4 ms·

I'd count among my favourites: * the Schroeder-Bernstein theorem in set theory * the fact that the axiom of choice is equivalent to the statement that every s

by sid0 16y ago

I'd count among my favourites: * the Schroeder-Bernstein theorem in set theory * the fact that the axiom of choice is equivalent to the statement that every set is well-ordered: how can something so intuitively correct and something so intuitively wrong be equivalent? * the general notion of dualism in category theory. Two for the price of one! * the ways fixpoints can be used in TCS: the Y combinator (heh), and the Kleene and Knaster-Tarski theorems. The best part is that they're all intuitively the same.

maayank 16y ago
Excellent choices. (The aforementioned sentence does not require use of AoC ;) In fact, the moments he describes in #3 are partly why I'm currently pursuing a Math and not a CS degree.
- sid0 16y ago
  As a guy pursuing a CS degree, I'm compelled to be a constructivist (what use is a proof if you can't get an algorithm out of it?) and reject both AC and the well-ordering theorem. Find me a well-ordering of the reals and then we'll talk. :) edit: speaking of proofs and algorithms, how could I forget the Curry-Howard correspondence. It just makes so much sense.
  - maayank 16y ago
    eh eh :) About an hour after taking a set theory exam it dawned on me that a proof I gave is correct iff the continuum hypothesis is true. The distance between a Fields medal and a F was never so small :) edit: you made me wiki it. Funny, I don't seem to know/remember it. Thanks, seems interesting :) Regarding theoretical proofs... What's the point of an algorithm if you don't propel its use through a viable product? I'm currently (concurrently with the studies) in a CS research position and it is a difficulty for me... Don't get me wrong, I love my job, but for me if there is any downside then this is it.
emillon 16y ago
«The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?»

by sid0 16y ago

maayank 16y ago

Excellent choices. (The aforementioned sentence does not require use of AoC ;) In fact, the moments he describes in #3 are partly why I'm currently pursuing a Math and not a CS degree.

sid0 16y ago
As a guy pursuing a CS degree, I'm compelled to be a constructivist (what use is a proof if you can't get an algorithm out of it?) and reject both AC and the well-ordering theorem. Find me a well-ordering of the reals and then we'll talk. :) edit: speaking of proofs and algorithms, how could I forget the Curry-Howard correspondence. It just makes so much sense.
- maayank 16y ago
  eh eh :) About an hour after taking a set theory exam it dawned on me that a proof I gave is correct iff the continuum hypothesis is true. The distance between a Fields medal and a F was never so small :) edit: you made me wiki it. Funny, I don't seem to know/remember it. Thanks, seems interesting :) Regarding theoretical proofs... What's the point of an algorithm if you don't propel its use through a viable product? I'm currently (concurrently with the studies) in a CS research position and it is a difficulty for me... Don't get me wrong, I love my job, but for me if there is any downside then this is it.

emillon 16y ago

«The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?»