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Set Theory and Foundations of Mathematics

settheory.net

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Re: Set Theory and Foundations of Mathematics

#11

Post by rectang »

dmead wrote: post #2

i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.

It'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.

Re: Set Theory and Foundations of Mathematics

#12

Post by valera_rozuvan »

vowelless wrote: post #5
gooseus wrote: post #3

Earlier quoted context omitted.

I 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 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.

> 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 fir…

Best 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/ .

Re: Set Theory and Foundations of Mathematics

#13

Post by Shaniqua »

banachtarski wrote: post #9
Shaniqua wrote: post #6

Earlier quoted context omitted.

But 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.

I 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).

It's no big deal if you don't admit universal set or anything other than ZFC.

Re: Set Theory and Foundations of Mathematics

#14

Post by tel »

gooseus wrote: post #3
dmead wrote: post #2

i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.

I 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 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.

> 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.

Re: Set Theory and Foundations of Mathematics

#16

Post by defen »

Shaniqua wrote: post #6
vowelless wrote: post #5

Earlier quoted context omitted.

> 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 fir…

But 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.

> 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

Re: Set Theory and Foundations of Mathematics

#17

Post by AnimalMuppet »

tel wrote: post #14
gooseus wrote: post #3

Earlier quoted context omitted.

I 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 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.

> 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.

Sartre 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.

Re: Set Theory and Foundations of Mathematics

#18

Post by sbilstein »

vowelless wrote: post #5
gooseus wrote: post #3

Earlier quoted context omitted.

I 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 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.

> 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 fir…

I 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.

Re: Set Theory and Foundations of Mathematics

#19

Post by acrefoot »

rectang wrote: post #11
dmead wrote: post #2

i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.

It'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…

If 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.

Re: Set Theory and Foundations of Mathematics

#20

Post by dmead »

acrefoot wrote: post #19
rectang wrote: post #11

Earlier quoted context omitted.

It'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…

If 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.

this 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.

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