Lebesgue Measure
We have seen that deknife can break up a unit interval into infinitesimals by cutting it aleph-null number of times. But what if, if we want to cut even more, say aleph-one number of times. The result is, for lack of a better name, what we could call figments. But look at the problem we have created. If we ask AC to go and collect an infinitesimal, it can look for the corresponding binary sequence and fetch it for us. But what does it do, if it is asked to fetch a figment. IST flatly says that AC cannot do it. Classical set theory says it can. Because, IST puts this restraint on AC, there are no sets in IST which are not Lebesgue measurable. See the frame below.