1. [OT] How many integers?
rforno at tutopia.com wrote:
> You know, the set of *all* integers (abstract integers, not
> Euphoria nor C nor computer integers) is infinite, but larger than
> the set of *even* integers, that is also infinite.
Actually, the set of even integers (let's call it ZE) and the set of all
integers (Z) are the same size (countably infinite). The standard way
to prove this is to use some function to map from one set to the other,
showing that there is the sets map each other completely with a
one-to-one relationship (bijective):
Then for any z in Z, there is a ze in ZE such that z = ze / 2. In other
words, for any normal integer, I can use even integers to 'count' the
normal integers, and so the sets must be the same size (btw, the set of
rational numbers is the same size, but any interval of real numbers will
be bigger).
Matt Lewis
2. Re: [OT] How many integers?
Hello Matt:
>
> rforno at tutopia.com wrote:
> > You know, the set of *all* integers (abstract integers, not
> > Euphoria nor C nor computer integers) is infinite, but larger than
> > the set of *even* integers, that is also infinite.
>
> Actually, the set of even integers (let's call it ZE) and the set of all
> integers (Z) are the same size (countably infinite). The standard way
> to prove this is to use some function to map from one set to the other,
> showing that there is the sets map each other completely with a
> one-to-one relationship (bijective):
>
> Then for any z in Z, there is a ze in ZE such that z = ze / 2. In other
> words, for any normal integer, I can use even integers to 'count' the
> normal integers, and so the sets must be the same size (btw, the set of
> rational numbers is the same size, but any interval of real numbers will
> be bigger).
>
> Matt Lewis
I say, Russian is better for such the things 
We say "power of a set", not "size of a set".
Regards,
Igor Kachan
kinz at peterlink.ru
3. Re: [OT] How many integers?
Hello again Matt:
>
> rforno at tutopia.com wrote:
> > You know, the set of *all* integers (abstract integers, not
> > Euphoria nor C nor computer integers) is infinite, but larger than
> > the set of *even* integers, that is also infinite.
>
> Actually, the set of even integers (let's call it ZE) and the set of all
> integers (Z) are the same size (countably infinite). The standard way
> to prove this is to use some function to map from one set to the other,
> showing that there is the sets map each other completely with a
> one-to-one relationship (bijective):
>
> Then for any z in Z, there is a ze in ZE such that z = ze / 2. In other
> words, for any normal integer, I can use even integers to 'count' the
> normal integers, and so the sets must be the same size (btw, the set of
> rational numbers is the same size, but any interval of real numbers will
> be bigger).
Then, Russians say "multitude" or "host", not "set".
So, "power of the normal integers host" forces to think
about infinity first of all.
And "host" itself may be small - as "set" is something
small first of all.
Russian is good for such the things, isn't it? 
Regards,
Igor Kachan
kinz at peterlink.ru
4. Re: [OT] How many integers?
Hello Matt:
> Igor Kachan wrote:
>
> > I say, Russian is better for such the things
> > We say "power of a set", not "size of a set".
>
> If I were being correct, I really should have said
> the "cardinality of the set."
> To my [native speaking american-english] ear, "size"
> sounds better than "power".
> (Plus my "proof" actually contains a lot of hand
> waving that might not be clear to someone not
> familiar with set theory.)
> ...and then Igor wrote:
> > Then, Russians say "multitude" or "host", not "set".
> > So, "power of the normal integers host" forces to think
> > about infinity first of all.
> > And "host" itself may be small - as "set" is something
> > small first of all.
>
> "Power of the normal integers host" sounds like you're
> talking about an army or something to me. :)
Ok, try "Power of the normal integers multitude"
> Then again, the "power of the host" of
> Russian words I know is a very small number.
What a problem? Learn Russian and you'll be all set.
Do you see that famous "set" again?
"multitude" is much more clear term for this subject.
Russian is Russian, no?
> > Russian is good for such the things, isn't it?
Do you see Russian is really good?
> Well, I've heard that you guys have some pretty sharp
> mathematicians, so I won't disagree with you
> (though IMHO it was a Frenchman who was probably the
> greatest pioneer in this area--Cauchy--so I wonder
> what the French would say?).
This is a German source, Cantor's theory,
he was the first setman-hostman-multitudeman
> Guess it's another case of imperfect translations.
Yes, my "host", "power", "multitude" are just the
reverse translation of that Cantor's theory terms
from Russian into English.
> I still like english, BTW. :p
>
> Matt Lewis
Ok, and I still like Russian, BTW. :b
Regards,
Igor Kachan
kinz at peterlink.ru
