Timeline for The axiom of choice as a consequence of a stronger semantics?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2017 at 7:47 | comment | added | David Roberts♦ | ... a section of the (onto) map with codomain $u$. (in one version, anyway, @Joel) The particular flavour of non-classical logic will determine what counts as a function automatically. | |
| Jul 11, 2017 at 23:20 | comment | added | Joel David Hamkins | That there is a function $f:x\mapsto y$ of the relevant kind, and this would depend on the context. In computability theory, one wants the function to be computable; in ZFC set theory, one wants it merely to exist as a set; in GBC, as a class. In constructive mathematics, one wants....(kindly fill in the blank). | |
| Jul 11, 2017 at 23:04 | comment | added | Andrej Bauer | How would you make the notion of uniformity mathematically precise? | |
| Jul 11, 2017 at 11:34 | comment | added | Joel David Hamkins | I had just meant that the (global) axiom of choice is the assertion that from the ordinary reading of $\forall x\exists y$ we may deduce the uniform reading, that is, the assertion that there is a function passing from $x$ to such a $y$. I don't have any specific references to suggest, but I have seen this issue discussed in various contexts, and the issue of uniformity of algorithms is quite commonly discussed. | |
| Jul 11, 2017 at 4:10 | vote | accept | Pace Nielsen | ||
| Jul 11, 2017 at 4:10 | comment | added | Pace Nielsen | Your post is getting at the heart of what I was asking about. Thank you! As a follow-up question, is your statement "the axiom of choice can be interpreted as the assertion that all existence claims...are uniform" an informal statement of belief, or a technical statement about soundness of a certain semantic. If the second, is there a place where this is proven? [I expect not, as this seems to be quite specialized.] | |
| Jul 10, 2017 at 20:56 | comment | added | Joel David Hamkins | I swapped $x$ and $y$ from the use in your question, in light of the common mathematical practice to use $x$ for the independent variable and $y$ for the dependent variable. The question here is about the existence of a function $y=f(x)$ for which $P(x,f(x))$. | |
| Jul 10, 2017 at 20:54 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |