01 / Existence
What counts as existing?
One object.
Several ways to exist.
Existence profile
01 / CollectionThe cube root of two
A precisely specified real number. Existence does not guarantee a ruler-and-compass construction.
Classical existence
✓ Yes · in ℝ
Specified as a real root in classical real analysis.◇ Curated
Algebraic over ℚ
✓ Yes · degree 3
A nonzero polynomial over ℚ has this object as a root.✓ Computed
Computable
✓ Yes
Rationals are exact; real roots of these integer polynomials admit bisection with rational bounds.◇ Curated
Ruler & compass
× No
A constructible real number must have degree a power of 2. Degree 3 fails this necessary condition.✓ Computed
Explicitly definable
✓ Yes
A rational expression, or a polynomial together with an isolating condition, specifies one value.◇ Curated
Structural characterisation
With a specified setting
The ambient field and the choice of a root matter; a polynomial alone need not distinguish conjugate roots.◇ Curated
✓ Computed
From a root to an obstruction
Primitive minimal polynomial over ℚ · degree 3
- Set α = ∛2. Then α³ = 2.
- Exact integer check: 2 is not a perfect cube. By the rational-root theorem, this degree-3 polynomial has no rational root.
- A degree-2 or degree-3 polynomial without a rational root is irreducible over ℚ. The minimal degree is therefore 3.
- A constructible real number must have degree a power of 2. Degree 3 fails this necessary condition.
- A degree that is a power of 2 is necessary, but by itself is NOT sufficient for arbitrary algebraic numbers.
Open the numerical picture
A numerical illustration, not a proof or a certified error bound.
What would a construction require?
An algorithm can approximate the root to any specified precision. A finite ruler-and-compass construction is obstructed.
Interpretation / Discussion
Which kind of construction should count as existence?
A finite geometric construction and an algorithm giving successively better approximations ask for different kinds of evidence. Change the allowed operations: does your notion of existence change with them?
A curated prompt for inquiry. Changing this lens never changes a mathematical result.