Timeline for Creating tag "small-uncountable-cardinals"
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2020 at 12:56 | history | edited | YCor | CC BY-SA 4.0 |
updated (acknowledgement that the tag I criticized is useful)
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| Mar 13, 2019 at 22:42 | comment | added | YCor | @TarasBanakh I created a usage guidance to the new tag, it's great if you have a look. | |
| Mar 13, 2019 at 11:34 | comment | added | YCor | @TarasBanakh great, I agree with you're last comment (on "small") — btw sorry readers for the double superposed discussion :) | |
| Mar 13, 2019 at 11:34 | comment | added | Taras Banakh | Very good. It is also correspond to my understanding what it is. | |
| Mar 13, 2019 at 11:33 | comment | added | YCor | @TarasBanakh Great, I've created a short usage guidance for infinite-combinatorics, which will hopefully be improved by better acquainted users. | |
| Mar 13, 2019 at 11:31 | comment | added | Taras Banakh | @YCore Anyway "small" relates to cardinalities of objects appearing in Classical Matehmatics: not very far from the continuum. If you try to label each natural mathematical object with its complexity in the Comulative Hierarchy of the Universum, you will discover that the most difficult objects of Functional Analysis and Differential Equations (like functions spaces of spaces of operators) have complexity not more $\omega+13$ (maybe $\omega+15$). Which means that the cardinality of such natural objects is at most $\beth_{15}$. This can be informally considered as the realm of small cardinals. | |
| Mar 13, 2019 at 11:26 | comment | added | Taras Banakh | "infinite-combinatorics" descibes rather methods and results involving combiknatorial properties of infinite sets. It is situated somewhere in the union of Set Theory and Combinatorics. Eventually this tag also can be created to point at the essense of a problem (which concerns combinatorial properties of infinite sets). | |
| Mar 13, 2019 at 11:22 | comment | added | YCor | @TarasBanakh I meant "strongly inaccessible" in my post (this is what I learnt as "inaccessible"). But the interval $[...,c]$ is a bad interval for defining a tag, since $\aleph_2$ or $2^{\aleph_1}$ would be small in some models and not small in others. Also I don't think the categories in $[c,q[$, for $q$ the smallest strongly inaccessible (if it exists) deserves to be considered separately. I don't think there a common accepted definition of "small" and I think it's better this way. | |
| Mar 13, 2019 at 11:21 | comment | added | Taras Banakh | So, up to my understanding of the realm of uncountable cardinals it can be divided into three categories: "small" (those in the interval $[\omega_1,\mathfrak c]$, "intermediate" (larger than $\mathfrak c$ but existing in ZFC, and finally "large" -- which do not necessarily exist in ZFC. | |
| Mar 13, 2019 at 11:19 | comment | added | Taras Banakh | Thank you for supporting the idea of introducing the tag "small-uncountable cardinals". Concerning the boundary between small and large cardinals is very precise: the existence of small cardinals follows from ZFC whereas the existence of large cardinals (starting with strong inaccessible) is a stronger assumption than the consistency of ZFC. More precisely this is a boundary between large cardinals and non-large. The boundary between small and non-small is $\mathfrak c$, the cardinality of continuum. | |
| Mar 13, 2019 at 10:48 | history | answered | YCor | CC BY-SA 4.0 |