infinite number of beers on the wall

infinite number of beers on the wall
take one down
pass it around
infinite number of beers on the wall

0 bottles of beer on the wall
0 bottle of beers
take one down
pass it around
65,535 beers on the wall

>>1
What happens if you run this infinitely?

>>3
stack hash error

KoG

I've seen this posted on /sci/ once and wondered what it implied. I concluded that it's likely that we're dealing with some set theory with the axiom of choice (because you can pick a beer) and the cardinality of the set of beers is at least ℵ₀ (infinite and maybe countable).

An infinite set is an infinite number of things...
You can't count them...

Because you would need an infinite number of numbers... =)

>>7

An infinite set can be either countable or uncountable.  Theres a difference between the two.

I assume the idea behind it is that infinite will never become finite no matter how much you take out of it

>>7
Naturals, integers, rationals are countable (N,Z,Q).
Reals are uncountable (R).
You can get to higher ordinals by using the power set operation.
To show that there is a difference between uncountable and countable sets, you should look up Cantor's diagonalization argument which shows that the set of reals has a greater cardinality than the set of integers (and a generalized version of it can be used to show that for power set operations).

Countable could also be considered as enumerable. For example, for natural numbers, you can start with an initial element (0), take a successor function which when applied to a number, produces its "successor" (also a natural). If the successor of 2 numbers are equal, the numbers are equal (this is how you define the successor function, as an axiom; it's usually s(x)=x+1 if you want a more familiar form). Also, 0 is never the result of a successor function's application (initial element). This may make it seem that the set of naturals takes the shape of {0,s(0),s(s(0)),s(s(s(0))), ...}, and is an countable/enumerable infinite set. Now to show that this set contains all naturals, you either need an induction schema axiom (such as with Peano Arithmetic) or an axioms like this (in the weaker Robinson Arithmetic):
Given a natural y, it's either 0 or it's the result of the successor function (y=0 ∨ ∃x (Sx = y)). This axiom would be a theorem if an induction axiom schema was present.

Induction axioms can have many forms, but here's 2 of them:
Given a predicate P(n). If P(0) is true and P(k) implies P(s(k)) then P(n) is true for all natural numbers n. At times this axiom seems a self-evident truth as if you know P(0) is true and P(k)->P(s(p)) (where s is as defined previously(successor or 1+)), we could see that P(0),P(1),P(2), ... and so on will all be true, but without the axiom we can't say that for any n it is true, hence this is where the countable infinity truly appears (and it's also the cause of the whole consistency problem - or Godel's incompleteness theorem).

A different way of stating the induction axiom would be like this: A is a set. 0 is in A, for any natural number n, if n is included in A then s(n) is in A, then A contains all natural numbers.

One could go on here about isomorphisms between countable sets and models, but that would be me getting carried away more than I should.

>>10
I occasionally wonder if professors or grad students browse 4chan

>>7
http://www.youtube.com/watch?v=JPrHlySlHTY

Now take your tomfoolery elsewhere, heathen.

>>12
I feel smarter


#include <iostream>
#include <limits>
using namespace std;

int main() {
    float beers = numeric_limits<float>::infinity();

    cout << beers << " number of beers on the wall" << endl
         << "take one down" << endl
         << "pass it around" << endl
         << --beers << " number of beers on the wall" << endl;

    return 0;
}

>>14 This problem is better suited to Lisp!

>>15
This problem is BEST suited to Haskell.

I tried to allocate a variable for how many beers I would need on the wall to start with but it caused a buffer overflow.

>>15-16
First you have to implement http://en.wikipedia.org/wiki/Ordinal_arithmetic
It would even be more interesting if you could make some sort of theorem prover which could derive them from more base principles/axioms (makes me wonder how incredibly difficult this would be).
Then after you "waste time" doing that (the result may very well be some AGI the likes we have yet to see, since I don't think theorem provers could do this without a lot of guidance, and even then, set theory is quite a weird overcomplicated axiomatic system), all you need to do is implement something like >>14's where beers equals to Aleph Null (which is still Aleph Null after decrementing).

>>7
No.

>>12
I understand the diagonalization argument applied to real numbers which prove they're uncountable.

I also understand how rational numbers are countable, writing them in a matrix like that.

But why can't you apply the diagonalization argument to rational numbers?

>>20
Disregard that, after pressuring Google much harder than I should have had to I finally found a good explanation.

Why do search engines still suck so much? They're optimizing for ad revenue and not for result quality, right?