Mapping ordinals to the reals
Seán O'Leathlóbhair
preserving maps to the non-negative real numbers. Here are some
examples. I will use w for omega.
All the finite ordinals can be mapped to the reals using the simple
map n -> n but that is not very exciting.
If we compound this map with x -> 1 - 1/(x + 1) then we have mapped
all the finite ordinals into the range [0,1). Now we can map w -> 1
and we have gone a little beyond the first trivial case.
It is easy to go a bit further. We can now map w, w + 1, w + 2, etc
into [1, 2) using a similar map as the finite ordinals into [0, 1). w.
2 naturally maps to 2, w.2 + 1, w.2 + 2 etc map into (2, 3). And the
scheme easily extends up to, but not including w^2.
We can go yet further by compounding again with the map x -> 1 - 1/(x
+ 1) and all the ordinals up to w^2 are mapped into [0,1) leaving 1
available for w^2. Using similar tricks, we can map ordinals up to
w^3 into [1, 2), up to w^4 into [2,3) etc.
So, now we have mapped up to, but not including w^w.
Obviously we can still go further by compounding with x -> 1 - 1/(x +
1).
Now comes the question: How far can we go?
1. Any particular countable ordinal? This would seem likely using
induction.
2. All countable ordinals? Not so obvious and even if it was possible
then probably not by my scheme. Maybe an existence proof is possible
even if a constructive one was not.
3. Uncountable ordinals? Well there are enough reals to hope to go
beyond the countable. I guess that there is no hope of a constructive
proof. If we assume AC then the non-negative reals can be well
ordered so that I guess that narrows down the answer. There is still
a little wriggle room since there are multiple ordinals with the same
cardinality but obviously I cannot go beyond the cardinality of the
reals.
--
Seán O'Leathlóbhair
G. A. Edgar
> 2. All countable ordinals? [...]
Yes.
More generally, any countable ordered set can be mapped in an
order-preserving way, into the rationals with the usual ordering.
>
> 3. Uncountable ordinals? [...]
No uncountable ordinal can be mapped (with order preserved) into the
reals.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Aatu Koskensilta
> In order to illustrate the ordinal numbers, I was considering order
> preserving maps to the non-negative real numbers.
>
Up to any countable ordinal.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Seán O'Leathlóbhair
wrote:
Thanks for the prompt reply.
> > 2. All countable ordinals? [...]
> Yes.
> More generally, any countable ordered set can be mapped in an
> order-preserving way, into the rationals with the usual ordering.
A good point about the rationals. All of my examples were maps to
just the rationals.
However, your answer seems to be to my question 1 rather than 2. For
2, I wanted a single map from all the countable ordinals to the reals
(or rationals). In other words, up to, but not including, the first
uncountable ordinal.
> > 3. Uncountable ordinals? [...]
>
> No uncountable ordinal can be mapped (with order preserved) into the
> reals.
Ah, I slipped there. Because (assuming AC), the reals could be well
ordered, I thought that would give a solution for the reals. Of
course, the mistake was forgetting this would be a very different
order to the usual one with no guarantee of an order preserving map to
the reals with the usual order.
You suggest that it can be proved that this is impossible. That
sounds vaguely familiar. Is it hard? Since, it sounds a little
familiar, it may be in one of my own books.
--
Seán O'Leathlóbhair
Arturo Magidin
=?iso-8859-1?B?U2XhbiBPJ0xlYXRobPNiaGFpcg==?= <jwla...@yahoo.com> wrote:
>On 6 Jun, 12:44, "G. A. Edgar" <e...@math.ohio-state.edu.invalid>
>wrote:
[.can one embed an ordinal into [0,oo) with an order preserving map?.]
>> > 3. Uncountable ordinals? [...]
>>
>> No uncountable ordinal can be mapped (with order preserved) into the
>> reals.
>
>Ah, I slipped there. Because (assuming AC), the reals could be well
>ordered, I thought that would give a solution for the reals. Of
>course, the mistake was forgetting this would be a very different
>order to the usual one with no guarantee of an order preserving map to
>the reals with the usual order.
>
>You suggest that it can be proved that this is impossible. That
>sounds vaguely familiar. Is it hard? Since, it sounds a little
>familiar, it may be in one of my own books.
