Timeline for answer to Reasons to believe Vopenka's principle/huge cardinals are consistent by Joel David Hamkins
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 25, 2010 at 1:34 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
Struck out "hot air"
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| Jun 25, 2010 at 1:31 | comment | added | Joel David Hamkins | 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. | |
| Jun 24, 2010 at 22:14 | comment | added | Timothy Chow | Joel, what you call "hot air" may not be convincing arguments that large cardinal axioms are true, but they have value in pointing us towards the right axioms to consider. That's all that we can expect of them, and they fulfill that role quite admirably. So I don't see why you disparage them for failing to fulfill a function that they could not be expected to fill. | |
| Jun 24, 2010 at 12:08 | comment | added | Joel David Hamkins | 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. | |
| Jun 24, 2010 at 6:49 | comment | added | Andreas Blass | 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)? | |
| Jun 24, 2010 at 5:19 | comment | added | Harry Gindi | (that last $\phi$ should be a $\varphi$). | |
| Jun 24, 2010 at 5:19 | comment | added | Harry Gindi | 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)? | |
| Jun 24, 2010 at 4:36 | comment | added | Joel David Hamkins | It should still be elementary embedding, even for graphs. This just means that there is a map $\pi:M_\alpha\to M_\beta$ on the vertices, so that $M_\alpha\models\varphi(\vec x)$ if and only if $M_\beta\models\varphi(\pi(\vec x)$. So $pi$ is first-order truth-preserving on the graphs $M_\alpha$, $M_\beta$. | |
| Jun 24, 2010 at 4:30 | comment | added | Harry Gindi | Dear Joel, what is the analog of "elementary embedding" for graphs? | |
| Jun 24, 2010 at 3:26 | comment | added | John Stillwell | 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. | |
| Jun 24, 2010 at 3:09 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |