<h1>0 is finite</h1>

We use dots rather than parentheses. "X and Y" is denoted "X · Y"

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Hypothesis: The real number 0 is finite

Proof:

i We define the real 0 as the null set, symbolized ∅ or { }.

ii. Infinite set is defined according to the following strict implication:

B ⊂ A · B bijective with A: ←→ :A infinite

iii. The following strict implication holds for a finite set.

If no B exists that is bijective with A, then A is finite.

So if A = ∅, then no set B is a proper subset of A, since any B ⊆ A implies B = A. Hence the null set is finite. And, with the real number 0 defined as ∅, we have 0 finite.

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Comment: It may seem obvious that ∅ is finite, but that belief requires proof. The hypothesis cannot be proved without a definition of 0, which is derived from metamathematics.

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