21. Transfinite induction can be used not only to prove things, but also to define them.22. He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction. 23. I used to fish out metal slides from Transfinite 18 : 55, 24 January 2007 ( UTC) 24. This group of definitions is dyadic fractions; a wider universe is reachable given some form of transfinite induction. 25. Zermelo proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930. 26. In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. 27. All primary source information ( including photos ) was just edited out of the Sweetest Day page by Transfinite . 28. The only reason I can see to prefer it is if you're not comfortable with transfinite induction. 29. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of " infinity ". 30. The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers.
