De Zolt's Postulate

A solid cannot be equivalent to one of its own parts.

This postulate asserts that a solid cannot share the same volume as a portion of itself - that is, a subset that is strictly contained within the original object. Here, "equivalent" specifically means "having the same volume."

In essence, no solid or geometric figure can be volumetrically identical to any of its proper parts.

Example

If solid A is equivalent to a part of another solid B, then B must be larger than A:

$$ B > A $$

In other words, we can say that solid B takes precedence over A in size.

Note. De Zolt’s postulate is often discussed within the framework of Giuseppe Peano’s theory of geometric magnitudes and in the formal treatment of elementary geometry. It was introduced in the 19th century by Luigi De Zolt as a safeguard against certain paradoxes - most notably, the Banach - Tarski paradox - that arise in non-Euclidean settings or under the assumption of the Axiom of Choice. According to the Banach - Tarski paradox, assuming the Axiom of Choice, it is theoretically possible to decompose a solid sphere into a finite number of non-measurable pieces (which individually have no volume) and reassemble them into two identical copies of the original sphere - without any deformation. So, while De Zolt’s postulate may seem like a down-to-earth axiom, it serves as a critical bulwark against mathematically valid yet physically implausible conclusions.

It plays a foundational role in preserving the intuitive coherence of volume and the broader concept of geometric equivalence.

Its purpose is to preclude paradoxes in which a solid can be disassembled and reconfigured into something with a greater volume than the original.

A Practical Example

Consider a cube-shaped box with each side measuring 10 cm. Its volume is:

$$ V = 10^3 = 1000\ \text{cm}^3 $$

Now, imagine a smaller cube that fits inside this box - say, one with 5 cm sides:

$$ V_{\text{piccolo}} = 5^3 = 125\ \text{cm}^3 $$

Could these two objects possibly have the same volume? Of course not!

And that’s precisely the point De Zolt’s postulate is making.

Since 125 ≠ 1000, a cube cannot be volumetrically equivalent to any of its own parts.

Note. It may sound like an obvious statement - but that’s exactly what makes it so effective. It rules out absurdities that, while technically consistent within certain mathematical frameworks, defy both physical reality and common sense. For instance, without this postulate, one might claim that by slicing a box in a certain way and rearranging the pieces, it's possible to create two boxes identical to the original. De Zolt’s postulate draws a hard line, preventing such theoretical manipulations from leading to nonsensical outcomes.

Notes

Some reflections and side notes on De Zolt’s postulate.

• Two-Dimensional Version of De Zolt’s Postulate
  A two-dimensional analogue also exists:

  A polygon cannot be equivalent (i.e., have the same area) as one of its proper parts.

  The underlying idea remains the same, but it applies to area rather than volume.

• Is De Zolt’s Postulate a Thoughtful Safeguard or Overly Cautious Conservatism?
  The history of mathematics is full of breakthroughs that upended our intuition - from imaginary numbers to the infinite, from topology to quantum mechanics. Time after time, intuition has been challenged, reshaped, and rebuilt from the ground up. De Zolt’s postulate represents a conservative stance: a way to impose order by deliberately limiting what is deemed possible. It is a cautious, perhaps even restrictive choice - one that avoids confronting complexity by refusing to let it in.

  Its harshest critics see it as an epistemological retreat. Faced with the first hint of paradox, rather than probing deeper into the foundations of mathematical knowledge, it declares: “This can’t happen. Period.”

  That said, it’s important to remember the historical context. De Zolt was working in the 19th century, when mathematics was still solidifying its role as the rigorous language of the physical world. A postulate that tightly bound measurement to shape was considered essential. Only with the advent of modern analysis and formal logic in the 20th century did mathematics gain the tools to grapple with what had once seemed unimaginable.

And so on.