Understanding Countability, Decimal Expansions, and Cantor's Set Theory: A Critical Discussion for Advanced Mathematics

1. Understanding the Setup: Sequences and Countability

You have presented two main ideas:

1. A recursive sequence c(n) defined by c(0) = 2, c(1) = 3, and c(n+2) = -1 + 2^{c(n+1)}, with the assertion that each c(n) is an integer and always prime.
2. A discussion about decimal expansions between 0 and 1, the count of numbers with fixed decimal digits, and their relation to Cantor's theorem on countability and uncountability.

2. Countability of Decimal Numbers with Fixed vs. Infinite Digits

Let's clarify countability in the context of decimal expansions:

• For decimal numbers between 0 and 1 with exactly one decimal digit, the possible numbers are 0.0, 0.1, ..., 0.9, total = 10 numbers. This is a finite set.
• With exactly two decimal digits, numbers like 0.00, 0.01, ..., 0.99, total = 10² = 100, still finite.
• For n decimal digits, the total numbers are 10ⁿ, again finite for any finite n.

But when n approaches infinity, i.e., considering all infinite decimal expansions (numbers with infinitely many digits after the decimal point), the set is uncountably infinite. This is Cantor's diagonalization theorem: you cannot list all real numbers between 0 and 1 in a sequence indexed by natural numbers—the set is too large to be counted.

Why is 10ⁿ countable or uncountable when n → ∞?

• For any fixed finite n, 10ⁿ is finite.
• If you treat n as infinity, 10ⁿ is shorthand for the set of all infinite digit sequences. The cardinality (size) of the set of infinite sequences of digits 0–9 is that of the continuum, which is uncountably infinite.

3. Higher Dimensions and Decimal Points

When you mention two or three dimensions, you refer to points in the [0,1) × [0,1) square or [0,1)^3 cube, etc. The cardinality of these sets also equals the cardinality of the continuum because there exists a bijection (one-to-one and onto function) between these multi-dimensional real intervals and the real numbers in [0, 1). Thus:

• 10^{2n} in two dimensions represents decimal expansions with 2n digits total (e.g., n digits in x and n digits in y).
• This still approaches the cardinality of the continuum when n → ∞.

So, the dimensionality of the space of infinite decimal expansions does not increase the 'uncountability' beyond that of the continuum; they have the same cardinality.

4. Prime Numbers and Their Countability

You mentioned the sequence c(n) generates prime numbers increasing very rapidly, and compared it with counts like 10^{n^2}. Let me clarify this aspect:

• Prime numbers are countably infinite, as they are a subset of natural numbers.
• The prime counting function π(x) (number of primes less than or equal to x) satisfies π(x) ~ x / \, ext{ln}(x) for large x.
• The outputs of your sequence c(n) grow extremely fast (double-exponential style), which may generate very large primes quickly, but this does not change their countability.

5. Clarifying Cantor's Theory and Addressing the 'False Logic' Concern

Cantor's set theory around countability and uncountability is well-established and rigorously proven. Some common misunderstandings are:

• Mistaking large finite numbers with uncountable infinities: Growing very fast (like 10^{n^2}) is huge but still finite for finite n.
• Countability vs. size: The natural numbers and primes are infinite but countable. Real numbers with infinite decimal expansions are uncountable.
• Decimal expansions: Numbers with finite decimal expansions are countable. Including all infinite decimal expansions leads to uncountability.

The diagonalization argument no longer holds for finite decimal expansions but only when infinite expansions are included, which is essential to real numbers.

6. Summary and Final Remarks

• Countable sets: Finite or infinite sets that can be put into one-to-one correspondence with natural numbers (e.g., integers, primes).
• Uncountable sets: Sets (like real numbers between 0 and 1) that cannot be listed this way due to their size.
• Higher dimensions of real numbers do not increase cardinality beyond the continuum.
• The sequence generating primes with high exponents remains a countable set.
• Therefore, Cantor's uncountability results are logically sound and rigorously proven in mathematics, not a 'false logic' or taboo.

If you want, I can explain Cantor's diagonal argument step-by-step or delve deeper into any specific concept you mentioned.