Knowledge, arising as a structure over the elemental regime of information, itself becomes an elemental regime for further organization. Formalism arises as a structure over the elemental regime of expressed knowledge: it establishes explicit rules for storing, connecting, and transforming the traces through which knowledge is retained and becomes available. But its states, rules, and results of operations themselves exist in the mode of the trace. A formal system works with already available distinctions, and is thus fundamentally limited to the mode of the trace.
Within this mode, the system cannot guarantee the construction of a correct criterion of access to a distinction whose trace has not yet been part of it: such a criterion would itself already be a new trace of that distinction. To guarantee such a criterion, the system would already have to contain either that trace or a rule that unambiguously generates it. But the distinction would then have been provided for in advance. A guarantee of access to every new distinction would require the system to already contain all possible distinctions and criteria of access to them, excluding the very possibility of the new and thus the generative character of the world considered here.
A general intuition of incompleteness arises here: a fixed system of traces cannot contain in advance criteria of access to all possible new distinctions.
Extending the formalism, changing the rules, or allowing the system to reorganize its own structure does not remove this limitation, but merely shifts it one level higher: it is necessary to distinguish when and how reorganization should take place, while the existing system of traces cannot guarantee the adequacy of that reorganization to the new distinction.
Formalism can dynamically retain traces of distinctions, but the emergence of a new distinction means that the previous organization of the expression of knowledge requires continual revision.