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The Hardy hierarchy assigns increasingly fast-growing functions to ordinals, once fundamental sequences for limit ordinals have been chosen. A common convention isfor nonzero limits $\lambda$, where $\lambda [n]$ increases toward $\lambda$. The ordinal notation system and its fundamental sequences are part of the definition.

For finite $k$, this gives $H_k(n)=n+k$. Taking $\omega [n]=n$ gives $H_{\omega }(n)=2n$. At larger ordinals the recursive nesting produces much faster growth. The hierarchy compares the strength of termination arguments and induction principles in proof theory; changing conventions can shift the exact formulas at particular indices.

Mathematical logic studies formal languages, formal proofs, model theory, computability theory, and the foundations of mathematics. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of mathematics. Proof theory studies deductions; model theory studies interpretations in which statements hold.

Three questions organize the subject: is an expression a well-formed first-order formula, is it true in a specified first-order structure, and does it have a formal proof from specified axioms? These questions are distinct. For example, $\exists n(n+n=1)$ is a first-order sentence in an appropriate first-order language; it is false in the natural numbers but true in the rational numbers, with $n=1/2$.

The beauty of mathematical logic is that the same logical connectives and rules of universal quantification and existential quantification organize arguments across mathematics. Classical propositional logic uses two truth values; Intuitionistic propositional logic interprets logical implication through constructions on proofs, without unrestricted law of excluded middle. This proof-based framework is developed in Frank Pfenning, Natural Deduction, sections 1–3.