In article ,
David Madore wrote:

John Baez in litteris scripsit:

It's easy to map the ordinal omega^2 into the real numbers
in a one-to-one and order-preserving way. Here's an artist's
conception, which uses the second dimension to make things
easier to see:

http://math.ucr.edu/home/baez/omega_squared.png

Thanks for calling me an artist :-) but I don't think I deserve the
title. I created that image for Wikipedia [....]

Thanks! I didn't check to see who made it. The phrase "artist's
conception" was intended as a slight joke, since in pop science
magazines one often reads things like "here is an artist's conception
of romance among australopithecines" adorning pictures that required
a lot of imagination to draw - but this time, it was actually a
mathematically precise picture!

Which ordinals can we do this for?

If you're asking which ordinals are order-isomorphic to a subset of
the real numbers, the answer is simple (at least, assuming the axiom
of choice): exactly the countable ordinals.

Yay! Great! That's exactly what I was asking.

I had produced a graphical representation of epsilon_0, once, but it's
actually entirely uninteresting to look at, it's just a mess.

If you still have it around, I would be interested to see it - and maybe
even attach it to week236. I can see why it would be a mess, though.

I suppose drawing it bigger wouldn't help, but it might be fun to
take some large ordinal and draw it in your style on the scale of
this artist's conception of a hydrogen atom:

http://www.phrenopolis.com/perspective/atom/index.html

This may be the world's biggest webpage: it's 18 kilometers wide!
(That's 50 million pixels at 72 pixels per inch.)

I hadn't known my webbrowser could scroll that far. My wrist didn't
even get tired. So, it might be possible to draw omega^omega or
something and have it look interesting, even if epsilon_0 is too big.

In article ,
wrote:

For that, we'd need a generalization of finite sets whose cardinality
can be be complex.

Has anyone since done anything with the idea
http://groups.google.co.uk/group/sci...d3ff8198196ace
that the set of Motzkin paths has cardinality i?

Maybe you meant Motzkin *trees*. In case anyone is wondering,
these are rooted planar trees where each node has one or two
daughter nodes. The set M of Motzkin trees is equipped with
an obvious isomorphism

M = 1 + M + M^2

since every Motzkin tree is either a one-node tree, a node
connected by an edge to another Motzkin tree, or a node connected
by two edges to two Motzkin trees.

Using the techniques of Schanuel, Gates, Leinster and Fiore,
the "generalized cardinality" |M| of the set of Motzkin trees
satisfies

|M| = 1 + |M| + |M|^2

so

|M| = +-i

This sort of reasoning seems completely insane at first, but it
leads to many valid and interesting results; for details see

http://math.ucr.edu/home/baez/week202.html

ANYWAY:

Jeff Morton and I put a lot of work into this idea when we were
trying to categorify the quantum harmonic oscillator. The Motzkin
trees are a categorification of the Gaussian integers; the
"+-i" hints that Galois theory is relevant. We figured out how
to categorify the algebraic integers in any algebraic extension of
the rationals, getting an "algebraic extension" of the category
of finite sets. We figured out the beginnings of a theory that
associates a "Galois 2-group" to any such algebraic extension.
I was pretty excited about this, but Jeff was eager to reach ideas
connected to physics, and this seemed like a long way around.
In particular, one needs not just algebraic numbers but also
transcendentals to make sense of the "exp(-itH)" in quantum mechanics.

So, we dropped this project and came up with a much simpler
category of "U(1)-sets" whose "cardinalities" are complex:
a U(1)-set is simply a finite sets of points ("quanta") labelled
by phases. Here we are putting the phases in "by hand" instead
of seeing them emerge from category-theoretic considerations.
This is a bit unfortunate, but the advantage is that everything
works quite quickly and smoothly, and there's a clear physical
meaning to it all.

If anyone who knows categories, combinatorics and Galois theory
wants to become a math grad student at UCR and work on the project
Jeff and I dropped, they should contact me.

In article , wrote:

In article ,
John Baez wrote:

Logicians [...] know a lot about how much induction
up to different ordinals buys you. And apparently, induction up to
Gamma_0 lets us prove the consistency of a system called "predicative
analysis". I don't understand this, nor do I understand the claim
I've seen that Gamma_0 is the first ordinal that cannot be defined
predicatively - i.e., can't be defined without reference to itself.
Sure, saying Gamma_0 is the first solution of

phi_x(0) = x

is non-predicative. But what about saying that Gamma_0 is the union
of all ordinals in the Veblen hierarchy? What's non-predicative
about that?

The situation is somewhat akin to the situation with the Church-Turing
thesis, in that one is tentatively equating an informal notion
(predicativity or computability) with a precise mathematical notion.
Therefore there is no definitive answer to your question, and Feferman
himself has articulated potential objections to the "standard view"
that Gamma_0 marks the boundary of predicativity.

