
Transfinite Cardinal Arithmetic with Wolfram Alpha - ColinWright
http://blog.wolframalpha.com/2010/09/10/transfinite-cardinal-arithmetic-with-wolframalpha/
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powertower
If my definition of infinity is the set of all integers, why isn't this true -

(2 * infinity) / (3 * infinity) = 2/3

Even if the definition is something else, it's the same set, and it should get
canceled out - or don't even use it in like that, pretend it's a variable (x)
- same effect.

> For Cantor, it simply must be accepted that the inner and outer circles have
> the same number of points—our intuition that the outer circle has more
> points, according to Cantor, is just wrong.

We know that the outer circle circumference is larger than the inner circle
circumference. We can even find the ratio very easily.

How do you transition from that - to counting infinitely small dots on those
circles and saying they have the same number of those dots? I don't see the
connection from that exercise to the mathematical rules.

Normal logic dictates that there should be more real numbers between 1 and 10
than there are between 1 and 2. Because 1-10 contains all the numbers that are
in 1-2, and then some!

But that's not true!?

~~~
Daniel_Newby
> If my definition of infinity is the set of all integers, ...

The cardinality of the set of integers is aleph-0. Aleph-0 is one of many
transfinite cardinals that measure the size of "infinite" sets.

> (2 * infinity) / (3 * infinity) = 2/3

This can be done with limits:

lim{x -> ∞} (2x)/(3x) = 2/3.

~~~
gizmo686
To anyone studing calculus:

lim{x -> ∞} (2x)/(3x) != (2 * infinity) / (3 * infinity)

In fact, the infinities we are speaking of are two completely different
objects. The limit as x-> ∞ is explicitly defined seperatly from the limit as
x->(Real Number). In fact this definition makes no further reference to ∞, and
instead says that for any arbitrarily small, positive d, there is a real
number m such that x>m implies (2x)/(3x) is within m of 2/3\. The infinity we
are talking about when we say (2 * infinity) / (3 * infinity) = 2/3, (I
assume) is the size of the set of integers, which bears no relation to the
symbol ∞ used in limits.

Ultimately, the important thing to remember in situations like this is that if
something has not been defined as true, or proven to be true, you cannot
assume it. Many similarities (such as overloading operators) are chosen for
convenience, not because they are the same.

The problem is that most of the time these similarities were made because
there really is a strong parrallel, and you look like an idiot for wasting a
math class to prove that 0x=0. (In my case, those two 0s weren't even the same
object: the left one was a real number, the right one was a vector)

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pervycreeper
On the other hand, Alpha does not seem to have the facilities currently for
dealing with transfinite ordinal arithmetic, which is arguably much richer and
more interesting (there's not much you can do with cardinals without CH).

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gizmo686
Regarding the ending point on Godel. If we have proven that it is impossible
to disprove the continuum hypothesis, then would we be safe in assuming it to
be true? If such an assumption were to lead to contradictions, we would have a
proof by contradiction that the hypothesis is false. Of course, the same
argument could be made for assuming the hypothesis is false. Therefore,
doesn't this demonstrate that their are 2 constructions of math we can use:
one where it is true, and the other where it is false. And that both of these
constructions are internally consistent (assuming that the math with neither
assumption is itself consistent). At this point it just becomes a question of
which version is more useful/interesting.

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edtechdev
For a more intuitive explanation and history of transfinite cardinals and a
cognitive analysis of their development, see writings by Rafael Nunez,
including these two papers:
[http://www.cogsci.ucsd.edu/~nunez/web/SingaporeF.pdf](http://www.cogsci.ucsd.edu/~nunez/web/SingaporeF.pdf)
[http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf](http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf)

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ucarion
"Pursuing that challenge, Kurt Gödel showed that the known axioms of
mathematics are insufficient to disprove the hypothesis."

Is this related to Godel's Incompleteness Theorem? Are there any non-
mathematician's explanations as to how Godel proved this?

~~~
ultrafilter
There's no relation to his incompleteness theorems. Gödel proved that every
universe V of sets that satisfies the (ZFC) axioms has a subuniverse L that
satisfies both the axioms and the continuum hypothesis (CH). L is constructed
by transfinite recursion. At each stage of the construction, the only new sets
you admit are the definable subsets of the set of all sets you admitted in
earlier stages.

[http://en.wikipedia.org/wiki/Godel%27s_constructible_univers...](http://en.wikipedia.org/wiki/Godel%27s_constructible_universe)

Unfortunately, proving that L satisfies ZFC and CH is a very delicate and
technical matter. On the other hand, it can be fun because it's extremely
meta. One of the steps of the argument is proving that L is the L of L. I
learned the proof from a graduate textbook chapter titled "defining
definability."

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CurtMonash
My first thought was -- what's the execution time on the algorithms?

Yeah, yeah -- I know it's symbolic. But still ... :D

