Understanding Countability, Cantor's Diagonal Argument, and Dimensions in Mathematics

Introduction

Your question involves several important concepts in mathematics related to set theory, number theory, and infinity. Let's break down each part carefully:

1. Counting Decimal Numbers with n Digits

When considering decimal numbers between 0 and 1 with exactly n digits after the decimal point, the quantities are:

• With 1 decimal digit: 10 possibilities (0.0 to 0.9)
• With 2 decimal digits: 100 possibilities (0.00 to 0.99)
• With n decimal digits: 10ⁿ possibilities

These sets are clearly finite for any fixed n. However, when n approaches infinity, we consider all infinite decimal expansions of numbers between 0 and 1, which corresponds to the entire set of real numbers in [0,1).

2. Countability and Uncountability

A set is called countable if its elements can be listed in a sequence (like natural numbers: 1, 2, 3, ...) even if the list is infinite. Otherwise, it is called uncountable.

• The set of all finite decimal numbers with finite digits (for example, up to n) is finite and countable. This is because you can list all those numbers explicitly.
• The set of all rational numbers (ratios of integers) is countable, even though there are infinitely many. This is because rational numbers can be arranged systematically (e.g., using a two-dimensional grid of numerators and denominators and following a diagonal path).
• The set of real numbers between 0 and 1 is uncountable. This is a classical result proven by Cantor’s diagonal argument. The infinite decimal expansions of reals cannot be arranged in a complete list where none is missing.

3. Explanation of Cantor's Diagonal Argument

Cantor's diagonal argument shows that if you try to list all real numbers in [0,1) as infinite decimal expansions, you can always construct a new number not in the list by changing the nth digit of the nth number, ensuring the new number differs from every number in the list in at least one decimal place.

This argument applies to infinite sequences of digits but not to finite sequences. So, it does not contradict that the numbers with any fixed finite number of decimal digits are countable. It only applies when considering the entire infinite decimal expansions, meaning infinitely many digits.

4. Dimensions and Number of Points

When you consider points in 1D, 2D, 3D, or n-dimensional spaces, the count of points in those continuous intervals or cubes (like [0,1]) is uncountably infinite. For example:

• 1D interval [0,1]: uncountably infinite points (continuum cardinality), not 10ⁿ
• 2D square [0,1]×[0,1]: also uncountable — effectively same cardinality as 1D continuum because we can create bijections between unit intervals and squares
• 3D cube and higher dimensions follow similarly — the cardinality does not increase beyond the continuum

Note: 10ⁿ is finite for finite n and goes to infinity as n approaches infinity, but infinity here is countable if it can be put in one-to-one correspondence with natural numbers, which is not the case for uncountable reals.

5. Comparing Sequences and Cardinalities

For the recursive sequence you gave (c(0)=2, c(1)=3, c(n+2) = -1 + 2^{c(n+1)}), the values grow extremely fast (much faster than polynomial or exponential). However, this is a sequence of integers (not a set containing all integers up to that number), so it is still a countable sequence because it is indexed by natural numbers.