
A Century of Controversy Over the Foundations of Mathematics  - mariorz
http://www.cs.auckland.ac.nz/~chaitin/cmu.html
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saurabh
Absolutely fascinating read. I think I feel the same way prehistoric man would
feel looking at nature. Its a daunting task, but it feels like our journey has
just begun just like our ancestors; and its getting more beautiful. I wanna
hear Bach now.

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ced
I've been wondering about the Russell Paradox...

    
    
      S = the set of all sets not members of themselves
      x = 1 / 0
    

Isn't the problem for x the same as for S? Not all mathematical expressions
are well-defined, and likewise for all "set expressions".

Aside: There was a wonderful quote in Jaynes' _Logic of Science_ , decrying
the kind of airy mathematics that Chaitin is doing...

 _Should one design a bridge using theory involving infinite sets or the axiom
of choice? Might not the bridge collapse?_

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ubernostrum
The problem is that there is no X for which "X / 0" is defined, at least in
the sort of traditional mathematics you're talking about.

But "the set of all sets which meet criterion X" _is_ well-defined and is an
extremely common thing to encounter. Similarly, "X is a member of itself" is
well-defined and is fairly common.

So you end up with this definition -- the set of all sets which are not
members of themselves -- which just puts together these two common, well-
defined (at least, we _hope_ they're well-defined) concepts and ends up at a
contradiction.

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ced
Mwell, I'm not really satisfied by your answer. Both "1" and "0" are well-
defined expressions; division is also well-defined, but, in order to avoid
logical contradictions, we cannot meaningfully define the division of 1 and 0.
To me, that sounds exactly like what you described above, and I thought that
the axiom of choice was the analog of the "you can't divide by 0" rule.

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ubernostrum
Maybe look at it this way:

There is no X for which "X / 0" is defined.

There are many, many, many X (in fact, potentially all X such that X is a set)
for which "X is a member of itself" is defined.

In other words, division by zero universally leads to contradiction no matter
what you put in for X. But self-membership is perfectly well-behaved for lots
and lots of values, and only starts to get troublesome on certain particular
cases. This is curious: why should something which works well in many, many
cases end up at a contradiction based not on the logical form of the
statement, but rather on the particular values we substitute in place of its
variables?

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yread
Thank you, a very interesting lecture. The process physics he is linking to
are maybe even more interesting. Everything based on random fluctuations ...
hmm I'd better learn the statistics good!

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anonymousDan
Great link! I've read about a lot of this stuff in text books etc, but it all
fits together so much better when put in its proper historical context like
that. If I understand it correctly, his main result in algorithmic complexity
theory implies that it is impossible to prove the absence of a pattern in
data. Does this thus imply that there is no way for us to know when we know
everything?

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Alex3917
"So 'this statement is false' is false if and only if it's true, so there's a
problem."

I don't understand this. This seems to assume that the meaning of a phrase is
contained within the phrase itself. But if you instead assume that the phrase
is a pointer to meaning that is contained somewhere else then the paradox goes
away.

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anigbrowl
Well, that's a pretty large assumption. At least in american English, the
phrase 'this _____' is commonly understood to be self-referential in some
manner unless some other context has been previously established. For example:

This sentence contains ten words and is a true statement.

If you haven't read _Godel, Escher, Bach_ , now is the time.

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Alex3917
Self-referrential to the words themselves, or to the meaning of the words?

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ubernostrum
Depends on what sort of metaphysics of propositions you accept, whether you
want to deal with an extensional or an intensional language, and a whole bunch
of other stuff that generally only philosophers and some mathematicians have
cared about.

If you're interested, though, let me know and I'll see what I can dig up for
you from back when I was studying this stuff.

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TweedHeads
Is this correct in set theory?

ω/ω=1

1/ω=ε

ε*ω=1

ω=1/ε

If we assume for every positive number there is only one negative number then
this must be true: +∞/-∞=1

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mariorz
why is 1/ω=ε ? would it not tend to 0. perhaps I'm misunderstanding you, I
don't really grok set theory.

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TweedHeads
Sure it tends to zero which is ε, being ω the largest and ε the smallest
possible.

Never mind, just trying to understand Cantor's brilliant mind.

