I begin by discussing several of the existing ways of proving the validity of transfinite induction up to ε₀ and argue that it is at least conceivable that there is room for a new proof that is more constructive than any of them. An attempt which I pay particular attention to is that made by Mariko Yasugi (1982). The centrepiece of her theory is the so-called "construction principle", a principle for defining computable functionals. I argue that, in principle, it ought to be possible to set up a theory whose terms denote or range over functionals of a sort constructed by a similar principle, in which the accessibility (a term to be defined below) of ε₀ is provable, yet which dispenses with quantifiers as well as with some strong axioms which she uses in order to achieve the same result. My theory, described in chapter 2, is called TF (for "term-forms"). In chapters 3, 4 and 5, a proof of the accessibility of ε₀ in TF is presented. This thesis ends (chapter 6) with a proof of the computability of the functionals that can be represented in TF.