1. andrey_smirnov says:

Let �be the smallest natural number p such that p grains of sand form a pile. Hence, p-1 is not a heap. An axiom that follows from the formulation of the paradox condition:

Ah.: 1 < 2 < 3….

The theorem:

Th.: (p+x) + (p+y) = 2p + x + y. That is, the sum of two or more heaps always forms a heap.

Let's number all existing heaps and get the following infinite sequence: x_1, x_2, x_3…. x_p, x_(p+1)..

Please note: the aggregate (x_1, x_2, x_3… x_p) �contains exactly p elements. Therefore, it is a heap that is included in the sequence under consideration by defining that sequence as the class of all heaps. But its elements don't match any of the piles, so it turns out that we didn't number it. 3 initially, the class of all heaps is non-mutable.

Next: aggregate (x_1, x_2, x_3… x_p) this is some element of the x_k sequence (by definition and by construction). This collection has p elements, i.e. it is exactly a heap. But the aggregate is (x_1, x_2, x_3…. x_(p-1)) contains p-1 element – this is NOT a heap by our axiom. But this aggregate is obtained by combining two or more heaps, so it is also a heap (according to the theorem above). Therefore, this aggregate is and is not a heap at the same time. Therefore, we can say that the existence of the number p leads to contradictions.

The contradiction is removed if: 1) we will not consider any aggregate as a heap, 2) we will consider any aggregate as a heap.

2. boris_vilyevich says:

This is already an ethical issue. How bad do you have to do to become bad?� In ancient languages, the score is usually reduced to “one-two-many” (although the first numbers have many shades). In modern versions, “two or more”is used. It turns out that everything that is more than two is a bunch. And how to separate a bunch and a bunch? need criteria