Timeline for Independence and Category Theory
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Oct 18, 2010 at 14:39 | comment | added | Todd Trimble | Harry, I appreciate your clarification. Reading again the question, I agree it's possible OP didn't ask what he meant to ask. But I imagine that all his (I'll assume "his") worries will be soothed by attending closely to the theory-model distinction and to what precise statements are provable in the theory. (If enough people find different ways of putting this, that might tip the balance. Andrej put it very well and very forcefully, I agree.) | |
| Oct 18, 2010 at 14:14 | comment | added | Harry Gindi | @Todd: Yes, I was clarifying what the OP meant to ask (since I was arguing with him on IRC, and his question here was somewhat unclear). Everything you've said is virtuous, fair, and true. | |
| Oct 18, 2010 at 14:06 | comment | added | Todd Trimble | Well, I was focusing on OP's question about existence of limits and colimits, and the question of whether any model is complete. As Andrej said, "Because ZFC proves that all small diagrams have limits", and this was my answer too. I think we are all in agreement here. | |
| Oct 18, 2010 at 13:16 | comment | added | Harry Gindi | The point that the OP seems to be missing is that given any small category $X$ and any functor $X\to Set$, we can give a formula for the colimit (i.e. a set satisfying the universal property), which exists as a ZFC-set (by inspection). | |
| Oct 18, 2010 at 13:12 | comment | added | Harry Gindi | So far, that's all well and good. However, for some reason, the OP seems to think that this is a problem with category theory. We can't pin down the exact set $colim X$ without further axioms of ZFC. Category theory says that this set exists in the ordinary category of ZFC sets and further that we can characterize it up to unique isomorphism by a universal property. | |
| Oct 18, 2010 at 13:07 | comment | added | Harry Gindi | @Todd: The question that the OP is asking is how can we define the colimit over a category when the content of the category is independent of ZFC. I'll strip down his objection to a very barebones case: Let $X$ denote a set consisting of one representative per isomorphism class of whitehead groups of cardinality less than $\gamma$, and consider this as a discrete subcategory (not full, obviously) of the category of sets (denote the inclusion also by $X$). Consider the proposition: colim X is empty. Then this proposition is independent of ZFC. | |
| Oct 18, 2010 at 12:31 | history | edited | Todd Trimble | CC BY-SA 2.5 |
Replaced the word "which" by "will"
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| Oct 18, 2010 at 12:25 | history | answered | Todd Trimble | CC BY-SA 2.5 |