 As we saw above, ordinals below \(\varepsilon_0\) suffice for the proof-theoretic treatment of PA. For stronger theories segments larger than \(\varepsilon_0\) have to be employed, requiring new normal forms akin to Cantor’s normal form. Ordinal representation systems utilized by proof theorists in the 1960s first emerged in a purely set-theoretic context more than 50 years earlier. In 1904 Hardy wanted to “construct” a subset of \(\bbR\) of size \(\aleph_{1}\), the first uncountable cardinal. His method was to represent countable ordinals via increasing sequences of natural numbers and then to correlate a decimal expansion with each such sequence. Hardy used two processes on sequences: (i) Removing the first element to represent the successor; (ii) Diagonalizing at limits. For example, if the sequence \(1,2,3,\ldots\) represents the ordinal 1, then \(2,3,4,\ldots\) represents the ordinal 2 and \(3,4,5,\ldots\) represents the ordinal 3 etc., while the ‘diagonal’ \(1,3,5,\ldots\) provides a representation of \(\omega\). In general, if \(\lambda=\lim_{n\in\bbN}\lambda_n\) is a limit ordinal with \(b_{n1},b_{n2},b_{n3},\ldots\) representing \(\lambda_n&lt;\lambda\), then \(b_{11},b_{22},b_{33},\ldots\) represents λ. This representation, however, depends on the sequence chosen with limit λ. A sequence \((\lambda_n)_{n\in\bbN}\) with \(\lambda_n&lt;\lambda\) and \(\lim_{n\in \bN}\lambda_n=\lambda\) is called a fundamental sequence for λ. Hardy’s two operations give explicit representations for all ordinals \(&lt;\omega^2\).