  This is a study of S. Kripke's notion of fulfilment. Motivated by
Paris-Harrington statement, Kripke was looking for a proof of G\"odel's
Incompleteness Theorem which was model-theoretic, natural (without
self-reference), and easy. Fulfilment gives a versatile tool for both Proof and
Model Theory.
  We begin with short proofs to a number of classical results. With two new
results: there is an easily definable subring $R$ of the primitive recursive
functions such that for any non-principal ultrafilter $D$ on $\omega$, $R/D$ is
a recursively saturated model of Peano arithmetic; and for any r.e. theory
$\textsf{T}$ and for any given r.e. set, we can feasibly find a $\Sigma_1^0$
formula which semi-represents it in $\textsf{T}$.
  We then give a version of Herbrand's Theorem, and of the Hilbert-Ackermann
method of proving consistency, answering a problem of D.
Guaspari:\[\{\ulcorner\phi\urcorner\in\Pi^0_k:\phi\text{ is
}\Sigma^0_k\text{-conservative over $\textsf{PA}$}\}\]is a complete $\Pi^0_2$
set.
  We extend H. Friedman's method for results of such as
$\Sigma^1_2\textsf{-AC}$ is $\Pi^1_3$-conservative over
$(\Pi^1_1\textsf{-CA})_{<\varepsilon_0}\upharpoonright$, along with uniform
versions\[\forall\alpha<\varepsilon_0\;(\Pi^1_1\textsf{-CA})_{\alpha}\upharpoonright\,\vdash\textsf{RFN}_{\Pi^1_3}({\Sigma^1_2}\textsf{-AC})\]
Then there are some model-theoretic applications, starting with
non-$\omega$-models. We extend the theorem of D. Scott involving Weak K\"onig's
Lemma, and describe the order types of elementary initial segments of
recursively saturated models. For $\omega$-models, we extend Friedman's theorem
on minimal models of analysis, and develop indicators for countable fragments
of $\mathcal L_{\infty\omega}$, with some representability results in
$\omega$-logic.
  We close with an exposition of the Paris-Harrington statement.
