Understanding Countability of Finite and Infinite Decimal Expansions
Let me start by carefully clarifying the concepts step-by-step, based on your questions and points.
1. Finite Decimal Numbers With n Digits After the Decimal Point
- For decimal numbers between 0 and 1 with exactly n digits after the decimal, the total number of such numbers is exactly 10n.
- This is because each of the n decimal places can take 10 values (0-9), independently.
- For example:
- n=1 → 10 numbers: 0.0, 0.1, ..., 0.9
- n=2 → 100 numbers: 0.00, 0.01, ..., 0.99
2. Countability of Natural Numbers and Finite Extensions
- All natural numbers are countable by definition.
- Adding finite decimal digits after the decimal point corresponds to finite sequences of digits, which can be encoded as integers (for instance, by removing the decimal point and treating it as an integer). Hence, these finite decimal numbers correspond one-to-one with a subset of natural numbers and are also countable.
3. What Happens as n Approaches Infinity?
- When you let n → ∞, i.e., you allow infinitely many digits after the decimal point, the set of all such infinite decimal expansions between 0 and 1 is known to be uncountable.
- This result is a fundamental theorem proven by Cantor’s diagonal argument:
- You cannot list all infinite decimal expansions in a sequence — there will always exist an expansion not in the list.
- This means the number of infinite decimal expansions (i.e., real numbers in [0,1]) is of strictly larger cardinality than the natural numbers (countable sets).
4. Addressing the Diagonal Argument Objections
You mentioned that if you remove the first two digits of each infinite decimal, it looks like an infinite integer list, so the diagonal argument must be flawed. Let's clarify:
- An infinite decimal expansion like 0.123456... can be considered as an infinite sequence of digits. Removing first two digits simply shifts the sequence and does not convert it into a finite integer — it's still an infinite sequence.
- These infinite sequences cannot be put into a complete, one-to-one correspondence with natural numbers; hence, uncountability remains valid.
- Certainly, infinite decimal expansions can sometimes correspond to rational numbers (which have eventually repeating decimals), but the total set includes irrationals with non-repeating digits, making the set uncountable.
- The diagonalization argument explicitly constructs a decimal expansion differing from every number on a purported complete list, guaranteeing it is missing and proving uncountability.
5. On the Cardinality of Higher Dimensional Unit Cubes
- For a hypercube of dimension d, where each coordinate lies in [0,1], the set of points is equivalent to the product set [0,1]^d.
- The cardinality of [0,1]^d is the same as the cardinality of [0,1] itself for all finite d.
- For example, the set of points in 2D unit square or 3D unit cube has the same cardinality as the interval [0,1].
- If dimension d goes to infinity, the cardinality remains the cardinality of the continuum (same as the real numbers).
- The count of finite decimal expansions with n digits in d dimensions is roughly 10n*d. But as n,d → ∞, this count tends to infinity but remains countable for any finite n,d. Only for infinite sequences (infinite digits or infinite dimensions) does uncountability arise.
6. On Comparing 10^{n^2} and the Growth of Sequences like c(n)
- You presented a sequence defined recursively by c(0)=2, c(1)=3, and c(n+2)=-1+2^{c(n+1)}.
- This sequence grows rapidly (much faster than polynomial or exponential growth).
- Although the values can become huge quickly, this does not impact countability: the total number of natural numbers (and subsets like primes) is always countable regardless of how large the terms get.
- Prime numbers are a subset of natural numbers and are countable; no matter how fast the growth, you can't make an uncountable subset out of primes.
7. On Rational Numbers Being Countable
- This is well-known: rational numbers (fractions of integers) are countable.
- You described rationals in terms of integer pairs (x,y) with y ≠ 0, which is correct.
- Pairs of integers themselves are countable (since integers are countable and the Cartesian product of two countable sets is countable), so rationals are countable.
8. Are There Any True Logical Contradictions or “Taboos”?
The mathematics around countability and uncountability has been studied rigorously for over a century with consistent results.
- Cantor's diagonal argument is a well-accepted proof with no logical contradiction within standard set theory (ZFC).
- Claims that diagonal argument is a logical illusion or flawed come from misinterpretations, especially if infinite sequences are mistaken for finite sequences or vice versa.
- Regarding infinite cardinalities comparisons such as 10^n vs n, for finite n these are standard finite values, and as n grows without bound, comparing cardinalities relates to infinite sets, which are addressed using set theory cardinal arithmetic, not ordinary arithmetic.
9. Final Thought
In conclusion, the count of numbers with finite decimal digits is countable, but the set of infinite decimal expansions between 0 and 1 is uncountable.
Infinite dimensional cubes do not increase cardinality beyond the continuum.
Cantor's diagonal argument stands as a valid mathematical proof and is not disproved by reinterpreting infinite decimal sequences as infinite integers by removing decimal points.
If you want, I can provide detailed examples or further clarify any of these points.
Hope this discussion clarifies your questions! Feel free to ask more.