We were having an electronic meeting amongst professors when somebody mentioned successors. One professor challenged the other, since he believed that successors should not be the x +1s only, like we have, for instance, the number 0.41, and its successor should be 0.42 instead or something along these lines.
This is an interesting matter, since the sigmatoid successor, as far as I know. has been coined by Peano, and therefore involves EXCLUSIVELY the natural numbers. See: Successor
At the same time, the fellow would be right, and we should be able to represent the successor of any number.
When we think about it, we see that the successor of 0.41 does not exist, like we know that there is one, but we are unable to tell what it is, since any number we choose, so say 0.42, will be a mistake (we can take the average between both and come up with a new candidate to successor).
When we think about it, we see that the successor of 0.41 does not exist, like we know that there is one, but we are unable to tell what it is, since any number we choose, so say 0.42, will be a mistake (we can take the average between both and come up with a new candidate to successor).
In this way, the other professor, who, in this case, was Prof. Corcoran, would be right: x+1 = s(x).
We actually do not have a special term for the number that comes after real or rational numbers because of what I have just said. If we talk about the integers, however, we can easily use the same name and symbols that Dedekind has used. We should however, more than likely, explain that we are extending the meaning of the sigmatoid, since we must tie all that we can tie in Mathematics.
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