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i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.
by dmead 11y ago
i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.
- gooseus 11y agoI tried to find it in the OP, it sounded way over my head... this link explained it real fast. https://www.mathsisfun.com/sets/power-set.html https://www.mathsisfun.com/sets/power-set.html I still wish I could find the time to study all this foundational math... super interesting and I have a sneaky suspicion the answer to a great many philosophical questions are hidden in there.
- vowelless 11y ago> I have a sneaky suspicion the answer to a great many philosophical questions are hidden in there. Godel's incompleteness theorem had a profound affect on me that I find hard to quantify. I grew up starry eyed, thinking that our minds have virtually limitless capabilities and that mathematics could answer everything (naive, I know). Then I came across Russell's paradox and the incompleteness theorems and for the first time (late high school) got confronted by the limits of our 'tools' (with proof!). And then encountering Heisenberg's uncertainty principle was just depressing and I took to computer science. I think better than just looking up seemingly random mathematical concepts is to go through the history of the development of set theory.
- Shaniqua 11y agoBut Russel's Paradox is easy to fix. Let x be a set. Then, V = {x| x not in x} is the set that causes Russel's Paradox. We can easily define a new set S = {x| p(x) and x in U} where p(x) is some property and U is the universal set. Then S easily fixes the contradiction.
- voidz 11y agoWhy is this downvoted? Is the statement incorrect?
- banachtarski 11y agoYes
- voidz 11y agoAh, I see you explained. Thanks. I love the subject, but I'm still a neophyte.
- Shaniqua 11y agoBy the way, I forgot to ask do you object to my post about "killing" the Russel set because you don't accept the existence of universal set or because you don't understand how {x in S| x not x} helps here? If it's the former, can you assume that U exists and explain how {x in S| x not x} solves the problem? So that I know you're not simply "saying things".
- banachtarski 11y agoI have no idea where you got this from but your statements don't follow any form of logic I'm familiar with. If V = {x | x not in x}, then if V contains itself, V is not in V (and vice versa) is an obvious contradiction. Your new set S doesn't help in the slightest. Fixing it is emphatically "not easy" and mathematicians generally rely on the ZFC axiomation (although several other possibilities were proposed).
- Shaniqua 11y agoIt's no big deal if you don't admit universal set or anything other than ZFC.
- johntb86 11y agoIt took 16 years (1901-1917) to get from russell's paradox to a set theory which didn't allow it, but which was able to create a lot of interesting and useful sets (ZF). So it seems like a big deal. And we still can't talk about "the set/collection/whatever of all sets" in the language of ZFC, so we're still missing out.
- defen 11y ago> universal set But now you're talking about non-standard set theory[1] which is fine but you are kind of side-stepping the issue. [1] https://en.wikipedia.org/wiki/Axiom_of_regularity https://en.wikipedia.org/wiki/Axiom_of_regularity
- Shaniqua 11y agoWell, it's not that difficult to fix it in ZF. If you define S to be S = {x in U: x not in x}, then it simply means S is not in U.
- dvt 11y agoHaving a universal set [in naive set theory] is a sufficient condition for Russel's paradox. See Naive Set Theory[1] bottom of page 6. This is why we can have no Universe in any consistent set theory. Edit: @mafribe makes the point that there are some set theories that can still have universal sets by culling other features that ZF-style set theories have. I was mostly referring to ZF-style set theory (hence my citation). Indeed, one could even make a ZF-style set theory paraconsistent and still have Universal sets. [1] http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Logica%20y%20Conjuntos/Paul%20R.Halmos%20-%20Naive%20Set%20Theory.pdf http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Al...
- mafribe 11y agoHaving a universal set is a sufficient condition for Russel's paradox. That's not true. There are set-theories, e.g. Quine's NF [1] which allow universal sets, and other things like the set of all ordinals, that are forbidden in ZF-style set-theories. The problem in ZF is caused by unlimited comprehension. NF circumvents this by restricting comprehension. Tom Forster [2] has written a great deal about set theories with universal sets, including the wonderful [3]. He makes the historical point that set theory was born with universal sets. [1] http://plato.stanford.edu/entries/quine-nf/ http://plato.stanford.edu/entries/quine-nf/ [2] https://www.dpmms.cam.ac.uk/~tf/ https://www.dpmms.cam.ac.uk/~tf/ [3] T. E. Forster, Set Theory with a Universal Set. http://ukcatalogue.oup.com/product/9780198514770.do http://ukcatalogue.oup.com/product/9780198514770.do
- dvt 11y agoTrue, I was mostly referring to ZF-style set theories (which is what the thread is mainly about). Your point could even be extended by saying that there are proofs for a paraconsistent ZF with a universal set. Your [3] link doesn't work by the way, I'm interested in reading Forster!
- mafribe 11y ago[3] works on my browser. Anyway, the link was to the publisher's page for the book. Here is another one: http://www.amazon.co.uk/Set-Theory-Universal-Exploring-Universe/dp/0198514778 http://www.amazon.co.uk/Set-Theory-Universal-Exploring-Unive... .
- Sirenos 11y agoIsn't Russell's paradox more like a problem with naive set theory? By analogy to a software framework, the paradox would be considered a bug in the framework and say nothing about how useful it is.
- tikhonj 11y agoIn a sense. But it's interesting because naive set theory appears so simple and natural: it's a concrete example of how human ideas that seem platonic and inviolate can actually be inconsistent. So it's less like finding a bug in a software framework and more like finding a bug in your understanding of how computers can work.
