  A constructive version of Hausdorff dimension is developed using constructive
supergales, which are betting strategies that generalize the constructive
supermartingales used in the theory of individual random sequences. This
constructive dimension is used to assign every individual (infinite, binary)
sequence S a dimension, which is a real number dim(S) in the interval [0,1].
Sequences that are random (in the sense of Martin-Lof) have dimension 1, while
sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is
shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is
a \Delta^0_2 sequence S such that \dim(S) = \alpha.
  A discrete version of constructive dimension is also developed using
termgales, which are supergale-like functions that bet on the terminations of
(finite, binary) strings as well as on their successive bits. This discrete
dimension is used to assign each individual string w a dimension, which is a
nonnegative real number dim(w). The dimension of a sequence is shown to be the
limit infimum of the dimensions of its prefixes.
  The Kolmogorov complexity of a string is proven to be the product of its
length and its dimension. This gives a new characterization of algorithmic
information and a new proof of Mayordomo's recent theorem stating that the
dimension of a sequence is the limit infimum of the average Kolmogorov
complexity of its first n bits.
  Every sequence that is random relative to any computable sequence of
coin-toss biases that converge to a real number \beta in (0,1) is shown to have
dimension \H(\beta), the binary entropy of \beta.
