Ha. There's even a wiki page on the paradoxes of set theory
http://en.wikipedia.org/wiki/Paradoxes_of_set_theory
If I recall correctly, a math professor once told me that it is not yet
proven if the cardinality of the power set of the natural numbers is larger
or smaller or equal than the
Am Donnerstag, 5. März 2009 13:09 schrieb Peter Verswyvelen:
Ha. There's even a wiki page on the paradoxes of set theory
http://en.wikipedia.org/wiki/Paradoxes_of_set_theory
If I recall correctly, a math professor once told me that it is not yet
proven if the cardinality of the power set of
On 5 Mar 2009, at 13:29, Daniel Fischer wrote:
In standard NBG set theory, it is easy to prove that card(P(N)) ==
card(R).
No, it is an axiom: Cohen showed in 1963 (mentioned in Mendelson,
Introduction to Mathematical Logic) that the continuum hypothesis
(CH) is independent of
Am Donnerstag, 5. März 2009 14:58 schrieb Hans Aberg:
On 5 Mar 2009, at 13:29, Daniel Fischer wrote:
In standard NBG set theory, it is easy to prove that card(P(N)) ==
card(R).
No, it is an axiom: Cohen showed in 1963 (mentioned in Mendelson,
Introduction to Mathematical Logic) that the
2009/3/5 Hans Aberg hab...@math.su.se:
GHC says that for any set x, there are no cardinalities between card x and
No it doesn't.
It says there is a syntax error in my code.
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Am Donnerstag, 5. März 2009 15:12 schrieb Daniel Fischer:
Yes, but the continuum hypothesis is 2^Aleph_0 == Aleph_1, which is quite
something different from 2^Aleph_0 == card(R).
You can show the latter easily with the Cantor-Bernstein theorem,
independent of CH or AC.
Just to flesh this up
On 5 Mar 2009, at 15:12, Daniel Fischer wrote:
No, it is an axiom: Cohen showed in 1963 (mentioned in Mendelson,
Introduction to Mathematical Logic) that the continuum hypothesis
(CH) is independent of NBG+(AC)+(Axiom of Restriction), where AC is
the axiom of choice.
Yes, but the continuum
On 5 Mar 2009, at 15:23, Daniel Fischer wrote:
Just to flesh this up a bit:
let f : P(N) - R be given by f(M) = sum [2*3^(-k) | k - M ]
f is easily seen to be injective.
define g : (0,1) - P(N) by
let x = sum [a_k*2^(-k) | k in N (\{0}), a_k in {0,1}, infinitely
many a_k =
1]
and then g(x)
Am Donnerstag, 5. März 2009 16:55 schrieb Hans Aberg:
On 5 Mar 2009, at 15:23, Daniel Fischer wrote:
Just to flesh this up a bit:
let f : P(N) - R be given by f(M) = sum [2*3^(-k) | k - M ]
f is easily seen to be injective.
define g : (0,1) - P(N) by
let x = sum [a_k*2^(-k) | k in N
On 5 Mar 2009, at 17:06, Daniel Fischer wrote:
Cantor-Bernstein doesn't require choice (may be different for
intuitionists).
http://en.wikipedia.org/wiki/Cantor-Bernstein_theorem
Yes, that is right, Mendelson says that. - I find it hard to figure
out when it is used, as it is so
Luke Palmer lrpal...@gmail.com wrote:
I don't think set theory is trivial in the least. I think it is
complicated, convoluted, often anti-intuitive and nonconstructive.
Waaagh!
I mean trivial in the mathematical sense, as in how far away from the
axioms you are. The other kind of
To wrap up:
While formalising, there is always a tradeoff between complexity of
the theory you're using and the complexity of it being applied to some
specific topic. Category theory hits a very, very sweet spot there.
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Peter Verswyvelen bugf...@gmail.com wrote:
Maybe this raises a new question: does understanding category theory
makes you a better *programmer*?
Possibly yes, possibly no. In my experience, you have to have a look at
how CT is applied to other fields to appreciate its clarity. Doing so,
you
On Wed, Mar 4, 2009 at 3:38 PM, Achim Schneider bars...@web.de wrote:
There's not much to understand about CT, anyway: It's actually nearly
as trivial as set theory.
You mean that theory which predicts the existence of infinitely many
infinities; in fact for any cardinal, there are at least
Peter Verswyvelen bugf...@gmail.com wrote:
Lambda calculus is a nice theory in which every function always has
one input and one output. Functions with multiple arguments can be
simulated because functions are first class and hence a function can
return a function. Multiple outputs cannot be
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