Mathematical Explanation: Dimensionality of Number Sets
Problem Statement
Consider a set of solutions to a function that are either:
- • Natural numbers: ℕ = {1, 2, 3, …}
- • Whole numbers: 𝕎 = {0, 1, 2, …}
- • Integers: ℤ = {…, −2, −1, 0, 1, 2, …}
These numbers exist on the real number line.
Question: Is this set "in one dimension," and why?
Key Concepts
What "One-Dimensional" Means
A one-dimensional space requires exactly one real coordinate to specify a point's location.
The real number line ℝ is the standard example: each point corresponds to a single real number.
Ambient Space vs. Set Dimension
Ambient space: The larger space containing the set (here, ℝ)
Set's intrinsic dimension: The dimensionality of the set itself as a topological space
Dimensional Analysis
Ambient Space Dimensionality
The real number line ℝ is one-dimensional because:
- ✓ Each point can be located with one coordinate
- ✓ It forms a continuous line with no width
- ✓ It satisfies the mathematical definition of 1D Euclidean space
Important Distinction
The sets ℕ, 𝕎, and ℤ are 0-dimensional as topological spaces:
- • They consist of isolated points
- • They contain no continuous intervals
- • Topologically, they are discrete sets
However, they are embedded in a 1-dimensional space (ℝ).
Answer to the Question
When we say a set is "in one dimension," we typically mean it is contained within a one-dimensional space.
The natural/whole/integer numbers are points on the real number line ℝ¹, which is definitively one-dimensional.
The type of numbers does not change their embedding space.
Yes, the set is in one dimension.
Why? (The Reasoning)
Because these numbers exist as specific locations on the real number line (ℝ¹), which is a one-dimensional Euclidean space.
Each point corresponds to a single real coordinate, satisfying the definition of one-dimensional embedding.
The fact that the set itself is topologically zero-dimensional (discrete points) doesn't change that it resides in a 1D ambient space.
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