As someone leaning towards Intuitionism myself, I believe in processes with operations that can be repeatedly applied any number of times without breaking themselves apart; such as applying the "successor" operator to "0", to "successor(0)"... to build any particular natural number, given enough time and memory.
I don't believe in the existence of a set containing all natural numbers at the same time, except as an idea to talk about.
https://en.m.wikipedia.org/wiki/Intuitionism