3 ms·

Banach–Tarski paradox This relies on declaring that 2x == x for sufficiently large x. Which is clearly bullshit: they're not the same, you just can't tell the

Banach–Tarski paradox This relies on declaring that 2x == x for sufficiently large x. Which is clearly bullshit: they're not the same, you just can't tell the difference. cantor sets The wikipedia page doesn't make clear what's supposed to be contrary to intuition about this. Unless you mean the 2x == x issue wrt cardinality? gabriel's horn This one isn't counterintuitive at all; smaller things always have more surface area. . Calculus with infinitesimals works by discarding (symbolic) terms known to be (numerically) below epsilon. Adding things to infinity works the same way. Both of these will be intimately familiar to anyone who works with algorithmic complexity ("big-O notation") or the IEEE 'float' or 'double' data types.

steven2012 12y ago
I'm pretty sure it's not just "sufficiently large x" but rather as x approaches infinity. The properties as you approach infinity are completely different, and counterintuitive, to simply a very very large number.
- gus_massa 12y ago
  For the Banach–Tarski paradox it's not enough to have an x that is a \HUGE number of elements, you need an actual infinite. For this and most (all?) of the unintuitive results that use the Axiom of Choice, it is even not enough to have a countable number of elements, like the integer numbers or the rational numbers. You need a bigger infinite like the number of elements in the real number. In this case the number of points in a sphere of real numbers in R^3. (More info about the different types of infinites: http://en.wikipedia.org/wiki/Cardinal_number http://en.wikipedia.org/wiki/Cardinal_number )
- tbrownaw 12y ago
  The "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information and treat infinity as all the same. But they're not all the same, they're simply large enough that any non-infinite numbers are below epsilon. Much like how infinitesimals are all equal to zero (below epsilon for any non-infinitesimal), but if they were all the same calculus wouldn't work.
myhf 12y ago
In mathematics we don't say that something is "clearly bullshit." We say that it "requires the axiom of choice."

steven2012 12y ago

I'm pretty sure it's not just "sufficiently large x" but rather as x approaches infinity. The properties as you approach infinity are completely different, and counterintuitive, to simply a very very large number.

gus_massa 12y ago
For the Banach–Tarski paradox it's not enough to have an x that is a \HUGE number of elements, you need an actual infinite. For this and most (all?) of the unintuitive results that use the Axiom of Choice, it is even not enough to have a countable number of elements, like the integer numbers or the rational numbers. You need a bigger infinite like the number of elements in the real number. In this case the number of points in a sphere of real numbers in R^3. (More info about the different types of infinites: http://en.wikipedia.org/wiki/Cardinal_number http://en.wikipedia.org/wiki/Cardinal_number )
tbrownaw 12y ago
The "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information and treat infinity as all the same. But they're not all the same, they're simply large enough that any non-infinite numbers are below epsilon. Much like how infinitesimals are all equal to zero (below epsilon for any non-infinitesimal), but if they were all the same calculus wouldn't work.

myhf 12y ago

In mathematics we don't say that something is "clearly bullshit." We say that it "requires the axiom of choice."