There's also someone named Nik Weaver who has debated Feferman
on this subject:

http://www.cs.nyu.edu/pipermail/fom/...il/010472.html
http://www.math.wustl.edu/~nweaver/conceptualism.html

He seems to claim that Gamma_0 and even larger ordinals have predicative
definitions. However, I'm too ignorant to follow this debate.
Usually in physics I have a sense for when people are being reasonable
even if I don't follow the details. In this debate I can't even
do that.

Having said that, I'll also say that one of the reasons for the standard
view is that Gamma_0 marks the boundary of "autonomous progressions" of
arithmetical theories. The book by Torkel Franzen that you cited is
probably the most accessible introduction to this subject.

This summer I'm in Shanghai without any academic affiliation, so it's
hard to get that book. When I return to Riverside in the fall I'll
try to read it. But my curiosity is burning right now, so I'll take
the liberty of asking some more questions.

Roughly
speaking, the idea is that if anyone fully accepts first-order Peano
arithmetic PA, then implicitly he accepts its consistency Con(PA), as
well as Con(PA+Con(PA)), etc.

I assume that by "etcetera" you mean there's one theory like this
per ordinal. I browsed a paper by Franzen where he was trying
to explicate how these theories actually let you prove interesting
new stuff.

It's a bit mysterious: I imagine a guy sitting there thinking
"Peano arithmetic is true, so I know it's consistent, and I know
*that's* consistent too, and I know *that's* consistent...", and
so on - and after pondering this way for an transfinite amount of time,
all of a sudden he can do new stuff like prove that Goodstein
sequences approach zero!

I think Franzen was trying to dispel this naive conception.
He said the real action happens at limit ordinals, where
the interpretation of everything changes in some sneaky way.

But, my understanding of his comments like an impressionist
painting of a surreal painting - Dali's "Sacrament of the Last
Supper" as reworked by Monet.

(Hey, I managed to sneak a docahedron into the discussion!)

If one tries to articulate exactly what
is "implicitly" involved in accepting PA in this sense, then one can
make a plausibility argument that Gamma_0 is a natural stopping point.

It would be really great if you could say more about this
plausibility argument.

I think you have a better shot at grasping the underlying intuition via
this approach than by staring at Gamma_0 itself and trying to figure out
what is non-predicative about its definition.

Okay, I won't try to do that.

In article ,
Jim Heckman wrote:

On 26-Jul-2006, (John Baez)
wrote in message :

But as you might have suspected, not *all* ordinals can be written
in this way. For one thing, every ordinal we've reached so far is
*countable*: as a set you can put it in one-to-one correspondence
with the integers. There are much bigger *uncountable* ordinals -
at least if you believe you can well-order uncountable sets.

? Is that last a reference to the Well-Ordering Theorem (equivalent
in ZFC to the Axiom of Choice)? Of course, you do need the WOT to
prove that /every/ set can be well-ordered, but ZF alone proves the
existence of uncountable ordinals.

That's interesting; I don't know if I ever knew that! The last
time I really studied axiomatic set theory was decades ago.

Anyway, I can easily imagine reasonable people who are comfy up
to omega or epsilon_0 (say) but don't believe you can well-order
any uncountable sets. So, I didn't want to get into a fight by
claiming bluntly that there *are* uncountable ordinals, without
any sort of caveat. I didn't want to be advocating ZFC - but now
that you bring it up, I don't even want to be advocating ZF.

But, I don't want to argue *against* them, either.

In fact, these days to get my back up you'd need to take a fairly
drastic position, like my friend Henry Flynt, who argues that
"mathematical knowledge amounts to the crystallization of officially
endorsed delusions in an intellectual quicksand":

http://www.henryflynt.org/studies_sci/mathsci.html

In article ,
John Baez wrote:

At first these numbers seem to keep getting bigger! So, it seems
shocking at first that they eventually reach zero. For example,
if you start with the number 4, you get this Goodstein sequence:

4, 26, 41, 60, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...

and apparently it takes about 3 x 10^{60605351} steps to reach zero!

Kevin Buzzard pointed out a typo here. The sequence is:

4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...

Also, while I got the huge number above from this website:

http://curvebank.calstatela.edu/goodstein/goodstein.htm

he pointed out they actually say the sequence "can increase for
approximately 2.6 * 10^{60605351} steps", not that it reaches
zero at this point.

Kevin then worked out the details himself, and I checked his
calculations. We now seem to agree that the sequence reaches
zero at the kth term, where

k = 24 * 2^24 * 2^{24 * 2^{24}} - 2

or approximately

k = 6.9 * 10^{121210694}

Please check and see if we've done it right.

You may also enjoy trying to figure out what the folks at the
National Curve Bank meant, and whether *they* were right.

Here is Kevin's email, prettied up by me, but perhaps with some
mistakes added:

apparently it takes about 3 x 10^{60605351} steps to reach zero!

You write this as if it were some kind of mystery. I remember working
out this number explicitly when I was a graduate student! There is
some nice form for it, as I recall. Let's see if I can reconstruct
what I did.

If I've understood the sequence correctly, it should be (where "n)"
at the beginning of a line denotes we're working in base n on this
line, so strictly speaking it's probably the n-1st term in the sequence)

2) 2^2 = 4
3) 3^3-1 = 2.3^2+2.3+2 = 26 [note: base 3, ends in 2, and 3+2=5]
4) 2.4^2+2.4+1 = 41 [note: base 4, ends in 1, and 4+1=5]
5) 2.5^2+2.5 = 60 [we're at a limit ordinal here, note 3+2=4+1=5]
6) 2.6^2+2.6-1 = 2.6^2+6+5 = 83 [note: base 6, ends in 5]
7) 2.7^2+7+4 [note: base 7, ends in 4]
8) 2.8^2+8+3 [note: base 8, ends in 3, so we next get a limit ordinal at...]

In article ,
John Baez wrote:
I assume that by "etcetera" you mean there's one theory like this
per ordinal.
[...]
If one tries to articulate exactly what
is "implicitly" involved in accepting PA in this sense, then one can
make a plausibility argument that Gamma_0 is a natural stopping point.

It would be really great if you could say more about this
plausibility argument.

Let's look more closely at what the notion of "one theory like this
per ordinal" means. There's no difficulty figuring out what "Con(PA)"
means or how to express that statement in the first-order language
of arithmetic. Ditto with "Con(PA+Con(PA))". However, once you start
ascending the ordinal hierarchy, a difficulty appears. The language
of arithmetic doesn't let you talk about "ordinals" directly---that's a
set-theoretical concept. In order to express a statement like "Con(T)"
for some theory T, you need at minimum to be able to give some sort of
"recursive description" or "recursive axiomatization" of T (where here
I use the word "recursive" in the technical sense of recursive function
theory) in the first-order language of arithmetic. This observation
already yields the intuition that we're not going to be able to ascend
beyond the Church-Kleene ordinal, because we won't even be able to
figure out how to *say* "T is consistent" for a theory T that requires
that many iterations to reach from PA.

There are other problems, though, that potentially get in the way before
we reach the Church-Kleene ordinal. Once we realize that what we need is
a system of "ordinal notations" to "fake" the relevant set theory, we may
(if we are predicavists) worry about issues such as:

1. As we ascend the ordinal hierarchy, isn't it illegitimate to make a jump
to an ordinal alpha unless we've already proved, at the level of some
ordinal beta that we've already reached, that an ordinal of type alpha
exists?

2. And isn't it illegitimate to create sets by quantification over things
other than the natural numbers themselves and sets that we've already
created?

Condition 1 goes by the name of "autonomy" and condition 2 goes by the name
of "ramification." If one formalizes these notions in a certain plausible
manner, then one arrives at Gamma_0 as the least upper bound of theories
that you can get to, starting with (for example) PA.

One can of course wonder whether 1 and 2 above really capture the concept
of "predicativity." Some secondary evidence has accumulated of the
following form: Some argument that intuitively seems to be predicative but
that is not immediately seen to be provable in the Feferman-Schuette
framework is shown, after some work, to indeed be provable below Gamma_0.

It's still possible, of course, for someone---you mentioned Nik Weaver---to
come along and argue that our intuitive notion of predicativism, fuzzy
though it is, can't possibly be identified with the level Gamma_0. The
reason you can't seem to decide immediately whether Weaver's position is
nonsensical or not is probably because the critical questions are not
mathematical but philosophical, and of course it's usually harder to arrive
at definitive answers in philosophy than in mathematics.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

John Baez in litteris scripsit:
In article ,
David Madore wrote:
I had produced a graphical representation of epsilon_0, once, but it's
actually entirely uninteresting to look at, it's just a mess.

If you still have it around, I would be interested to see it - and maybe
even attach it to week236. I can see why it would be a mess, though.

Try URL: http://www.madore.org/~david/.tmp/eps0-2.ps.gz (gzipped
PostScript file). And if you replace ".ps.gz" by ".c" you have the
(pretty much unreadable) C program which might have been used to
generate it... except that it doesn't seem to generate exactly the
same thing, so I don't really know (I presume some parameter was set
to a different value). As the URL indicates, these files might not
stay long, but you're welcome to do what you wish with them.

