> 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 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.
> 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.
I 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.
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.
In 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.
> 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.
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.
Having 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.
Having 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.
True, 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!
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 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.
By 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".
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.