Set Theory and Foundations of Mathematics | Hacker News Reader

Set Theory and Foundations of Mathematics

settheory.net

121 pointsvalera_rozuvan58 comments

If you follow the link to his page "I'm upset...Here is why", you can see that this person has a significant degree of "crankiness". These people through their perhaps not deliberate obfuscation, often manage to trick the mathematically naive into thinking there is substance there - "it is so hard and confusing, it must be real math."

It is a disservice to have links like this on Hacker News.

The parent isn't saying the author is cranky, like ornery, he's saying he's a crank. The link being referred to [0] seems to be a classic piece of crankery meandering from the solution to global warming to perceived slights against the author on Internet forums and who knows what else.

The question then, knowing this, is whether the OP link is quality content?

[0] http://settheory.net/life

I tried, but I don't think I have it in me to read that wall of text. What specifically made you think he was cranky?

That's two links away. Do we really need to hold HN to such a high standard?

Honestly, Wittgenstein's philosophical critiques of set theory in relation to the foundations of mathematics still hold true today. Shame he doesn't get more recognition for his great work in logic.

Wittgenstein was primarily interested in the linguistic applications of logic. IMO, he wasn't on the same level as Whitehead, Russell, or Curry; let alone Hilbert, Gödel, and Zermelo.

And maybe it's just my experience, but Wittgenstein was always shoved down our throats in my undergraduate and graduate classes, whereas people may have never even heard of Jan Łukasiewicz or Stephen Kleene.

Ah, different communities with different goals. In CS, I've run into Kleene a lot and Wittgenstein never. (Although I'm sure people have heard of him!)

I'm intrigued but, upon Googling, pretty at sea with all the vocabulary. Is there an explain-like-I'm-slightly-above-five for Wittgenstein's critique of set theory? It seems like he doesn't like infinity very much?

Have you seen Victor Rodych's entry at the plato.stanford.edu site entitled "Wittgenstein's Philosophy of Mathematics"? Direct link: http://plato.stanford.edu/entries/wittgenstein-mathematics/ .

2.6 Wittgenstein's Intermediate Critique of Set Theory http://plato.stanford.edu/entries/wittgenstein-mathematics/#...

I guess it is mostly a matter of taste, like whether one favors functional programming or imperative programming. Set theory is being used successfully in formal systems, so in principle there is nothing wrong with it.

   Shame [Wittgenstein] doesn't get more recognition for his great work in logic.

Which great work in logic did W. produce?

He is known for some inchoate criticisms, e.g. he doesn't like Cantor's diagonal proof, but none of his criticisms have -- as far as I'm aware, lead anywhere interesting in logic.

The Tractatus Logico-Philosophicus. In it, Wittgenstein touts his logical atomism. I don't put much stock in the Tractatus, but some people build entire academic careers on it.

So who knows, maybe I'm missing something.

I don't see any particular contributions to logic in the Tractatus.

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.

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.

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.

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.

The book by DFW sounds interesting, but all I could find is "Everything and More: A Compact History of Infinity". Is that it?

Sorry yea, I misremembered the title when I was commenting.

> 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...

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.

Dennet also has a pretty good take down of Penrose in chapter 15 of Darwins Dangerous Idea.

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/ .

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.

Alas, not even safe there. Thanks a lot, Turing.

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.

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.

[1] http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Al...

   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.

[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

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!

[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-Unive... .

You can fix the Russel's Paradox in ZF as well.

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.

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.

> 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

Well, 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.

There is no U in ZF so I really don't understand what you are saying.

You can call U anything you want. It's by axiom schema of specification: for any set A, there some set B with a set C in B iff C in A.

Why is this downvoted? Is the statement incorrect?

Yes

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".

Ah, I see you explained. Thanks. I love the subject, but I'm still a neophyte.

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.

> 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.

Sorry, but "Where do we humans find meaning?" isn't even a mediocre philosophical question. It's navel gazing by angst ridden teenagers.

It was a bit tongue-in-cheek, but I'd argue that mathematical foundations (maybe not set theory interpreted strictly) does take aim at answers to "why is there something?" and "where do humans find meaning?". Possibly by unifying the questions!

> 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.

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?

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.

Only if you're biased towards a realist philosophical view. Though admittedly it makes many feel warm and fuzzy, there's really no evidence for it over positions like intuitionism.

I'm not, but I can't see how you cannot study set theory without eventually studying intuitionism.

Well, 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.

I studied set theory was I young as a student, but seeing something like this reminds me there's so much to learn, and re-learn.

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