- valera_rozuvan 11y agoBest article I read on the history of development of set theory is by José Ferreirós: "The Early Development of Set Theory". It can be found at http://plato.stanford.edu/entries/settheory-early/ http://plato.stanford.edu/entries/settheory-early/ .
- sbilstein 11y agoI just finished reading Godel's Proof (a non-textbook account of the proof recommended by Hofstader in GEB) and it was mind-blowing. I really recommend that and David Foster Wallace's Beyond Infinity. Beyond Infinity is great history of set theory and written in an exciting and fun style. I came across the ideas the opposite way...I loved math as child and as engineer but I always felt like systems of rationalization had to be limited in some way. I think I started feeling this way when I'd get into debates with other students in school about arguments and we'd have equally rational but contradictory viewpoints. Fast forward a few years later and I learn some genius had actually proved that we can't answer everything with these systems...or that there isn't a system that can say it all.
- valera_rozuvan 11y ago> Fast forward a few years later and I learn some genius had actually proved that we can't answer everything with these systems...or that there isn't a system that can say it all. So true...
- EvaK_de 11y agoThe book by DFW sounds interesting, but all I could find is "Everything and More: A Compact History of Infinity". Is that it?
- sbilstein 11y agoSorry yea, I misremembered the title when I was commenting.
- pjungwir 11y agoI felt similar amazement at Gödel's proof, and even suspect that Penrose must be on to something when he says the mind must therefore not be computable. . . . But the book Gödel's Theorem: An Incomplete Guide to its Use and Abuse by Törkel Franzén is very good at explaining that the conclusions we should draw are fairly limited, e.g. that the proof doesn't apply to anything other than arithmetic. I'm not sure I'm totally convinced, but it's a great perspective to dispel some of the more mystical interpretations.
- GeneTT 11y agoDennet also has a pretty good take down of Penrose in chapter 15 of Darwins Dangerous Idea.
- OJFord 11y ago> I grew up starry eyed, thinking that our minds have virtually limitless capabilities ... and for the first time (late high school) got confronted by the limits of our 'tools' ... was just depressing and I took to computer science. Alas, not even safe there. Thanks a lot, Turing.
- SchemeLisp 11y agoBack when I just started learning all this stuff, I used to confuse elements of a set and its subsets. For example, I'd be easily stumped when asked what the elements and subsets of {a} were. Also, it's good to know why the empty set is a subset of any set and the proof/s that there are 2^n elements in a power set.
- tel 11y ago> I have a sneaky suspicion... You have no idea. The more I study this stuff the more I feel that all philosophical questions might be somehow encoded in there.
- AnimalMuppet 11y agoSartre said that the great philosophical question is, why is there something instead of nothing? (This does not mean, why is there something in philosophy. Why is there something in the real world? Why does anything physically exist?) If your philosophy doesn't explain the real world, it isn't much of a philosophy. But I don't think set theory can explain that. Two other great philosophical questions are: Where do we humans find meaning? And, what is the basis for morals and ethics? How do we determine what actions are right and wrong? Set theory isn't going to answer those questions at all.
- mindcrime 11y agoTwo other great philosophical questions are: Where do we humans find meaning? And, what is the basis for morals and ethics? How do we determine what actions are right and wrong? And Camus said that the one truly interesting philosophical problem is suicide. http://www.camus-society.com/myth-of-sisyphus.html http://www.camus-society.com/myth-of-sisyphus.html Set theory isn't going to answer those questions at all. Indeed.
- anthony_d 11y agoSorry, but "Where do we humans find meaning?" isn't even a mediocre philosophical question. It's navel gazing by angst ridden teenagers.
- valera_rozuvan 11y ago> Two other great philosophical questions are: Where do we humans find meaning? And, what is the basis for morals and ethics? I think the best question is "What will humans do, when we find meaning to everything?". I hope that never happens.
- tel 11y ago
- rectang 11y agoIt's not really your fault, but defeating Google-gotcha interview questions is a very distorting motivation. I'd like to think that this material is worth studying because it will help you to achieve things you otherwise could not, whether at Google or elsewhere. But that leads to a different kind of study: deep dives into specialized subjects relevant for the task at hand, rather than attempting to maintain shallow coverage over a broad field.
- acrefoot 11y agoIf they specifically asked for "powerset" that might be considered a gotcha question, but a powerset is another way of thinking of binary enumeration[0], which isn't that rare. [0] by that I mean both the case of counting in binary, as well as enumerating all the ways you can have or not have some things--you might care about that when considering all the interactions of config flags you might have to deal with.
- dmead 11y agothis is exactly the case. had i just remember what a power set was instead of say something dumb like "all the n-k combination of these children something someting" it would have gone much better. and yes, it was a hard tree question.
- rwallace 11y agoWell, that's like saying you blew a Google interview question because you failed to recognize a black vine weevil, therefore keeping up with entomology is pretty important. One company having a buggy interview process that asks questions irrelevant to the job, at the very most means if you are applying for a job at that company you should spend a couple days cramming for the interview. It does not mean you should spend the precious hours of your life studying a field that's irrelevant to your job on an ongoing basis. To be clear, if you want to study set theory or entomology because you find them interesting for their own sake then far be it from me to criticize that decision. I'm just disagreeing with the argument that you should do so because of one company's buggy interview process.