Understanding Cantor's Theory on Countability and Infinity with Decimal Expansions and Higher Dimensions

Introduction

Let's carefully analyze your questions step-by-step, focusing on these key themes:

• The countability of numbers with finite decimal digits vs infinite decimal expansions
• Cantor's diagonal argument and uncountability of real numbers
• Prime number countability compared to natural numbers
• Multidimensional spaces and counting points with finite decimal expansions

1. Decimal Numbers with n Digits vs Infinite Digits

Consider numbers in the interval [0,1) represented with decimals:

• With 1 decimal digit: numbers look like 0.0, 0.1, ..., 0.9 → total of 10 numbers.
• With 2 decimal digits: 0.00, 0.01, ..., 0.99 → total of 100 numbers.
• With n decimal digits: there are exactly 10ⁿ numbers.

These finite-length decimal expansions (stopping after n digits) are perfectly countable because you can list them all in order (from 0 to 10ⁿ -1).

However, the real numbers between 0 and 1 include all infinite decimal expansions as well. That is, numbers with infinitely many digits after the decimal point.

There are infinitely many such infinite decimals. Cantor showed that this set of infinite decimal expansions is uncountable using his diagonal argument. This means you cannot enumerate all real numbers between 0 and 1, even though you can enumerate all numbers with any fixed finite number of decimal digits.

2. Cantor's Diagonal Argument and Uncountability

Cantor's diagonal argument is a proof that the real numbers between 0 and 1 cannot be placed into a one-to-one correspondence with the natural numbers, i.e., they are uncountable.

The logic in brief:

1. Assume that all real numbers (represented by infinite decimals) between 0 and 1 are listed.
2. Construct a new number by changing the nth digit of the nth number in the list.
3. This new number differs from each number in the list by at least one digit → it is not in the list.

This contradicts the assumption that all reals were listed, so the set of real numbers between 0 and 1 is uncountable.

Common Misconception:

The set of numbers with finite decimal expansions (any fixed finite number of decimal digits) is countable, but the limit as the number of digits goes to infinity does not yield a countable set. The infinite decimal expansions create an uncountably infinite set.

3. Counting Points in Higher Dimensions

If you consider points in a d-dimensional hypercube [0,1)^d, then:

• With n decimal digits per coordinate, each coordinate has 10ⁿ possible values.
• Total points with n decimal digits in d dimensions is (10ⁿ)^d = 10^n*d.

If d = n (dimension grows with the number of digits), then points count ≈ 10^n².

But note:

All these sets with finite digits remain countable, because you can arrange these points in a sequence (for example, lexicographically).

It is only when considering infinite decimal expansions in even one coordinate that the set becomes uncountable.

4. Prime Numbers and Countability

Prime numbers form a subset of natural numbers. Since:

• The natural numbers are countably infinite.
• A subset of a countable set is either finite or countable.

Therefore, the primes are also countably infinite.

Your formula for the recurrence:

c(0) = 2

c(1) = 3

c(n+2) = -1 + 2^{c(n+1)}

generates very large numbers quickly. However, it is not proven that all c(n) in this sequence are prime numbers, and the logic about the size of c(n) compared to 10^n² does not affect the countability of the set of primes.

5. Summary and Clarifications

• Finite decimal expansions with n digits give a finite, countable set of numbers: cardinality = 10ⁿ.
• Allowing infinite decimal expansions (infinite digits) creates uncountably infinite sets (the real numbers).
• Cantor's diagonal argument rigorously shows the uncountability of real numbers, a fundamental result in set theory.
• The prime numbers are a countably infinite subset of natural numbers.
• Higher-dimensional finite-digit points are also countable for fixed n and d.
• When dimensions grow with digits (d = n), the number of points grows as 10^n², still finite for fixed n and countable over all n.

6. On the Claim that Cantor’s Theory is False or Taboo

Cantor’s theory is a thoroughly vetted, foundational pillar of modern mathematics. It has withstood over a century of scrutiny and forms the basis of set theory and our understanding of infinity.

Common misunderstandings often arise from conflating finite decimal expansions with infinite expansions or misinterpreting what “countable” means:

• Countable means the elements can be put in one-to-one correspondence with natural numbers.
• Some infinite sets (like natural numbers or rationals) are countable.
• Others (like real numbers) are uncountable.

Your reasoning about counting integers, decimals, and higher dimensions does not contradict Cantor’s proofs but relates to finite approximations.

7. Additional Notes on Rational Numbers

The set of rational numbers (fractions) is countable, even though dense in real numbers.

This is because you can enumerate all pairs of integers (numerators and denominators) carefully (e.g., via Stern's diatomic sequence or arranging in a grid and enumerating diagonally).

Conclusion

Cantor's diagonal argument proves uncountability of real numbers. Finite decimal expansions correspond to countable sets, but infinite decimal expansions produce uncountability.

Prime numbers are countable subsets of natural numbers.

Higher-dimensional finite-digit points are countable for fixed dimensions and digits but grow rapidly in number as dimension and digits increase.

Your insights on countability reflect common confusions that arise from mixing finite and infinite sets — understanding these distinctions clarifies the nature of infinite cardinalities in mathematics.

If you want, I can guide you through Cantor’s diagonal argument step-by-step or help with formalizing the countability of rationals or primes.

Selam! I hope this clarifies your questions about countability, infinity, and Cantor's theory.