It may be better to start with 0 as the pre-eminent number rather than 1 and leave 1 undefined except to the extent the successor function needed it for incrementing. This would allow one to conclude that these numbers exist as a set within logic. The successor function would define these numbers existing as members of the set. As members of a set they are "distinct objects that make up that set" (Wikipedia) and hence exist independently of each other.
However, what the OP appears to desire is not to claim that any of these numbers exist outside of perhaps a pre-eminent number (0 or 1). This may be possible. Wittgenstein objected to the existence of these numbers, including the pre-eminent numbers. He would provide an example of how this might be done.
G. E. M. Anscombe describes Wittgenstein's position in comparison to Frege and Russell as follows: (page 126)
For Frege and Russell (natural) number was not a formal concept, but a genuine concept that applied to some but not all objects (Frege) or to some but not all classes of classes (Russell); those objects, or classes, to which the concept number applied were picked out from others of their logical type as being 0 and the successors of 0.
So it is not necessary to consider numbers as genuine concepts, that is, as something more than a formal concept in logic.
If one takes an approach like Wittgenstein's one may be able to avoid the need for these numbers existing except as pointing to "which term it is, which performance of the generating operation the term results from" (page 126).
For more detail on how Wittgenstein viewed numbers through their use as the exponents in any formal series, see pmfcolling's question: What does Wittgenstein mean when he says "there are no numbers in logic"?, the answers provided and Wittgenstein's Tractatus Logico-Philosophus 6.01 and following.
Anscombe, G. E. M. An Introduction to Wittgenstein's Tratatus. 1971. St. Augustine's Press.
Wikipedia contributors. (2019, April 19). Element (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 14:37, May 8, 2019, from https://en.wikipedia.org/w/index.php?title=Element_(mathematics)&oldid=893194907