
Set Theory and Foundations of Mathematics - valera_rozuvan
http://www.settheory.net/
======
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.

~~~
spacehome
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?

------
robobro
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.

~~~
shadytrees
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?

~~~
valera_rozuvan
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/](http://plato.stanford.edu/entries/wittgenstein-mathematics/) .

------
dmead
i blew a google interview question because i failed to recognize a powerset.
keeping up with this stuff is pretty important.

~~~
gooseus
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](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.

~~~
vowelless
> 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.

~~~
Shaniqua
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.

~~~
dvt
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...](http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Logica%20y%20Conjuntos/Paul%20R.Halmos%20-%20Naive%20Set%20Theory.pdf)

~~~
mafribe

       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/](http://plato.stanford.edu/entries/quine-nf/)

[2] [https://www.dpmms.cam.ac.uk/~tf/](https://www.dpmms.cam.ac.uk/~tf/)

[3] T. E. Forster, Set Theory with a Universal Set.
[http://ukcatalogue.oup.com/product/9780198514770.do](http://ukcatalogue.oup.com/product/9780198514770.do)

~~~
dvt
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!

~~~
mafribe
[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...](http://www.amazon.co.uk/Set-Theory-Universal-Exploring-
Universe/dp/0198514778) .

------
burritofanatic
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.

