Timeline for What kind of algebra is the class of ordered pairs equipped with the binary operation which forms them?
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 14, 2014 at 0:51 | comment | added | Joel David Hamkins | I would go along with that. | |
| Jul 14, 2014 at 0:41 | comment | added | Ioachim Drugus | Thus, an ordered pair can be defined in such a manner that it encrypts not only order but also some additional information. This is very interesting because (1) this justifies using pairing algebras in encoding formulas as sets as I described their intended use in my question (2) Some properties of sets can be expressed in the language of pairing algebra and equality, which is useful for an algebraic set theory. | |
| Jul 13, 2014 at 23:56 | comment | added | Ioachim Drugus | From your answer, it sound correct to name the algebras which I described, "pairing algebras". | |
| Jul 13, 2014 at 1:56 | comment | added | Joel David Hamkins | Yes, that is precisely what "flat" means for the flat pairing functions. They don't raise ranks on any infinite set. Another way to say this is that every $V_\theta$ for infinite $\theta$ is closed under pairs. (It is impossible to never increase rank, since the finite ranks $V_k$ cannot be closed under pairing (for $k>1$) on finite cardinality grounds, since $n^2$ is larger than $n$ for $n>1$. | |
| Jul 13, 2014 at 1:50 | comment | added | Mariano Suárez-Álvarez | Out of curiosity: is there an ordered-pair definition which does not raise the rank? | |
| Jul 13, 2014 at 1:50 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 230 characters in body
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| Jul 13, 2014 at 1:41 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
fixed typo
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| Jul 13, 2014 at 1:26 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |