Could someone please prove, using the principle of mathematical induction, that the set consisting of all natural numbers is a transitive set?
That is, prove that:
Do you agree that, for your formula is true?
Now assume that it is true for a we have to show that it's also true for its successor,
Let be an element of i.e. In the first case, the induction hypothesis states that In the second case, and then
So, using the mathematical induction principle, we proved that is a transtive set.
Here is a non-induction way to prove this. I know you asked for induction but I bring this approach up because it might help you when you do more stuff on ordinals. If is a set of ordinals then is a set of ordinals. Since is the union of all natural numbers and since each natural number is an ordinal it follows that is an ordinal.