I am not sure into which rubric to put this, but since there is some Model Theory here, I am putting it in this one.
First a technical question: the TEX button here doesn't give me the usual choices of what kind of symbol to use, only saying "wrap [tex] tags around selected text." Where do I differentiate between, say, subscript, superscript, Greek letters, quantifiers, and so forth?
So, until I learn that, I ask for patience with my clumsy notation.
First, I define the Cantor set informally:
A(0) = [0,1]
A(n+1) = the set of closed intervals obtained by taking out the open middle third of each interval contained in A(n), for natural numbers n.
The Cantor set = the points not removed at any step.
It is tempting to put
The Cantor set = the intersection of all A(n), but this would cause problems, as follows:
I concentrate on p = (the number corresponding to) the right end-point of the left-most interval of the Cantor set. If we stayed in the real numbers, we would have p=0, which is not what we want. So the intersection definition is not adequate. (Even if it were adequate, the following reasoning would still hold. Just covering my bases.)
Rather, p fulfills the following description:
for all natural numbers n, 0<p<1/3^n.
By the Archimedean property of the real numbers, p is not a real number, but by the Compactness Theorem, we assume the existence of a model which includes both real numbers and p, and all such infinitesimals. This new model can contain all the points of the Cantor set.
So far my reasoning. However, everywhere I look, the Cantor set is considered a subset of the real numbers. What is wrong here?