It's easy. If a is an ordinal, and f:a->[0,oo) is an order preserving
map, then for every r in a such that r+1 is in a, there exists a
rational q_r such that f(r)<q_r<f(r+1).
If a were uncountable, then it would contain uncountably many
successor ordinals, which would in turn give you uncountable many
rationals.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
Seán O'Leathlóbhair
> In article <1181131297.852530.29...@h2g2000hsg.googlegroups.com>,
>
> =?iso-8859-1?B?U2XhbiBPJ0xlYXRobPNiaGFpcg==?= <jwlaw...@yahoo.com> wrote:
> >On 6 Jun, 12:44, "G. A. Edgar" <e...@math.ohio-state.edu.invalid>
> >wrote:
>
> [.can one embed an ordinal into [0,oo) with an order preserving map?.]
>
> >> > 3. Uncountable ordinals? [...]
>
> >> No uncountable ordinal can be mapped (with order preserved) into the
> >> reals.
>
> >Ah, I slipped there. Because (assuming AC), the reals could be well
> >ordered, I thought that would give a solution for the reals. Of
> >course, the mistake was forgetting this would be a very different
> >order to the usual one with no guarantee of an order preserving map to
> >the reals with the usual order.
>
> >You suggest that it can be proved that this is impossible. That
> >sounds vaguely familiar. Is it hard? Since, it sounds a little
> >familiar, it may be in one of my own books.
>
> It's easy. If a is an ordinal, and f:a->[0,oo) is an order preserving
> map, then for every r in a such that r+1 is in a, there exists a
> rational q_r such that f(r)<q_r<f(r+1).
>
> If a were uncountable, then it would contain uncountably many
> successor ordinals, which would in turn give you uncountable many
> rationals.
Thanks. As you say nice and easy. A useful result that there is a
rational between any two reals. I remember that being surprising when
I was first learned it. It means that we could not use the ability to
interleave sets to define cardinality. The bijection definition of
having the same cardinality is easy to motivate, but intuition from
finite sets would also suggest that if two sets could be interleaved
then they would have the same size.
--
Seán O'Leathlóbhair
Rainer Rosenthal
> Thanks. As you say nice and easy. A useful result that there is a
> rational between any two reals. I remember that being surprising when
> I was first learned it. It means that we could not use the ability to
> interleave sets to define cardinality. The bijection definition of
> having the same cardinality is easy to motivate, but intuition from
> finite sets would also suggest that if two sets could be interleaved
> then they would have the same size.
Well there *are* people (ask WM) who still stick to their first
surprise and ignore that there are ways of improving one's knowledge.
Sad but true.
Best regards,
Rainer
Seán O'Leathlóbhair
I used to enjoy teaching infinite cardinal numbers. First, I would
show that various familiar sets, e.g. the natural numbers, the
integers, the rational numbers, etc are the "same size". When the
students finally got comfortable with that they would often start to
imagine that all infinite sets were the same size and it was time to
show that the real numbers are bigger. Some would get it immediately
but others would struggle for quite a while. Finally to really
confuse them, I could mention the continuum hypothesis.
--
Seán O'Leathlóbhair
Dave L. Renfro
> I used to enjoy teaching infinite cardinal numbers.
> First, I would show that various familiar sets,
> e.g. the natural numbers, the integers, the rational
> numbers, etc are the "same size". When the students
> finally got comfortable with that they would often
> start to imagine that all infinite sets were the
> same size and it was time to show that the real
> numbers are bigger. Some would get it immediately
> but others would struggle for quite a while. Finally
> to really confuse them, I could mention the continuum
> hypothesis.
To really shock them, first show them the diagonal
proof that card(B) < card(P(B)), where B is any set
and P(B) is the collection of all subsets of B.
Then indicate how you can iterate this any finitely
many times to the natural numbers to get the cardinal
beth_n for each positive integer n. Let beth_w be the
cardinality of the union of all these finite iterates
of the power set operation applied to the natural
numbers, and show why beth_w > beth_n for each positive
integer n. Discuss the countable ordinal numbers, such
as w^w, epsilon_0, epsilon_w, gamma_0, (w_1)^CK, etc.
