How can axiomatic foundations be laid out? How does one identify, when they form "foundations"?

This is a particular problem related to this:

Can social sciences have a non-subject -related axiomatic foundation? Why not?

Since if one wanted to discover such axiomatic foundation (for practical purposes and in order to have more consistency than without such foundations), then due to fundamental subjectivity (that thought, observation etc. are fundamentally subjective) it's impossible, since no subject can "reach" for anything general, there's only subjectivity.

But how can such foundations then be discovered? Or they cannot and should not even attempted except for "practical purposes"? But not for "consistent correctness"?

Case example:

Foundations of mathematics are those that are agreed upon as serving as foundations. However, in principle it's possible to select many other foundations, so there doesn't exist a single formulation. And thus foundations of mathematics are not really "generally consistent" (or are only in a particular system).

One take is:

Verificationist theory of meaning https://en.wikipedia.org/wiki/Verificationism

Is it sufficient? But does this apply to maths, since it requires "validation through empirical?" (or perhaps that comes from success in physics)?

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