Issues with Ordinals
Working in set theory, and I'm having issues with a couple of problems.
1) Given w is omega, the smallest ordinal, prove that w is not isomorphic to w + 1.
What I've done, is shown that since w is in w + 1, then w + 1 cannot be isomorphic to it, for it would be isomorphic to an initial segment of w + 1, a contradiction.
2) There exists 2Aleph naught well orderings of all natural numbers
No clue here where to start.
3) Show that the lexicographic product (N x N, <) is isomorphic to w * w. (Define the order on N x N by (n1, n2) < (m1, m2) iff n1 < m1 or (n1 = m1 and n2 < m2))
Again, not sure where to start.
Any help would be appreciated.
1) Given w is omega, the smallest ordinal, prove that w is not isomorphic to w + 1.
What I've done, is shown that since w is in w + 1, then w + 1 cannot be isomorphic to it, for it would be isomorphic to an initial segment of w + 1, a contradiction.
2) There exists 2Aleph naught well orderings of all natural numbers
No clue here where to start.
3) Show that the lexicographic product (N x N, <) is isomorphic to w * w. (Define the order on N x N by (n1, n2) < (m1, m2) iff n1 < m1 or (n1 = m1 and n2 < m2))
Again, not sure where to start.
Any help would be appreciated.