In addition to what Jess Mazer asks, it would also be of interest to know just what is known or believed about the infinite sets of those objects which mathematics does deal with. I've read in E.T. Bell that the infinity of curves or functions is greater than the infinity of the reals. An information scientist at another forum told me that the infinity of the hyperreal numbers is larger than that of the reals, & that the infinity of the surreals is larger than that of the hyperreals. Is this true? And does the infinity of curves or functions (of the standard or "archimedean" numbers) correspond to either the infinity of the hyperreals or that of the surreals? Is it known whether one could possibly define a larger set of numbers than the surreals? One also hears that anything that can be done with nonstandard numbers can be done with standard numbers, as long as it doesn't pertain to the difference between.them. Is the difference between them still regarded as not leading to an!
ything of interest?
- Ben Udell
If you have a function on the integers that takes each integer and assigns it either a one or a zero, the total number of such functions is equal in cardinality to the reals--just think of writing each real in binary, with each successive digit labeled by the next integer. This set is also equal in cardinality to the power set (set of all subsets) of the integers--for every function that assigns each integer a one or a zero, you can make a subset that includes only the integers assigned a one by that function, and so there'll be a one-to-one relationship between the set of all subsets of the integers and the set of all these functions.
There's a general theorem in set theory that the power set of any set must have a higher cardinality than the set itself, based on a diagonalization argument. Therefore, by the same type of reasoning, it must be true that the set of all possible functions defined on the reals that assign each real a one or a zero must have a higher cardinality than the reals themselves. Similarly, the set of all meta-functions defined on the set of all the functions I just described must have a still higher cardinality, and so on. I don't know the answer to the questions about curves or hyperreals though. And of course, all this is assuming you "believe" in the hierarchy of infinite cardinalities, which as I said in my last post I feel a bit iffy on.
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