>But what do you mean by "ancient"?

>As far as I recall, treatments of infinity date from Cantor.

NO AT ALL Dear Peter. Obviously You the layman in the history of mathematics

Arithmetica (Greek: Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD.[1] It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Equations in the book are called Diophantine equations. The method for solving these equations is known as Diophantine analysis.

https://en.wikipedia.org/wiki/Diophantus

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.

https://en.wikipedia.org/wiki/Euclid%27s_theorem

>But then set theory as a foundation for mathematics is only one of a number of possibilities. Category theory, topos theory, ... there are plenty of alternatives.

Dear Peter. Obviously You the layman in category theory 

The notions of natural number in ZFC and in category theory are equivalent

Category Theory Lecture Notes

Daniele Turi

http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf

Jaykov's question is not very precise, so a precise answer is not possible. Proof by contradiction (which dates back to Zeno of Elea) is an important tool for showing a set is infinite: e.g. proving the reals are uncountable using Cantor's diagonalization argument, or the primes are infinite as per Euclid. Cantor's ideas were revolutionary and were not accepted quickly by the mathematical community. Even today some authors of popular articles tacitly assume all infinite sets are countable. A notable (living) mathematician (Zeilberg) questions the ontology of Cantor and infinite sets, in http://www.math.rutgers.edu/~zeilberg/Opinion108.html . The mathematical establishment, in general, eschews such philosophical matters.

http://www.math.rutgers.edu/~zeilberg/Opinion108.html

>Jaykov's question is not very precise

It meant  axiom of infinity of ZFC.

The small part of mathematical community accept only Heyting arithmetic.Heyting arithmetic adopts the axioms of Peano arithmetic (PA), but uses intuitionistic logic as its rules of inference

In my joint work with A. Piękosz "Quasi-metrizability of bornological universes in ZF", Journal of Convex Analysis 22 (2015), you can find three definitions of countable sets, not equivalent in some models for ZF. One of them, which I like best, does not refer directly  to numbers. Namely, a set X is D-countable iff each D- infinite subset of X is equipollent with X. D-infinite is for Dedekind infinite sets. Every D-infinite and simultaneously D-countable set is equipollent with the set of all D- finite ordinal numbers of  von Neumann in ZF. I use such definitions for my understanding physics.  Regards, Eliza Wajch

@ Peter Breuer

Peter Breuer:"Yes of course the notions of natural numbers in category theory and ZF are equivalent. That's the point! CT and ZF are alternative foundations for mathematics. They both can (and cannot, in that they can model anything the other can or cannot) give the same theories for the natural numbers.  They both contain models of the other. What is your point?"

On my point CT and ZF both are not appropriate foundations for mathematics. 

Any appropriate foundations for mathematics should be completely protected from crash.

 What is your point?

 @Eliza Wajch

 Eliza Wajch: "I use such definitions for my understanding physics. "

For contemporary physics also useful Colombeau's infinite numbers. Maybe you know. 

@ Peter Breuer:"Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20)."

Ok. But it meant at least potential infinity.

My opinion is that the subject of infinite sets is closed in the mathematics area. 

We present a new look to infinity: New philosophical view merging infinity to the imaginary number. There are warnings in the Kabbalah, the Jewish book of mysticism, about peering into this aspect of mathematics.

Article New philosophical view merging infinity to the imaginary number