Exponent Base of One
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Theorem
Let $x$ be an ordinal.
Then:
- $1^x = 1$
Proof
The proof shall proceed by Transfinite Induction on $x$.
Basis for the Induction
\(\ds 1^0\) | \(=\) | \(\ds 1\) | Definition of Ordinal Exponentiation |
This proves the basis for the induction.
Induction Step
The inductive hypothesis supposes that $1^x = 1$ for some $x$.
\(\ds 1^{x^+}\) | \(=\) | \(\ds 1^x \times 1\) | Definition of Ordinal Exponentiation | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \times 1\) | Inductive Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Ordinal Multiplication by One |
This proves the induction step.
Limit Case
The inductive hypothesis supposes that $\forall y \in x: 1^y = 1$.
It follows that:
\(\ds \forall y \in x: \, \) | \(\ds 1^y\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \bigcup_{y \mathop \in x} 1^y\) | \(=\) | \(\ds \bigcup_{y \mathop \in x} 1\) | Indexed Union Equality | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 1^x\) | \(=\) | \(\ds 1\) | Definition of Ordinal Exponentiation |
This proves the limit case.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.31$