--
David A. Madore
,
http://www.madore.org/~david/ )

Maybe you meant Motzkin *trees*.

No, I did mean paths:

"A001006 Motzkin numbers: number of ways of drawing any number of
nonintersecting chords among n points on a circle.

Also number of Motzkin n-paths: paths from (0,0) to (n,0) in an n X n
grid using only steps U = (1,1), F = (1,0) and D = (1,-1)."

http://www.research.att.com/~njas/sequences/A001006

But of course there are loads of combinatorial interpretations,
including the trees you mention (which was actually the way I found the
link between Leinster and Fiore's construction and the Motzkin
numbers.)

If anyone who knows categories, combinatorics and Galois theory
wants to become a math grad student at UCR and work on the project
Jeff and I dropped, they should contact me.

Now, that's an excellent offer.

On 29-Jul-2006, (John Baez)
wrote in message :

In article ,
Jim Heckman wrote:

On 26-Jul-2006, (John Baez)
wrote in message :

But as you might have suspected, not *all* ordinals can be written
in this way. For one thing, every ordinal we've reached so far is
*countable*: as a set you can put it in one-to-one correspondence
with the integers. There are much bigger *uncountable* ordinals -
at least if you believe you can well-order uncountable sets.

? Is that last a reference to the Well-Ordering Theorem (equivalent
in ZFC to the Axiom of Choice)? Of course, you do need the WOT to
prove that /every/ set can be well-ordered, but ZF alone proves the
existence of uncountable ordinals.

That's interesting; I don't know if I ever knew that! The last
time I really studied axiomatic set theory was decades ago.

Anyway, I can easily imagine reasonable people who are comfy up
to omega or epsilon_0 (say) but don't believe you can well-order
any uncountable sets. So, I didn't want to get into a fight by
claiming bluntly that there *are* uncountable ordinals, without
any sort of caveat. I didn't want to be advocating ZFC - but now
that you bring it up, I don't even want to be advocating ZF.

But, I don't want to argue *against* them, either.

OK, but I'd be interested to know which ZF axioms your "imagine[d]
reasonable people" don't believe. Or is their problem with
mathematical logic?

[...]

--
Jim Heckman

In article ,
David Madore wrote:

John Baez in litteris scripsit:

David Madore wrote:

I had produced a graphical representation of epsilon_0, once, but it's
actually entirely uninteresting to look at, it's just a mess.

If you still have it around, I would be interested to see it - and maybe
even attach it to week236.

Try URL: http://www.madore.org/~david/.tmp/eps0-2.ps.gz (gzipped
PostScript file). And if you replace ".ps.gz" by ".c" you have the
(pretty much unreadable) C program which might have been used to
generate it... except that it doesn't seem to generate exactly the
same thing, so I don't really know (I presume some parameter was set
to a different value). As the URL indicates, these files might not
stay long, but you're welcome to do what you wish with them.

Thanks. Could it be that this PostScript file shows a picture,
not of epsilon_0, but of omega^omega?

There's a countable sequence of really big lines. The first one
is clearly 0. The second is clearly omega. It looks to me like
the third could be omega^2... and so on.

Hmm, but maybe the third really big line is omega^omega.
Clearly if you're drawing epsilon_0 we should see big lines for omega,
omega^omega, omega^omega^omega and so on.

Between the second and third really big lines I see a countable
sequence of "pretty big" lines. I thought these were omega 2,
omega 3, and so on... leading up to omega^2. But now, looking more
carefully, I see some fine structure which suggests they could be
higher ordinals, perhaps leading up to omega^omega.

ANYWAY:

If any hacker out there creates nice pictures of omega^omega
and/or epsilon_0, I'll put them on my website. If you do
both and I think they're really nice, I'll also give you a
signed copy of the new (corrected) version of "Gauge Fields,
Knots and Gravity", as soon I can buy it from World Scientific
(I got my copy a while ago, so it should be coming out soon.)
Or, if you prefer, some other book of comparable price.

Fine print: if a bunch of people attempt this, I'll give prizes
for the one or two that look the best. To be cool, the picture
should be in David Madore's style:

http://math.ucr.edu/home/baez/omega_squared.png
http://www.madore.org/~david/.tmp/eps0-2.ps.gz

unless you can think of something better. The main problem
with the second picture above is that the fine structure gets
too small too fast, so it's hard to see what's going on.

To be *really* cool, the picture will be an impressively tall
or wide webpage, along these lines:

http://www.phrenopolis.com/perspective/atom/index.html

The world's largest webpage deserves to be a picture of infinity,
not a mere hydrogen atom. Why should physicists have all the fun?