Timeline for Is each of the infinite statements of the Generalized Continuum Hypothesis independent?
Current License: CC BY-SA 4.0
13 events
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| Jun 11 at 15:02 | comment | added | Joel David Hamkins | If you are tempted to ask further questions about this topic, I would suggest that math.stackexchange may be a more suitable forum. | |
| Jun 11 at 14:10 | vote | accept | SarcasticSully | ||
| Jun 11 at 14:09 | answer | added | Joel David Hamkins | timeline score: 6 | |
| Jun 11 at 14:03 | history | edited | SarcasticSully | CC BY-SA 4.0 |
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| Jun 11 at 13:58 | history | edited | SarcasticSully | CC BY-SA 4.0 |
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| Jun 11 at 13:48 | comment | added | Joel David Hamkins | @SarcasticSully Regarding the principle in your edit, see mathoverflow.net/a/6594/1946. | |
| Jun 11 at 13:48 | history | edited | SarcasticSully | CC BY-SA 4.0 |
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| Jun 11 at 13:45 | comment | added | SarcasticSully | @C7X oh, yeah I'd missed infinite ordinals. To clarify, my claim wasn't that $k$CH implies $k+1$CH, it was the converse: that $k+1$CH should imply $k$CH. I got that from assuming not-$k$CH (that is, there does exist such a set), and proving that its power set must have a cardinality that contradicts $k+1$CH. That said, I did make an assumption listed in an edit that I'm not as sure about as I was. | |
| Jun 11 at 13:37 | history | edited | SarcasticSully | CC BY-SA 4.0 |
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| Jun 11 at 8:16 | comment | added | Alessandro Codenotti | As already mentioned, by Easton's theorem the aleph function can do pretty much anything at regular cardinals. There are some restrictions at singular cardinals, for example if GCH holds below a singular $\kappa$ with uncountable cofinality, then GCH must also hold at $\kappa$ | |
| Jun 11 at 6:21 | comment | added | C7X | The generalized continuum hypothesis is the even stronger statement that for all ordinals $\alpha$, there is no set $\mathbb S$ where $\beth_\alpha<\vert\mathbb S\vert<\beth_{\alpha+1}$. About the proof that $k$CH implies $k+1$CH, I claim the proof is not correct: why should $\beth_k<\vert\mathcal P(\mathbb S)\vert$ necessarily hold? I also claim that Easton's theorem shows that you can consistently have any desired pattern of $k$CH successes and failures. | |
| S Jun 11 at 5:51 | review | First questions | |||
| Jun 11 at 6:06 | |||||
| S Jun 11 at 5:51 | history | asked | SarcasticSully | CC BY-SA 4.0 |