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Set Theory and Foundations of Mathematics

settheory.net

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Re: Set Theory and Foundations of Mathematics

#41

Post by tel »

AnimalMuppet wrote: post #17
tel wrote: post #14

Earlier quoted context omitted.

> 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…

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!

Re: Set Theory and Foundations of Mathematics

#42

Post by mafribe »

robobro wrote: post #23

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.

   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.

Re: Set Theory and Foundations of Mathematics

#43

Post by mafribe »

dvt wrote: post #35
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.

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 citat…

   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

Re: Set Theory and Foundations of Mathematics

#44

Post by dvt »

mafribe wrote: post #43
dvt wrote: post #35

Earlier quoted context omitted.

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 citat…

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 wi…

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!

Re: Set Theory and Foundations of Mathematics

#45

Post by dvt »

mafribe wrote: post #42
robobro wrote: post #23

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.

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.

Re: Set Theory and Foundations of Mathematics

#46

Post by Shaniqua »

dvt wrote: post #35
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.

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 citat…

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

Re: Set Theory and Foundations of Mathematics

#47

Post by johntb86 »

Shaniqua wrote: post #13
banachtarski wrote: post #9

Earlier quoted context omitted.

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.

Re: Set Theory and Foundations of Mathematics

#48

Post by OJFord »

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 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.

Re: Set Theory and Foundations of Mathematics

#49

Post by rwallace »

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.

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.

Re: Set Theory and Foundations of Mathematics

#50

Post by gver »

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.

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