Timeline for Is this Variation of the Continuum Hypothesis Inconsistent with ZFC or ZF?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2017 at 21:38 | comment | added | Keith Millar | I mainly mentioned NCH because it seems like this is a "variant" of NCH (where cardinal exponentiation speeds up) | |
| Oct 15, 2017 at 11:12 | comment | added | Morteza Azad | Just as a reminder, I would like to mention that despite its tempting appearance and intuitive background from natural numbers, the Natural Continuum Hypothesis (NCH) is inconsistent with ZFC due to Konig's lemma. However, it sounds interesting to me that you considered the cofinality version of CH independently. | |
| Oct 14, 2017 at 18:20 | comment | added | Yair Hayut | Under CFCH, $2^{\aleph_n} = \aleph_{\omega_{n+1}}$ and therefore $2^{\aleph_{\omega}} = \aleph_{\omega_\omega}^\omega = \aleph_{\omega_{\omega + 1}}$. By the theory of pcf, we get that if $a$ is the set of all cardinals between $\aleph_{\omega}$ and $\aleph_{\omega_\omega}$, the possible cofinalities of $\prod a$ is the regular cardinals in the range between $\aleph_{\omega_\omega + 1}$ and $\aleph_{\omega_{\omega + 1}}$ which has cardinality $|a|^+$. A recent paper of Gitik claims that it is consistent, but it is open if one can obtain this type of pcf structure in accessible cardinals. | |
| Oct 14, 2017 at 15:44 | comment | added | Keith Millar | Ah, so ZFC + CFCH is even stronger. | |
| Oct 14, 2017 at 11:48 | comment | added | Goldstern | Easton's theorem implies that ZFC + "for all regular $\kappa$, $2^\kappa = \aleph_{\kappa^+}$" is equiconsistent with ZFC. Other variants, such as "the $\kappa^+$-th fixpoint" or $\aleph_{\aleph_{\kappa^+}}$ are possible, too. - But it is well known that ZFC+ non-SCH is much stronger than ZFC, in terms of consistency strength. | |
| Oct 14, 2017 at 6:01 | history | edited | Keith Millar | CC BY-SA 3.0 |
Fixed grammar
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| Oct 14, 2017 at 5:36 | history | edited | Keith Millar | CC BY-SA 3.0 |
Edited with fixed definition of SCH
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| Oct 14, 2017 at 4:33 | review | First posts | |||
| Oct 14, 2017 at 5:26 | |||||
| Oct 14, 2017 at 4:31 | history | asked | Keith Millar | CC BY-SA 3.0 |