Then, I add the following operation.
We count a decimal number whenever we arrive at the halfway point. We write the first digit of the decimal number to a notebook. When I write the number in the notebook, we write it after deleting a written number by then. Therefore, only single digit is always written in the notebook.
If the last number exists t is not infinity when we arrive at the point B. If the last number does not exist, what is written in the notebook, when we arrive at the point B?
However, even if we use the above new thought experiment, actual infinity has no contradiction, because the movement from each position to each position which we divided the distance endlessly to infinity is done at the same time in actual infinity. Though the last number, that we write in the notebook, does not exist, we arrive at the point B.
Infinity means actual infinity or potential infinity. The mainstream of mathematics is actual infinity now. Actual infinity has no contradiction. However, I guess that actual infinity does not exist physically.
In potential infinity, the circle ratio π is uncertain within the range of an error epsilon. This is similar to the uncertainty principle of position and momentum in quantum mechanics. Then, I would like to call mathematics of the uncertain number quantum number theory or quantum mathematics. The quantum number theory is explained in the following article. I hope it helps you.
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