Barbers in Real Villages
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Suppose there may be a village where the barber shaves all the other men that do not shave themselves. So the barber shaves himself. No contradiction. It is ok for the barber to shave himself.
Approximation of Total Natural Functions through Primitive Recursive Functions on Finite Sets
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In this post we state that if $f \colon \mathbb{N} \to \mathbb{N}$ is a total function, and $A \subseteq \mathbb{N}$ is a finite set, then $\exists g: \mathbb{N} \to \mathbb{N}$ a primitive recursive function such that: $$f|_A = g|_A. \quad \square$$
On the Non-Existence of Russell's Set
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In this post we prove that
$$\neg \exists R \forall x \; x \in R \Leftrightarrow x \notin x.$$Indeed,
$$\forall R \exists x \; x \in R \Leftrightarrow x \in x,$$because
$$\forall R \; R \in R \Leftrightarrow R \in R. \quad \square$$A Divisibility Criterion with 41
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In this post we write natural numbers in decimal.
Let $n$ be a 100-digit natural number.
Let us form 20 of 5-digit numbers with the digits of $n$ in the order in that they appear in the decimal representation of $n$ (left to right).
Then $n$ is a multiple of 41 if and only if the sum of the 20 numbers mentioned above is a multiple of 41. (This works because $10^5 \equiv 1 \pmod{41}$.)