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    $\begingroup$ As a historical footnote, I'd like to mention that arguments for incompleteness, in the form "if an inaccessible $\kappa$ exists then its existence is not provable" (because $V_\kappa$ satisfies "there is no inaccessible") actually preceded Gödel's theorems. Kuratowski gave such an argument in 1925 and Zermelo in 1928, though neither was rigorous by today's standards. $\endgroup$ Commented Jun 24, 2010 at 3:26
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    $\begingroup$ This is bound to be stupid, but when $M_\alpha$ is a graph, what does $M_\alpha \models \varphi(\vec{x})$ mean? Also, what is $\phi$, and what are its source and target (if that even makes any sense)? $\endgroup$ Commented Jun 24, 2010 at 5:19
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    $\begingroup$ Joel, your remark about graphs instead of arbitrary structures is nice for non-logicians. An alternative way to be nice to them is to replace "elementary embedding" with mere "embedding". (It makes no real difference, since you can Skolemize the structures.) But I don't see that both sorts of niceness can be done simultaneously. Does anyone know an equivalent formulation of Vopenka's principle that uses mere embeddings but for a "simple sort of structure (simpler than a graph plus all its Skolem functions)? $\endgroup$ Commented Jun 24, 2010 at 6:49
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    $\begingroup$ Harry, when $G$ is a graph, then $G\models\varphi$ simply refers to first-order satisfaction for assertions $\varphi$ in the language of graph theory, where you can quantify over vertices and use the edge-relation. You can express things like "G is triangle-free" and "G has total diameter 5" this way, but other assertions, such as "G is connected," are not first order expressible. Andreas, that is a very interesting point. $\endgroup$ Commented Jun 24, 2010 at 12:08
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    $\begingroup$ I think I shall delete those words, since I surely don't mean to offend anyone. I just meant to express my frustration with the justifications that are sometimes provided for large cardinals on the basis of reflection ideas, in those instances when to the contrary they are attempts to justify our confidence in the axioms. It is, of course, the reflection principles themselves, suitably formalized, that carry a large cardinal interest. $\endgroup$ Commented Jun 25, 2010 at 1:31