PROOF 15
Prove.
a⁰ = 1
On the positive reals.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
For a natural number n, aⁿ is a multiplied n times.
Proof.
(1)a¹ = a, and for natural n, aⁿ⁺¹ = aⁿ × a.convention
(2)This definition has meaning only when n is a natural number. a⁰ has no meaning yet.convention
∎
Definition D2
a⁰ is the value fixed so that the law of exponents also holds at exponent 0.
Proof.
(1)A1 holds when m and n are natural numbers.A1
(2)D1 assigns no value to a⁰. The value must be fixed anew.D1
(3)There is one criterion — keep the law of exponents intact through integer exponents.A2
∎
Lemma L1
a⁰ × a = a
Proof.
(1)0 + 1 = 1A6
(2)a⁰⁺¹ = a⁰ × a¹D2, A2
(3)a⁰⁺¹ = a¹(1), E
(4)a¹ = aD1
(5)a⁰ × a¹ = a⁰ × a(4), E
(6)a⁰ × a = a(2), (3), (4), (5)
∎
Theorem T
If a > 0, then a⁰ = 1
Proof.
(1)a⁰ × a = aL1
(2)Since a > 0, a⁻¹ exists.A3
(3)(a⁰ × a) × a⁻¹ = a × a⁻¹(1), (2), E
(4)a × a⁻¹ = 1A3
(5)(a⁰ × a) × a⁻¹ = a⁰ × (a × a⁻¹)A4
(6) = a⁰ × 1 = a⁰(4), A5
(7)a⁰ = 1(3), (4), (6)
∎
Corollary C
The value of 0⁰ is not determined by this argument.
Proof.
(1)Suppose the stage is widened to allow 0 as a base.assumption
(2)L1 gives 0⁰ × 0 = 0.L1, (1)
(3)For every x, x × 0 = 0.A7
(4)Whatever value 0⁰ is given, (2) holds.(2), (3)
(5)A3 provides no 0⁻¹ for the base 0, so line (3) of T cannot be set up.A3
(6)The value of 0⁰ is not narrowed down.(4), (5)
∎
Remark
Can A2 be removed from this proof?
Fix a⁰ to a value other than 1 and see what breaks.