Indicate how one gets beth_w^w, beth_gamma_0, etc.
Then get to beth_w_1, the w_1'th iterate of the power
set operation applied to the natural numbers.
Then get to cardinals like beth_w_w, beth_beth_w^w,
beth_beth_beth_3w, etc. Finally, show them cardinals
(identified as initial ordinals) d such that beth_d = d.
And if you have time, mention that there are cardinals
e such that e is the e'th cardinal (in their usual order)
such that beth_d = d.
Dave L. Renfro
Seán O'Leathlóbhair
I would normally do the power set proof to show that there was no
biggest cardinal. The syllabus would peter out about that point. I
don't recall much coverage of ordinals which is maybe why I need to
revise them a bit.
Do you feel that the large sets you just mentioned are more shocking
than the undecidability (*) of CH? I still rank CH as one of the most
surprising things ever (in a maths context anyway). I guess that I
have a tendency to be a Platonist so I feel that CH should be
decidable and hence something must be wrong with our axioms. CH was
more surprising to me than say Banach-Tarski but maybe only because I
encountered CH much earlier.
(*) The spell checker did not like "undecidability" and wanted to
change it to "undesirability". I am not sure whether I agree. Would
I like CH to be true or false? I don't know.
--
Seán O'Leathlóbhair
Dave L. Renfro
> I would normally do the power set proof to show that there
> was no biggest cardinal. The syllabus would peter out about
> that point. I don't recall much coverage of ordinals which
> is maybe why I need to revise them a bit.
>
> Do you feel that the large sets you just mentioned are more
> shocking than the undecidability (*) of CH? I still rank CH
> as one of the most surprising things ever (in a maths context
> anyway). I guess that I have a tendency to be a Platonist
> so I feel that CH should be decidable and hence something
> must be wrong with our axioms. CH was more surprising to me
> than say Banach-Tarski but maybe only because I encountered
> CH much earlier.
In place of the ordinals (which, for an elementary class,
is probably not reasonable to bring up), you could mention
that not only is there no highest cardinality (this only
tells us there are infinitely many different cardinal
numbers) but, in fact, for each cardinal number b, there
are MORE THAN b many cardinal numbers. This is
something I used to wonder about a lot when I was young
(which was well before the internet; now, anyone just
google for the information), namely whether there is
actually an uncountable number of different infinities.
All the books I knew about when I was in middle school
(1971-73; books by authors such as Isaac Asimov, George
Gamov, and Irving Adler, which I could find at the public
library where I lived) either said there was more than
one type of infinity or they said there was an unlimited
number of infinities. I was puzzled as to why, after
talking about how some infinite sets are uncountable,
the author didn't mention whether the infinitely many
kinds of infinity was an uncountable infinitely many
or just a countable infinitely many (or tell us that
no one knew the answer, if that was the case). Even today,
it seems to me that this is a natural question, but I
almost never see an elementary discussion of cardinal
numbers in which this question (are there uncountably
many different infinities) is brought up.
I'm not sure if I'm surprised that CH is independent
or not, so I guess (by default) I'm not. And I probably
knew about CH being independent (if not what this actually
meant) fairly young, age 14 or 15. The Banach-Tarski
paradox never bothered me all that much either, because
I've seen so many wild things that can happen when
non-measurable sets and functions are involved. For
example, there exists a pairwise disjoint collection
of continuum many (not just 2, not just 58, not just
infinitely many, not just aleph_1 many, but continuum
many) subsets of [0,1], each having outer measure 1.
I find this quite a bit more exotic than the Banach-Tarski
paradox. Luzin and Sierpinski proved this result about
the continuum many sets in 1917, by the way, and the paper
is on the internet at
ftp://ftp.bnf.fr/000/N0003118_PDF_422_424.pdf
Dave L. Renfro
Rainer Rosenthal
> I'm not sure if I'm surprised that CH is independent
> or not, so I guess (by default) I'm not.
Independent of whether CH is true or not, this seems to
be a pretty nice remark :-)
Cheers,
Rainer Rosenthal
r.ros...@web.de
--
I'm not sure if I'm surprised that CH is independent or
not, so I guess (by default) I'm not. (Dave L. Renfro)
Seán O'Leathlóbhair
I haven't taught any of this for decades and I am not likely to teach
it again. So, this is just for personal amusement.
