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.
[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.
and yes, it was a hard tree question.
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.
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.
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.
So true...
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.
Alas, not even safe there. Thanks a lot, Turing.
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.
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/Al...
Having a universal set is a sufficient condition for Russel's paradox.
[1] http://plato.stanford.edu/entries/quine-nf/
[2] https://www.dpmms.cam.ac.uk/~tf/
[3] T. E. Forster, Set Theory with a Universal Set. http://ukcatalogue.oup.com/product/9780198514770.do
Your [3] link doesn't work by the way, I'm interested in reading Forster!
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).
But now you're talking about non-standard set theory[1] which is fine but you are kind of side-stepping the issue.
If you define S to be S = {x in U: x not in x}, then it simply means S is not in U.
You have no idea. The more I study this stuff the more I feel that all philosophical questions might be somehow encoded in there.
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.
I think the best question is "What will humans do, when we find meaning to everything?". I hope that never happens.
And Camus said that the one truly interesting philosophical problem is suicide.
http://www.camus-society.com/myth-of-sisyphus.html
Set theory isn't going to answer those questions at all.
Indeed.
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.