Set Theory and Foundations of Mathematics
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Set Theory and Foundations of Mathematics
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Re: Set Theory and Foundations of Mathematics
#2i blew a google interview question because i failed to recognize a powerset. keeping up with this stuff is pretty important.
Re: Set Theory and Foundations of Mathematics
#3i 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.
Re: Set Theory and Foundations of Mathematics
#4i 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.
Back 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.
Re: Set Theory and Foundations of Mathematics
#5i 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 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.
Re: Set Theory and Foundations of Mathematics
#6Earlier 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…
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.
Re: Set Theory and Foundations of Mathematics
#7Earlier 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.
Why is this downvoted? Is the statement incorrect?
Re: Set Theory and Foundations of Mathematics
#8Earlier 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…
Isn'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.
Re: Set Theory and Foundations of Mathematics
#9Earlier 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.
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).
Re: Set Theory and Foundations of Mathematics
#10Earlier 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.
Why is this downvoted? Is the statement incorrect?
Yes