It is a pity that I did not have that suggestion when I was teaching
it. It would probably have been a bit beyond the syllabus but not
very much and would probably have been interesting and comprehensible
to some at least. I always like to go a bit further than necessary.
The response of a student to this was quite revealing. Some would not
like to go an inch beyond the syllabus, if it was not going to come up
in a paper, they did not want to know. Others would be happy to
wander far beyond the syllabus if I captured their interest. You can
probably guess which group got the best final results. Was it the
efficient ones who did not waste time learning stuff beyond the
syllabus?
Does "not just aleph_1 many, but continuum many" mean that you think
CH is false or just hedging your bets?
--
Seán O'Leathlóbhair
Dave L. Renfro
> It is a pity that I did not have that suggestion when
> I was teaching it. It would probably have been a bit
> beyond the syllabus but not very much and would probably
> have been interesting and comprehensible to some at least.
I'm not saying that it has to be proved. One could simply
point out that there are uncountably many different infinities
(cardinal numbers). It's the absence of even just pointing
this out that I was mostly complaining about. In books,
especially semi-popular treatments, it would only take
one sentence or a footnote. In an undergraduate "transition
to advanced mathematics" course, it would only take 20 seconds
(maybe let this be the closing statement for one class lecture).
> Does "not just aleph_1 many, but continuum many" mean
> that you think CH is false or just hedging your bets?
What I mean is that their construction provably gives continuum
many (regardless of the status of CH), unlike some constructions
that are provably aleph_1 many. For example, the number of alternate
iterations of the operations "complement" and "countable union"
needed to generate the smallest sigma-algebra from an arbitrarily
specified collection of subsets of the reals (take unions at the
limit ordinals) is provably aleph_1. Also, the number of iterations
of the pointwise limit operation (take pointwise limits of the
union of the previous collections at limit ordinals) needed to
obtain the Baire functions (smallest collection of functions
f:R --> R closed under pointwise limits and containing the
continuous functions) from the continuous functions is provably
aleph_1. Or, every analytic set is the union of some collection
of at most aleph_1 many Borel sets. In this last case, if
aleph_1 = c, then the result is trivial, since any subset of
the reals is a union of at most c many singleton sets.
I'm not especially interested in the various areas of study in
which "cardinal characteristics of the continuum" (google the
phrase if you're interested) are used to provide more precisely
stated results than aleph_1 many or continuum many (although when
I'm aware of something of this nature in something I'm doing,
I'll usually make a note of it), but in the case of "uncountable"
and "cardinality c", I'll definitely make note of whether
"uncountable" or "cardinality c" applies, and state the
result for the version that gives the stronger statement
relative to what I'm looking at. For example, in the hypothesis
of a statement, "uncountable" gives rise to a stronger
statement, and in the conclusion of a statement, "cardinality c"
(or "aleph_1 many") gives a stronger statement [1]. Of course,
you may not be able to prove the statement for the stronger
version, in which case you don't assert the stronger version.
[1] Unless the conclusion is "fewer than cardinality c", in which
case "fewer than uncountable" gives a stronger statement,
so it'd probably be best to say each statement needs to be
looked at individually, rather than my trying to give
some meta-mathematical guide-lines.
Dave L. Renfro
Seán O'Leathlóbhair
> Seán O'Leathlóbhair wrote (in part):
>
> > It is a pity that I did not have that suggestion when
> > I was teaching it. It would probably have been a bit
> > beyond the syllabus but not very much and would probably
> > have been interesting and comprehensible to some at least.
>
> I'm not saying that it has to be proved. One could simply
> point out that there are uncountably many different infinities
> (cardinal numbers). It's the absence of even just pointing
> this out that I was mostly complaining about. In books,
> especially semi-popular treatments, it would only take
> one sentence or a footnote. In an undergraduate "transition
> to advanced mathematics" course, it would only take 20 seconds
> (maybe let this be the closing statement for one class lecture).
I understood that. In the same way, I would mention CH but not give
any proofs. If I come to teach this again, I will bring the subject
up. Maybe start by asking the students what they expected. Ask at
the end of one class and see what they think in the next class.
