Talk:Reinhardt cardinal

consistency
This link says that Reinhardt cardinals have been proven inconsistent: http://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent/41030#41030 Tom.Q.Ellis (talk) 19:24, 24 April 2011 (UTC)


 * Kunen proved they are inconsistent with ZFC. They may or may not be consistent with just ZF; that's an open problem. &mdash; Carl (CBM · talk) 19:35, 24 April 2011 (UTC)

V
What is V? It's not defined in the article. Crasshopper (talk) 22:28, 11 February 2013 (UTC)


 * The von Neumann universe. See http://en.wikipedia.org/wiki/Cumulative_hierarchy. 86.185.216.197 (talk) 12:39, 28 March 2014 (UTC)

Quantifying over formulas of ZFC
Contrary to what the article says, ZFC can quantify over all formulas of ZFC. Even familiar relatively weak fragments of arithmetic can quantify over formulas of any first order language and prove useful things about them. This is how Godel's incompleteness theorems are proved. What ZFC cannot do is formulate a satisfactory truth predicate for all sentences of ZFC. More precisely ZFC can define the Godel numbers of strings of symbols and can quantify over them and it proves existence of a set of all Godel numbers of formulas. ZFC can even give correct truth definitions for certain limited classes of formulas, and for each such class it proves existence of the set of all true sentences of that class. But no formula of ZFC can correctly define truth for all sentences of ZFC. So ZFC cannot state or prove existence of a set of all true sentences of ZFC. A refinement of that fact leads to the subtleties about defining Reinhardt cardinals in ZFC. Colin McLarty (talk) 13:19, 22 October 2014 (UTC)

Rupert McCallum has withdrawn his claim that the existence of a Reinhardt cardinal is inconsistent with ZF
On January 24, 2018, Rupert McCallum withdrew his claim that Reinhardt cardinals are inconsistent with ZF. See: Discussion of McCallum’s paper on Reinhardt cardinals in ZF. --RJGray (talk) 02:26, 22 February 2018 (UTC)