Is it true, and is it provable that given the set of algebraic numbers, and a set of finite transcendental numbers, it is possible to form the set of real numbers using only a coutably infinite numbers of operations?

Similarly, can all real numbers be formed by the operation

Thanks. Here's another similar one that seems bound to be false, but is it possible to form all real numbers from a finite number of operations between all algrebraic numbers and a finite number of transcendental numbers? It seems as though the two sets would have different cardinality, but I can't prove this with any rigor.

Brad

You're right, they would have different cardinality. In fact, even if you allow a countably infinite number of operations but only a finite number of operations used in forming any particular number, then this will not include all the reals. The reason is that a countable union of countable sets is countable, and the reals are uncountable. Let A₀ ={the algebraic numbers and a countable number of transcendental numbers (note that this is slightly more than you started with)} and A_n+1 be the set formed by adding, multiplying or dividing by two numbers in A_n . Each A_n is clearly countable and the union of all the A_n gives you all the numbers which can be formed using a finite number of operations on algebraic and a countable number of transcendental numbers. However, it is a countable union (because it is indexed by natural numbers n) of countable sets. So it is a countable set.

By the way, now that I look at it, the first thing you posted was slightly misleading. In fact, doing a countable number of operations on what I called A₀ above (including infinite summations) would not form all reals, for the same reason as above (a countable number of operations on a countable set only gives a countable set). Even though you can form each real with a countable number of operations it is impossible to form all reals simultaneously with a countable number of operations.