Filling up holes in the sand

Probability Level 3

The function $\log_{\cdot}(\cdot): \mathbb{Q}^{+} \times \mathbb{Q}^{+} \longrightarrow \mathbb{R}$ . While the codomain is the real numbers $\mathbb{R}$ , can the image be $\mathbb{R}$ ?

Yes No Undecidable in ZFC

2 solutions

Jake Lai
May 3, 2015

Cantor's diagonalisation argument implies that there cannot exist a bijection from a countably infinite set (in this case, $\mathbb{Q}^{+} \times \mathbb{Q}^{+}$ ) to an uncountably infinite set (in this case, $\mathbb{R}$ ).

Abhishek Sinha
May 3, 2015

If possible, assume there exists positive rationals $q_1,q_2$ such that $\log_{q_1}q_2=\sqrt{2}$ Hence we would have $q_2=q_1^{\sqrt{2}}$ By Gelfond-Schneider theorem the right hand side is transcendental, thus resulting in contradiction.

Nice solution!

Jake Lai - 6 years, 1 month ago

Filling up holes in the sand

2 solutions

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