Infinite Sets…I see it, and now, I believe it.
I have been exploring the idea that infinite sets might not exist. This is suggested by Prof. N. Wildberger found in his youtube sermons here. It is interesting that the preaching ties the existence of an infinite set to our ability to write it down. Is this some kind of extreme construtivist teaching or something?
Indeed Georg Cantor must have been considered a heretic when he preached on the idea of infinite sets in the late 19th century. Today, for sure it is part of math’s orthodox doctrine.
It so happened that I am studying model theory too and as I reflected on this, I asked myself what model theory might say about this issue.
Getting out my copy of D. Marker’s Model Theory: An Introduction I accidentally stumbled on his Example 1.2.1 (Infinite Sets) and it goes like this…
Consider the language , that means no functions, relations nor constants. Consider now the theory of where we have, for each , the sentence . This says that there are at least distinct elements. Then a structure is a model for this theory iff the carrier set/universe of , called is infinite.
This is not obvious and the example showed no proof. However, I will try to show why must be infinite.
We only prove in this direction. For a contradiction, assume there is a model but is finite.
being finite implies it is of finite cardinality , i.e., . We know that
, there such that . For example, . I think this step is often called “without loss to generality”;-)
Then we can always set . Consider now the definition of satisfiability, a model of . By definition of satisfaction this implies . This says that , which says there is an th element in . This is a contradiction, for we said that the number of elements of is . This implies that can not be satisfied by the structure , thus it is not a model, a contradiction.
must be infinite.
The proof for () follows a similar line.