Timeline for Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
Current License: CC BY-SA 2.5
8 events
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| Mar 9, 2011 at 14:36 | comment | added | Joel David Hamkins | Andrej, if we have a combinatory logic on a larger power set, then can't we simply cut down to an elementary substructure of any smaller infinite size? If so, wouldn't this provide the desired full answer to question (4)? That is, given any $Z$, first go to a larger power set, get the universal binary operation there, and then take an elementary substructure of size $|Z|$, to impose such an operation on $Z$ itself. And now implement Terry's idea on the constants. | |
| Mar 5, 2011 at 20:35 | comment | added | Joel David Hamkins | Andrej, please don't worry about it! Enjoy your family life, and update the answer when you have time, anytime. You've already provided so much food for thought for us. | |
| Mar 5, 2011 at 20:32 | comment | added | Andrej Bauer | This will take a while because I also have to put a six-year old to bed (and six-year olds specialize in not going to bed). | |
| Mar 5, 2011 at 20:12 | comment | added | Andrej Bauer | Ok, I will edit my answer so that we don't need constants. | |
| Mar 5, 2011 at 20:11 | comment | added | Andrej Bauer | Every powerset is a (total) combinatory algebra because it is a model of $\lambda$-calculus, so at least for powers of $2$ we get a similar result. Reals numbers are a power of two (in the insane world of classical mathematics with axiom of choice). | |
| Mar 5, 2011 at 20:03 | comment | added | Joel David Hamkins | Andrej, Thanks! I like this answer very much. But you still seem to need the constants, so it doesn't answer the strong version of question 1. But perhaps one can omit the constants somehow with a complicated term. Another point is that it appears that we can relativize your idea to have an oracle, and thereby get an answer to question 3 in the case of countable rings (using the ring operations as an oracle). With a view to question 4, is there any reason to expect such operations on uncountable sets? | |
| Mar 5, 2011 at 20:01 | comment | added | Todd Trimble | Dang it! :-) I was just setting out to write something about combinatory logic. :-) | |
| Mar 5, 2011 at 19:55 | history | answered | Andrej Bauer | CC BY-SA 2.5 |