

Levels of Infinity - DiabloD3
http://www.xamuel.com/levels-of-infinity/

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xyzzyz
Yeah, the cardinals are really mind blowing -- but only as long as you
consider them to be "real", that is, if you hold a Platonic view on reality
and consider mathematical objects (that is, sets) to really exist. This naive
dream is quickly crushed by introductory model theory, where you learn that
there are exists a set 'S' (a lot of them actually) which can be regarded to
be "the set of all sets",and a special relation 'e' between the elements of
this set which represents a set membership. Together, the set and the relation
form a structure we call a model of set theory. All true statements about sets
are also true, if we restrict ourselves to our model and replace the usual set
membership relation with our special relation.

So, in our model 'S' there is some element 'o', such that there is no 'x' in
'S' such that 'x e o' -- that is, no 'x' is in a relation 'e' with 'o'. If we
take into account that relation 'e' is supposed to represent set membership in
our model, we notice that this object 'o' represents an empty set in our
model, the existence of one is ensured by usual set theory axioms. Since every
object in the set theory is a set, so is our 'o', and although it represents
an empty set in our model, it does not need to be in fact empty -- it is
enough that it is empty in the sense of our special membership relation 'e'.

Apart from "empty set" 'o', in our model there are representatives for all the
usual sets we know -- the set of natural numbers, reals, functions from
naturals to {1, 2} (and {1, 2} as well) -- and under our relation 'e' they
behave in exactly the same way as usual sets behave under usual set membership
relation.

So far, our models looks exactly like the whole universe, only smaller. But,
thanks to some results of model theory (precisely, the Lowenheim-Skolem
theorem), we can impose another restruction on our model -- we can require it
to be countable.

Now this is really mind blowing -- our model behaves just like the whole
universe and yet there is only as many elements in it as there are natural
numbers! Sounds quite paradoxical -- you could ask, but what about Goedel's
theorem? It sure has to hold, because this is a model of set theory, but yet
there are exactly as many reals as naturals in our model -- countably many.
How is that even possible?

Well, the answer is quite obvious -- the concept of "cardinality" is not
absolute. When we say that two sets have the same cardinality, we mean that
there is a bijection between their elements. Since our model is countable,
there is a bijection between elements representing natural numbers and the
ones representing reals. But there is no paradox -- this bijection is not an
element of our model. It shows that that naturals' and reals' representatives
are in bijection, but only in the whole universe. When we restrict ourselves
to our model an we ask for an element representing such a bijection, none
exists. While there are exactly as many 'reals' as there are 'natural numbers'
in our model, we cannot see it from inside.

Now, maybe this is the case with our regular numbers -- they are the same in
number, but it is impossible for us to see. But the longer one think about
this problem, the less sense the question makes. The simplest solution is to
accept the fact that sets have no real existence whatsoever, that the
mathematician does not explore structure of some abstract constructs, but only
manipulates the strings on paper in some defined way. This approach is not
very romantic, but is the only way I know of escaping from problems and
paradoxes brought by Platonic view on reality.

~~~
harshavr
Nice explanation, though I dont see how this leads you into becoming a
formalist at the end. That countable models exist for the set theoretic
universe, seems like part of the phenonmena of infinite structures having
perfect reflections in subsets. This is, of course, much more breathtaking
than the more usual facts like the natural numbers having the same cardinality
as even numbers since cardinality is transcended and the whole universe
collapses into a countable set.

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aik
I wonder if others feel the same way:

I think one reason I'm so fascinated by certain abstract mathematical concepts
(like this one), in addition to certain theoretical physics concepts that prod
at the root of how everything works, is in the hope that someday I will come
across something that will trigger some new all-encompassing form of
understanding inside me like nothing else ever has. Something, but not
necessarily, of eternal consequences.

~~~
TeMPOraL
I think I have similar feelings. The deep reason for me to read about advanced
mathematics, theoretical physics and science in general is the hope that one
day this accumulated knowledge will let me understand the world we live in in
a better way; a hope of getting a framework of thought that allows to
experience and do things that previously I couldn't.

