# Tag Info

5

If by "$=$" you mean $\beta$-equality, then the answer is yes, $MX=X$ for all $X$ is a stronger property than $MM=M$. For example, let $$A := \lambda a.aa(aa)$$ (to save parentheses, I am using the standard left-associative notation for application; in your notation, the above term would be $\lambda a.a(a)(a(a))$) and take $$M := AA.$$ We clearly ...

3

The non-constructive version of Pataraia's theorem is called the Bourbaki-Witt fixed point theorem. I learned it from Davey and Priestley's Introduction to Lattices and Order, and Wikipedia gives the following references: Nicolas Bourbaki (1949). "Sur le théorème de Zorn". Archiv der Mathematik. 2: 434–437. doi:10.1007/bf02036949. Ernst Witt (1951). "...

1

Here is a detailed outline, not entirely made rigorous. Setting $b_n \stackrel{\rm def}{=} \frac{1}{6}-Y_n$, with $b_0 = 1/6$, we have $$b_{n+1} = \frac{4}{3}\left( \sqrt{1+\frac{3}{2} b_n - \frac{9}{2} b_n^2} - 1\right)\tag{1}$$ (I like to set things near zero.) By induction, $b_n \geq 0$ for every $n$, and a simple computation shows that \$b_{n+1} - b_n \...

Only top voted, non community-wiki answers of a minimum length are eligible