As much as I know (and please correct me if I am wrong) no fundamental mathematical systems like ZF or Peano axioms include an examination of our cognition's ability to count. In my opinion, if we ignore our abilities to count, then there is a reasonable chance that our fundamental axioms include hidden assumptions. For example, let's examine this situation: On the table there is a finite unknown quantity of identical beads > 1 and we have: A) To find their sum. B) To be able to identify each bead. Limitation: we are not allowed to use our memory after we count a bead. By trying to find the total quantity of the beads (representing the discreteness concept) without using our memory (representing the continuum concept) we find ourselves stuck in 1, so we need an association between continuum and discreteness if we want to be able to find the bead's sum. Let's cancel our limitation, so now we know how many beads we have, for example, value 3. Now we try to identify each bead, but they are identical, so we will identify each of them by its place on the table. But this is an unstable solution, because if someone takes the beads, put them between his hands, shakes them and put them back on the table, we have lost track of the beads identity. Each identical bead can be the bead that was identified by us before it was mixed with the other beads. We shall represent this situation by: ((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c)) By notating a bead as 'c' we get: ((a XOR b),(a XOR b),c) and by notating a bead as 'b' we get: (a,b,c) We satisfy condition B but through this process we define a universe, which exists between continuum and discreteness concepts, and can be systematically explored and be used to make Math. What I have found through this simple cognition's basic ability test is that ZF or Peano axioms "jump" straight to the "end of the story" where cardinal and ordinal properties are well-known, and because of this "jump" Infinitely many information forms that have infinitely many information clarity degrees, are simply ignored and not used as "first-order" information forms of Math language. In my opinion, any language (including Math) is first of all an information system, which means that fundamental properties like redundancy and uncertainty MUST be taken as first-order properties, but because our cognition's abilities were not examined when ZF or Peano axioms were defined, both redundancy and uncertainty were not included in our logical reasoning or in our fundamental axiomatic systems. Also in my opinion, through this simple test we get the insight that any mathematical concept is first of all the result of cognition/object (abstract or non-abstract) interactions. My paper http://us.share.geocities.com/complementarytheory/ONN.pdf [Broken] is a reexamination of some fundamental mthematical concepts that are based on this insight.