Theorem: Closed intervals are not countable sets.
Prove by reducing to absurdity:
Suppose it is a countable set, then there is a closed interval with a closed interval in the middle and a closed interval in the middle, and so on. In this way, the closed interval sequence with monotonically decreasing middle is obtained. Therefore, from the closed interval set theorem in mathematical analysis, we can know the existence. Obviously, but for any existence, we deduce the contradiction, so it is uncountable. Proved it.
In the future, we will call an equivalent set with continuous potential. Here are some theorems related to continuous potential and some important sets with continuous potential.
Any interval has a continuous potential.
Theorem 1: Any interval has a continuum potential. In particular, the set of real numbers has a continuous unified potential. The proof of this theorem is relatively easy, and readers can try it themselves.
N-ary sequence
Let be a positive integer greater than. If the items in a series consist only of this number, it is called a meta-series. If you are in the middle.
Only when the finite term is not, it is called a finite element sequence; Otherwise it is called infinite element sequence.
Lemma: Let an infinite set be at most a countable set, then.
Proof: suggested establishment
Because it is an infinite set, the countable subset that can be taken is countable at this time, so the lemma proof can be obtained.
Theorem: If, then the variable sequence of the whole continuous system potential.
Proof: First, it is easy to prove that the finite sequence of elements is countable. So from the above lemma, we only need to prove that it is equivalent to the infinite element sequence.
To this end, there are unique positive integer derivatives, unique derivatives, unique derivatives, and so on. Generally speaking, it is followed by
The way to get the result is an infinite sequence of n elements. In this way, I can't draw a conclusion that a mapping from the above formula to the whole infinite sequence is a bijection for every easy knowledge, so that the whole infinite sequence has a continuous system potential. Theorem proved.
Note: If, then the formula is the usual decimal notation.
All subsets of countable sets have continuous system potentials.
Proof: Let be a nonempty subset of all positive integers. Definition and manufacture
Obviously, bijection between the whole subset and the whole binary sequence has been established. The latter has continuous potential, which makes the whole subset of countable set in continuous potential. Theorem proved.
Influence of direct product on continuum potential
Theorem: The direct product of several sets with continuous potential at most has continuous potential.
Proof: Suppose that each is the whole of a binary sequence and their direct product. In order to prove this theorem, we only need to prove that it is equivalent to the whole of a binary sequence.
At this point, it is a binary sequence for each order. According to the above law, it is a mapping to a total binary sequence, which is obviously bijective, so it is equivalent to a total binary sequence. Theorem proved.
Note: Careful readers may find that it seems to have the same effect as the diagonal rule used in the process of proving that countable sets are still countable sets. Diagonal rule is a common means in mathematical analysis. Readers who have studied mathematical analysis or advanced mathematics can think about where we use the diagonal rule.
Inference 1: There is a continuum potential in both plane and space.
Inference 2: All real number sequences have continuous unified potential.