I have been trying to remember when I learnt quite how many infinities
there were and whether I ever wondered whether or not there were
countably many. I have failed. I can remember first learning that N,
Z, Q etc were the same size, that R was bigger, that there was no
biggest, but I cannot remember when I learnt more or what I thought
about the subject in the meantime.
> > Does "not just aleph_1 many, but continuum many" mean
> > that you think CH is false or just hedging your bets?
>
> What I mean is that their construction provably gives continuum
> many (regardless of the status of CH), unlike some constructions
> that are provably aleph_1 many. For example, the number of alternate
> iterations of the operations "complement" and "countable union"
> needed to generate the smallest sigma-algebra from an arbitrarily
> specified collection of subsets of the reals (take unions at the
> limit ordinals) is provably aleph_1. Also, the number of iterations
> of the pointwise limit operation (take pointwise limits of the
> union of the previous collections at limit ordinals) needed to
> obtain the Baire functions (smallest collection of functions
> f:R --> R closed under pointwise limits and containing the
> continuous functions) from the continuous functions is provably
> aleph_1. Or, every analytic set is the union of some collection
> of at most aleph_1 many Borel sets. In this last case, if
> aleph_1 = c, then the result is trivial, since any subset of
> the reals is a union of at most c many singleton sets.
I see, a nice distinction.
> I'm not especially interested in the various areas of study in
> which "cardinal characteristics of the continuum" (google the
> phrase if you're interested) are used to provide more precisely
> stated results than aleph_1 many or continuum many (although when
> I'm aware of something of this nature in something I'm doing,
> I'll usually make a note of it), but in the case of "uncountable"
> and "cardinality c", I'll definitely make note of whether
> "uncountable" or "cardinality c" applies, and state the
> result for the version that gives the stronger statement
> relative to what I'm looking at. For example, in the hypothesis
> of a statement, "uncountable" gives rise to a stronger
> statement, and in the conclusion of a statement, "cardinality c"
> (or "aleph_1 many") gives a stronger statement [1]. Of course,
> you may not be able to prove the statement for the stronger
> version, in which case you don't assert the stronger version.
I will watch for that in future.
> [1] Unless the conclusion is "fewer than cardinality c", in which
> case "fewer than uncountable" gives a stronger statement,
> so it'd probably be best to say each statement needs to be
> looked at individually, rather than my trying to give
> some meta-mathematical guide-lines.
I hope that it would be fairly obvious which choice gives the stronger
statement in a particular case. Have you noticed how common this care
is? In other words, have you often seen claims that are weaker than
they could have been? E.g. a claim that something is merely
uncountable when the proof would have supported a conclusion of
continuum many.
--
Seán O'Leathlóbhair
Dave L. Renfro
> I have been trying to remember when I learnt quite
> how many infinities there were and whether I ever
> wondered whether or not there were countably many.
> I have failed. I can remember first learning that
> N, Z, Q etc were the same size, that R was bigger,
> that there was no biggest, but I cannot remember
> when I learnt more or what I thought about the
> subject in the meantime.
For me this was an obvious question from the two
main (only?) treatments of cardinal numbers that
I knew about and had read when I first learned
about the idea (6'th or 7'th grade, back around
1970 or 1971). There were two paperback books that
I had gotten around this time, during one of my
family's shopping trips to Charlotte, NC (about 40
miles away from where I then lived), that had a few
pages about cardinal numbers in them. [The books were
purchased at something called a "Candy Kitchen",
a combination book/magazine/coffee/breakfast/candy
store, a type of store that disappeared from the
entire region within the next 5 to 8 years.]
One was Isaac Asimov's "Realm of Numbers". The other
was George Gamov's "One, Two, Three, ..., Infinity".
I still have both of them, too. Asimov's book is
the first one that I got and read.
At the very end of Asimov's book there are two or
three pages where he discusses countability of the
rationals and uncountability of the reals. At one point
Asimov says that the cardinal numbers are aleph_0,
aleph_1, etc. (I don't remember if he mistakenly
identifies aleph_1 with the continuum.) Then he says
that a fitting way to end the book is not "the end",
but as "aleph_infinity" (the symbols, in large type size;
instead of oo for "infinity", he might have used "omega",
I don't remember now).
