Pairing Sets
Today we continue building our mathematical universe, one set at a time. But first, a clarification is in order. Last time, I introduced two axioms that we expect to hold in our universe and I said that based solely on them, we can't prove that any other set exists except the empty one. What I meant by this is that the universe of sets which has in it only the empty set does satisfy these two axioms. Of course, we can't possibly represent all of Mathematics in such universe, so we must exclude it from consideration. We do so by introducing new axioms, which then restrict the universes which are acceptable for our work. So, you should think of what we are currently doing is trying to zero in on the "right universe". I will just put it out there as a teaser for future posts that the questions of meaning of this "right universe" have caused whole bunch of debates, both among mathematicians and philosophers. But now, back to our work. I'll just go ahead and ...