Timeline for Characterization of transitive closure graphs
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2011 at 17:53 | vote | accept | Hans-Peter Stricker | ||
| May 5, 2011 at 14:52 | comment | added | Joel David Hamkins | But in any case, any axiomatization of a system capable to performing elementary coding or arithmetic (such as this) will be liable to the Incompleteness phenomenon, so we cannot write down a complete axiomatization of the theory of $V_\omega$ in any case. | |
| May 5, 2011 at 14:51 | comment | added | Joel David Hamkins | Well, you don't need all those axioms when you're doing finite set theory, and as you have observed, there are much simpler axiomatizations to get to the theory you want. For example, there are axiomatizations based on the concept of adjunction (adding a single element to a given set). I think people like to look at ZFC-fin still with most of the ZFC axioms, however, out of analogy with the models of full ZFC. | |
| May 5, 2011 at 14:10 | comment | added | Hans-Peter Stricker | What I wonder: How can a class of objects with so seemingly weak and simple properties like (1)-(3) provide a model for a theory which has about ten independent axioms, only two of which are directly related to (1)-(3)? | |
| May 5, 2011 at 13:14 | comment | added | Hans-Peter Stricker | I see X at the top, too, and the empty set at the bottom, but the arrows pointing upwards. "As you like it." | |
| May 5, 2011 at 13:00 | comment | added | Joel David Hamkins | So in my answer, you should exchange "children" and "parent" in the labeling procedure. | |
| May 5, 2011 at 12:58 | comment | added | Joel David Hamkins | Oh, I see. That part is fine then. I usually picture the graph with $X$ at the top, and each node would be the parent of its elements, which has a certain easy and coherent sense. In any case, the inductive labeling idea still works, and shows you how to turn the graph into a set. | |
| May 5, 2011 at 12:52 | comment | added | Hans-Peter Stricker | ad (2): Sorry for the confusion, but since I assume an arrow from x to y iff x is an element of y, x is a "parent" of y iff x is an element. | |
| May 5, 2011 at 12:51 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 63 characters in body; added 2 characters in body
|
| May 5, 2011 at 12:45 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |