Mathematical Philosophy a Study of Fate And Freedom Lectures for Educated Laym

Cover Mathematical Philosophy a Study of Fate And Freedom Lectures for Educated Laym
Mathematical Philosophy a Study of Fate And Freedom Lectures for Educated Laym
Cassius Jackson Keyser
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Consider the numbers in the row !> 3> i> • • • (&d infinitum), t, 0.
Let S' be a sequence having the class of thes° numbers for its field and let S' be such that, if a is an S' predecessor of b, then a and b are in the row, a on the left of b, and that any number in the row is an S' predecessor of all the numbers on its right and an S' successor of all the numbers on its left. S', as you see, is now completely determined — MORE ABOUT LIMITS 275 we know all the couples constituting it; and you n
...ote that its field F is the same as that of S above. Let V be the same as before and let k be the same as in the above paragraph, in which was found to be an S £-limit of V . The question is: is an S r &-limit of V\ The answer is no, as you readily see; for choose a ^-neighborhood d of 0, where d is, say, \ ; any F term /' differing by more than null (zero) from 0, if it be in the chosen neighborhood, is, as you see, a predecessor of |; and so y> though it is be- tween such t' and 0, is not in the chosen neighborhood; accordingly, as said, is not an S' &-limit of V .

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