Look at the literature on Zeno machines / hypercomputation

The phrase "Zeno machines" let me to the literature I wanted; thanks, Jeff! Incidentally, apropos of Andy Latto's suggestion about ordinal time: in the article "Rotor Walks and Markov Chains" (arXiv:0904.4507), which is related to my work on infinite decimals, Ander Holroyd and I explicit use countable ordinals to index the steps of a computation (though curiously only the "linear ordinals" of the form a omega + b turn out to be relevant). In a sense, Zeno machines are just one way of embedding ordinals into R. Jim Propp