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:

Thank you.

Printable View

- Feb 22nd 2009, 05:41 AMgeorgelSet theory, transitive sets
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:

Thank you. - Feb 22nd 2009, 06:30 AMclic-clac
Hi

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. - Feb 22nd 2009, 07:10 AMThePerfectHacker
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.

- Feb 23rd 2009, 06:29 AMgeorgel
Thanks to both of you, I will probably have another question or two in the next couple of days.. :)