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CallMeV
I first got into transfinite numbers through a book by Rudy Rucker, "Infinity
and The Mind." I tried explaining this subject to a Doctoral candidate once,
at around 2am - and he accused me of having come off my meds.

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simias
It's interesting, however there is something I quite don't understand:

"[...]we can take 0 and 1, promote them above all the other naturals, and get
a new order on the naturals, which looks like (2,3,4,…,0,1). This looks like a
shifted-copy of ω, followed by exactly two new things on top. Its order type:
ω+2. Similarly we can get ω+3, ω+4, and so on."

I don't understand how that's "infinity + 1" or "infinity + 2" when it's the
same number of element, just in a different order.

~~~
btilly
The answer lies in the difference between a _cardinal_ and an _ordinal_. A
cardinality is a size, an answer to "how many". As in 1, 2, 3... An ordinal is
a position. As in first, second, third...

For finite numbers they are trivially equivalent. No so for infinite. If you
add 1 to ω you haven't changed the cardinality at all. But if you append a new
number after ω then that new number is in a position which doesn't exist
anywhere in ω, and therefore it must be a larger ordinal.

So because cardinals and ordinals are different, there is no problem with ω+2
being a larger ordinal than ω, even though it has the same exact cardinality.

~~~
defrost
Also of some note is the fact that ω+1 has something that ω does not, it has a
single unique element that is greater than all other elements, something that
ω doesn't have.

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spjwebster
For a fictional, tripped-out exploration of the various levels of infinity,
try Rudy Rucker's White Light:

<http://en.wikipedia.org/wiki/White_Light_(novel)>

I picked this up at a end of line book shop when I was 13 or 14, along with a
bunch of other Wired Press books, and it blew my mind.

~~~
arethuza
I remember watching Open University program when I was about that age that
worked through Cantor's diagonal argument - I vividly remember being
spellbound that such a seemingly simple argument could has such a profound and
counter intuitive result.

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Jun8
It is a famous anecdote that Cantor was not quite as surprised when he came up
with his "diagonal proof" (reals are uncountable) but when he saw that the
points on the plane can be mapped to points on the line, i.e. the infinities
are the same. There just seems to be more points in R^2 than R, this is highly
counterintuitive.

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SeoxyS
These mathematics seem highly dubious to me. It's like they just found a bunch
of different ways to describe the same thing, Infinity.

Yes 0 is 0. I could also describe it as (5 * 4 * 3 * 0), but how does that
help?

When you think about it, there is no such thing as infinity. It's just a word
and concept used to describe an amount bigger than you're able able to imagine
or comprehend. It's something we use because we need to give a name to the
"limit" to the far end of a spectrum.

~~~
cdavid
Those are very conventional mathematics, and most core concepts have been
settled for a century now. Cantor in particular paved the way toward a better
understanding of many related concepts in that area.

As for the significance of those differences: I can think about two areas
where this is fundamental (but there are many more). First, the natural
numbers (0, 1, ...) vs rationals vs real numbers is really big in terms of
"how many" numbers you have. You can prove than the set of natural numbers and
the set of rational numbers is roughy the same size (they are both countable).
Now, as you may know, there are some irrational numbers, like square root (2),
pi, e, etc... But how many ? The answer is "many more than rationals". One way
to define the real numbers (rationals + irrationals) is as limits of a set of
rational numbers - i.e. for any real number, you can find a set of rationals
which get arbitrarily close to an irrational (we say that rational numbers are
dense for the set of real numbers).

Another field where you see the difference: topology vs probability. Roughly
speaking, probability is a function which for a set gives you a number between
0 and 1. The key property of probability is that P(A U B U ....) = P(A) + P(B)
+ ... where A, B, ... are disjoint sets. Another way to look at it is that
probability is a special case of the notion of set measure, which is the
mathematical way of talking about volume, length, etc... There is a key
restriction when defining measure, that is you cannot take arbitrary unions of
sets, like say as many as real numbers - actually, you can prove that you
_cannot_ build measures if you want arbitrary (uncoutable) unions. This is a
naive way to describe the Banach-Tarski paradox, which says that you can split
a ball into two new balls which are identitical to the original one.

