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Set Theory and Foundations of Mathematics

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Re: Set Theory and Foundations of Mathematics

#31

Post by dvt »

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.

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.

Re: Set Theory and Foundations of Mathematics

#32

Post by amelius »

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 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.

Re: Set Theory and Foundations of Mathematics

#33

Post by pjungwir »

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 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.

Re: Set Theory and Foundations of Mathematics

#34

Post by tikhonj »

dvt wrote: post #31
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.

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!)

Re: Set Theory and Foundations of Mathematics

#35

Post by dvt »

Shaniqua wrote: post #6
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…

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...

Re: Set Theory and Foundations of Mathematics

#36

Post by tel »

vezzy-fnord wrote: post #25
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.

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.

Re: Set Theory and Foundations of Mathematics

#37

Post by valera_rozuvan »

shadytrees wrote: post #30
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?

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/ .

Re: Set Theory and Foundations of Mathematics

#38

Post by tikhonj »

Sirenos wrote: post #8
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…

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.

Re: Set Theory and Foundations of Mathematics

#39

Post by valera_rozuvan »

shadytrees wrote: post #30
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?

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

Re: Set Theory and Foundations of Mathematics

#40

Post by Shaniqua »

defen wrote: post #29
Shaniqua wrote: post #21

Earlier quoted context omitted.

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.
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