Can infinity be divided yet still remain whole?

Question

Accepted Answer

In the context of mathematical set theory and transfinite arithmetic, the concept of "dividing" infinity depends entirely on how the operation is defined. Because infinity is not a standard number, it does not obey the typical rules of arithmetic used for finite integers.

1. Cardinality and Subsets
In set theory, if you divide an infinite set (such as the set of all natural numbers, $\mathbb{N}$) into smaller subsets, those subsets can often retain the same cardinality (size) as the original set.

Example: If you take the set of all natural numbers $\{1, 2, 3, 4, ...\}$ and "divide" it into two sets—the set of even numbers $\{2, 4, 6, ...\}$ and the set of odd numbers $\{1, 3, 5, ...\}$—both resulting sets are still countably infinite. 
   The Result: You have partitioned the original infinite set, yet each "piece" remains infinite. In this sense, the infinity remains "whole" because it maintains its cardinality.

2. The Nature of $\infty$ in Calculus
In standard calculus, infinity is not treated as a value that can be divided. If you attempt to divide infinity by a finite number, the limit remains infinite:
$\frac{\infty}{n} = \infty \quad (	ext{where } n 	ext{ is a finite constant})$
Because infinity is a boundary condition or a direction rather than a quantity, it cannot be diminished by division. It remains "whole" because it is not a quantity that can be partitioned into smaller, finite portions.

3. Philosophical Perspectives
From a philosophical standpoint, whether infinity remains "whole" depends on the definition of wholeness:

Holism: If "whole" implies a complete, indivisible unit, then infinity is arguably the only entity that can be divided infinitely without losing its essential nature. 
   Fractals: In geometry, fractal objects exhibit self-similarity. If you "divide" or zoom into a fractal, you find the same infinite complexity at every scale. Here, the structure remains "whole" regardless of how many times you subdivide the space.

Summary
Mathematically, infinity can be divided into smaller sets, but those sets often possess the same magnitude as the original. It remains "whole" in the sense that it does not lose its infinite property; it does not "run out" or become finite simply because it has been partitioned.