Timeline for Arguments against large cardinals
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| Nov 26, 2015 at 22:27 | comment | added | Joseph Van Name | Todd Trimble. Let me also mention that this obstruction to tameness at the first measurable cardinal also holds in topology for similar reasons. For example, every complete uniform space is realcompact if and only if there is no measurable cardinal. In particular, a discrete space is realcompact if and only if its cardinality is below the first measurable cardinal. Every extremally disconnected $P$-space is discrete if and only if there are no measurable cardinals. Also, every topological group is the fundamental group of a compact space if and only if there is no measurable cardinal. | |
| Jul 24, 2013 at 21:40 | comment | added | Todd Trimble | I somewhat disagree with "Assuming that they don't exist, on the other hand, doesn't seem to lead to very interesting results." For example, the inexistence of measurable cardinals entails the truth of certain duality principles, several examples being given by Lawvere here: facultypages.ecc.edu/alsani/ct99-00%288-12%29/msg00128.html . Another example by Andreas Blass is that the identity functor on $Set$ is the unique exact functor, provided there are no measurable cardinals. In these ways, measurable cardinals could be viewed as obstructions to "tameness". | |
| Oct 29, 2010 at 14:57 | history | answered | Timothy Chow | CC BY-SA 2.5 |