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Set Theory and Foundations of Mathematics

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Re: Set Theory and Foundations of Mathematics

#21

Post by Shaniqua »

defen wrote: post #16
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.

> 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.

Re: Set Theory and Foundations of Mathematics

#22

Post by mindcrime »

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…

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.

Re: Set Theory and Foundations of Mathematics

#24

Post by Shaniqua »

banachtarski wrote: post #10
voidz wrote: post #7

Earlier quoted context omitted.

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".

Re: Set Theory and Foundations of Mathematics

#25

Post by vezzy-fnord »

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.

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.

Re: Set Theory and Foundations of Mathematics

#26

Post by anthony_d »

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…

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

Re: Set Theory and Foundations of Mathematics

#27

Post by valera_rozuvan »

sbilstein wrote: post #18
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…

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…

> 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...

Re: Set Theory and Foundations of Mathematics

#28

Post by valera_rozuvan »

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…

> 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.

Re: Set Theory and Foundations of Mathematics

#29

Post by defen »

Shaniqua wrote: post #21
defen wrote: post #16

Earlier quoted context omitted.

> 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.

Re: Set Theory and Foundations of Mathematics

#30

Post by shadytrees »

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.

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?
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