Timeline for (Non?)-linearity of the consistency strength ordering in ZF
Current License: CC BY-SA 2.5
6 events
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| Oct 3 at 11:34 | comment | added | plm | Actually, looking at the presentation by JD Hamkins cited in another answer here, jdh.hamkins.org/wp-content/uploads/… one can find on the penultimate page an illuminating explanation for why large cardinal axioms are linearly ordered by consistency strength. He explains that to produce examples of incomparable statements one needs to modify arithmetic truth, but usual large cardinals are studied by methods that preserve arithmetic truth, so it is researchers' preferences that force the apparent linear ordering. | |
| Oct 3 at 9:48 | comment | added | plm | Is there a formal definition of "large cardinal" ? In any case one can come up with many intuitive explanations of why they would be linearly ordered. The most compelling to me is that cardinal properties that are incomparable to the "trunk" are rare and disconnected from other maths: one must use a purpose-built algorithm to produce any. This would be a phenomenon remotely similar to that of natural proofs in complexity theory: "natural" properties of boolean functions, which are the ones researchers easily come upon because they are common, thus intuitive, are useless to prove lower bounds. | |
| Mar 30, 2011 at 19:39 | comment | added | Alex Lupsasca | You are entirely right; I stand corrected. | |
| Mar 30, 2011 at 19:39 | history | edited | Alex Lupsasca | CC BY-SA 2.5 |
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| Mar 28, 2011 at 1:00 | comment | added | Ed Dean | Terminological note: "Orey sentence" isn't another name for a double jump sentence. It refers instead to when there is no jump: $T + \sigma \equiv T + \neg\sigma \equiv T$. | |
| Mar 28, 2011 at 0:32 | history | answered | Alex Lupsasca | CC BY-SA 2.5 |