Proof that at least one number exists: Take for our logic first order logic with equality[1]. Take for our definition of the word "number" the Peano axioms[2].

Proof that at least one number exists: Take for our logic first order logic with equality[1]. Take for our definition of the word "number" the Peano axioms[2]. The first Peano axiom is "0 is a natural number." Define "N" to be the unary relation "is a natural number." In our formal language, we write the first Peano axiom as "N0". By the principle of existential generalization[3], we have "∃x Nx", which we read as "there exists x such that x is a natural number." Q.E.D. I assume you find the above deeply unsatisfying. Most people do. Yet I find it clarifies the matter entirely! If a person's conception of "number" is consistent with the Peano axioms, and if that person's use of the word "exists" includes the existential logical quantifier, then for that person "numbers exist" is a theorem. If a person uses other definitions, the matter may be otherwise. [1]: https://plato.stanford.edu/entries/logic-classical/ https://plato.stanford.edu/entries/logic-classical/ [2]: https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms [3]: https://en.wikipedia.org/wiki/Existential_generalization https://en.wikipedia.org/wiki/Existential_generalization