1. Introduction to Countability
A set is called countable if its elements can be put into one-to-one correspondence with the natural numbers \( \mathbb{N} = \{0, 1, 2, 3, ...\} \). Intuitively, this means you can "list" all elements without missing any.
2. Finite Decimal Expansions and Their Countability
Consider decimal numbers between 0 and 1 with exactly n digits after the decimal point:
- With 1 decimal digit: the set is \( \{0.0, 0.1, ..., 0.9\} \), so 10 numbers.
- With 2 decimal digits: the set is \( \{0.00, 0.01, ..., 0.99\} \), which has \(10^2 = 100\) numbers.
- With n decimal digits: the count is \(10^n\).
Since \(10^n\) is finite for any finite \(n\), this set is clearly finite and hence countable.
3. What Happens as \( n \to \infty \)?
If we consider decimal expansions with infinitely many decimal digits, this corresponds to the set of all real numbers between 0 and 1 (including irrationals). This set is uncountable. Why?
- Each number is represented as an infinite sequence of digits.
- The set of all infinite sequences over a fixed finite alphabet (here digits 0 to 9) has the cardinality of the continuum.
- Cantor's diagonal argument shows no listing (no countable enumeration) can capture all infinite decimal expansions. By constructing a new number that differs in the nth digit from the nth listed number, we ensure some number is always missing in any countable list.
Therefore: While \(10^n\) is finite for any fixed \(n\), taking the limit as \(n \to \infty\) means dealing with infinite sequences, which form an uncountable set.
4. Addressing Cantor's Diagonal Argument and Is It False Logic?
Cantor's diagonal argument is logically rigorous and widely accepted in mathematics.
- It applies specifically to infinite sequences (or infinite decimal expansions), not to finite ones.
- The construction ensures the new infinite sequence differs from every sequence in the list at least at one digit.
- This demonstrates no enumeration of infinite decimals can list all reals — they are uncountable.
There is no contradiction or "false logic" here; it follows precisely from the definitions of countability and infinite sequences.
5. Multi-Dimensional Intervals and Counting Points
Consider the unit hypercube \([0,1]^n\):
- Each coordinate ranges in \( [0,1] \).
- If restricting coordinates to finite decimal expansions of length \(n\), each coordinate has \(10^n\) possible values.
- Therefore, the total number of possible points with \(n\) decimal digits after the decimal in all dimensions is \( (10^n)^n = 10^{n^2} \).
- Since this is finite for finite \(n\), the set is countable.
- But if you allow infinite decimal expansions across all \(n\) dimensions, the set is uncountable (has cardinality of the continuum).
6. Comparing Sets \(a = 10^{n^2}\) and Recursively Defined Sequence \(b = c(n) = -1 + 2^{c(n-1)}\)
The sequence \(c(n)\) grows extremely fast and can outgrow \(10^{n^2}\) for large \(n\). However, the size of \(c(n)\) (a number) does not determine the countability of sets of real numbers.
Prime numbers, though infinite, are countable, and their growth rate is characterized by \(p(n) \sim n \ln n\) (the nth prime roughly behaves like \(n \ln n\)). This doesn't conflict with set cardinality but relates to density and size growth.
7. Countability of Rational Numbers
Rational numbers are indeed countable. This can be seen by parameterizing rationals as pairs of integers:
- Every rational is \(x = p/q\), where \(p, q \in \mathbb{Z}\), \(q \neq 0\).
- Since \(\mathbb{Z}\times \mathbb{Z}\) is countable (enumerate pairs \((p,q)\) in a grid fashion), rationals are a countable subset.
8. Summary and Clarification
- Decimal numbers with finite digits after the decimal point are finite and countable.
- The set of real numbers between 0 and 1 includes numbers with infinite decimal expansions — an uncountable set.
- Cantor's diagonal argument is a correct and rigorous proof that infinite sequences over finite alphabets are uncountable.
- Dimension counting with finite decimal digits follows basic product counting principles (e.g., \(10^{n^2}\) for \(n\)-dimensions with \(n\)-digit accuracy).
- The transition from finite to infinite digit expansions causes a jump from countable to uncountable cardinality.
- Prime numbers, rational numbers, and other infinite discrete sets remain countable.
- Claims of "false logic" or "taboo" are misunderstandings of the precise definitions and arguments in set theory.
9. Final Note
Mathematics, especially set theory and cardinality, relies on careful definitions of sets and counting techniques. Cantor's theory is foundational and consistent. Misinterpretations often come from conflating finite approximations with infinite objects or mixing finiteness with infinitude.