I think that it is not necessary for the infinite number of numbers to exist in my mind. All I need to have in my mind is S(n) = n+1. — Ludwig V

This is Aristotle's finitism. Finitism is like this: if we put you in a spaceship that has an odometer, you will never see any but a finite number on it. You'll never see an infinity symbol, though you never stop moving forever and ever.

Per Aristotle, infinity exists in potential. The actual is always finite. Set theory, by handling infinity as a set, appears to be defying finitism. This is an unresolved issue in phil of math. Someday it may result in a shift in thinking about set theory.