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The number of essences that exist is finite. The proof for establishing this fact, is called Burhan Al-Tatbiq. It can be expressed in the following manner:

• If an infinite number of essences existed, then it would be possible for there to be two groups of existents, each infinite in size[1], while one is lesser in size than the other.
• It is impossible for there to be two groups of existents, each infinite in size, while one is lesser in size than the other.
• Therefore an infinite number of essences do not exist.

### Premise 1: If an infinite number of essences existed, then it would be possible for there to be two groups of existents, each infinite in size, while one is lesser in size than the other.

Suppose that an infinite number of essences existed. In such a scenario, the set of all existent essences would be comprised of an infinite number of existents. Now consider the half of this set, which would also be comprised of an infinite number of existents[2].

The second set is part of the first, so it is necessarily lesser than it in size. And yet, both sets are comprised of an infinite number of existents.

### It is impossible for there to be two groups of existents, each infinite in size, while one is lesser in size than the other.

If two groups are comprised of an infinite number of existents each, then those groups are equal in size. On the other hand, if one group is lesser in size than the other, then the two are unequal in size.

Given the above: to suppose that there are two groups of existents, each infinite in size, while one is lesser in size than the other, is to suppose that the two groups are both equal and unequal in size. This is contradictory, and is therefore impossible.

### Therefore an infinite number of essences do not exist.

The two premises are true, so the conclusion necessarily follows. Therefore, the number of essences that exist is necessarily finite.

[1] By “size” in this context, we mean group size, or cardinality. As in: the number of existents in each group.

[2] Since half of an infinite amount, would also be infinite.