The foundations of balance sheet and the inequality of the basic accounting equation from the viewpoint of set theory

The basics of financial accounting in the balance sheet and the basic accounting equation are revisited from the viewpoint of axiomatic set theory and predicate logic. The conceptual distinction between assets and claims on the assets are pointed out to allow for the application of set theory; next, it follows an application of the axioms of the theory. This application leads to obtain two sets of monetary units, which represent assets and claims (Liabilities and Equity) on the assets. These sets are properly built, according to the applications of the axioms, and comprise all the grouped items, still with financial meaning, of the lower levels in the balance sheet. An equality relation is applied to the classified balance sheet, according to these sets, to determine the equality of assets to the union of liabilities and equity; the application resulted in an inequality. This inequality is interpreted with the restriction of the application of the set theory. The special case where the basic accounting equation holds is described; however, this case has no financial meaning.