It was natural to wonder, and I did, whether there were
just countably many infinities or uncountably many
infinities when I saw this.
At the end of the first (first two?) chapter(s) of
Gamov's book, Gamov discusses how the real numbers
form a larger infinite set than the integers or the
rationals (and incorrectly labels this larger set
with aleph_1) and the set of curves in the plane forms
a still larger infinity (incorrectly identified with
aleph_2; also, this is not correct if "curve" means
continuous images, or even Borel measurable images),
but no one has been able to come up with a concrete
set of objects with more than aleph_2 many elements
in it. Thus, we find ourselves in much the same situation
as some primitive tribe that he began the book with,
a tribe that had no words for numbers greater than three
(I think; maybe it was "two" or "four", but I seem to
remember what I'm about to say, and the title fits better
if it was "three"). If anything, we're worse off because
we can't even get past "two". Within a couple of years,
maybe sooner, I realized that Gamov made a mistake with
the "aleph_1 = cardinality of the reals" identification
and also that all you have to do is form the power set
of the collection of curves to get a larger cardinality.
I don't remember if Gamov says that the cardinalities
continue, even though we don't have any examples of sets
with these cardinalities, but I think he did because
I seem to remember seeing/reading something about the
"endless progression of infinities" in Gamov's book
and wondering whether there were uncountably many of
them. Both authors seemed (in my memory now; I'm not
at home where I can look at the books to be sure) to
list the cardinal numbers, or imply that they can be
so listed, which suggested to me that there were only
countably many of them because, in all the discussion
prior to this, every set that was countable was shown
to be so by showing how its elements could be put into
a list. As I got older (mid to late high school), I began
wondering why this obvious question didn't occur to Asimov
and Gamov, or at least if it did, why they didn't bring
it up. I still wonder about this, because in both books
it seems to be an unavoidable question given how they
each dealt with the subject.
> I hope that it would be fairly obvious which choice gives
> the stronger statement in a particular case. Have you
> noticed how common this care is? In other words, have
> you often seen claims that are weaker than they could
> have been? E.g. a claim that something is merely
> uncountable when the proof would have supported a
> conclusion of continuum many.
The first time I became especially aware of the
"uncountable" vs. "continuum many" distinction was
in the early stages of my Ph.D. research (1991-92).
I was studying a lot of subtle issues in real analysis
and eventually wrote a dissertation about various
exotic non-differentiability properties of continuous
functions (properties that hold for larger than co-meager
subsets of the space C[0,1], more specifically), and
I found myself sometimes seeing a paper where someone
stated a theorem for "cardinality continuum", someone
later saying "uncountable" (Saks 1932 Fund. Math paper
"On the functions of Besicovitch in the space of
continuous functions" is a frequent example), or the
other way around (I don't have an example of this, but
I'm sure I've seen it), which lead to a lot of confusion
on my part when I was still new to non-textbook literature
and feeling a little unsure about myself on certain
subjects I didn't feel I knew well enough.
Eventually I came to realize that pretty much any
of the exceptional sets that one encounters in "basic"
real analysis were Borel, and for Borel sets, the
properties "uncountable" and "cardinality continuum"
are equivalent. However, this explanation doesn't
work for literature in the 1870's to 1910's period,
and thus some authors would prove "uncountable" and
then later someone would rework the proof a bit to
get "cardinality continuum". Even more often, I came
across proofs during this era when the author actually
proved "cardinality continuum" (because the set was
shown to contain a perfect set, or the set was shown
to be in 1-1 correspondence with another set known
to have cardinality continuum, or the set was shown
to be uncountable by a dyadic tree argument, etc.) but
only stated the result as "uncountable", and then I'd
see a several-decades-later paper in a (usually of
lesser prestige) journal where the later author claimed
to have strengthened the earlier author's result by
proving cardinality continuum (clearly having only read
a statement of the earlier result, maybe from a secondary
source such as a JFM review, but not the original author's
proof). An example that comes close to this is
"On properties of derivatives of continuous functions"
by Alexander Abian and Paula Kemp [Simon Stevin 66
(1992), 19-27]. I say "close", because I'm not sure
if the sets they prove to have cardinality continuum were
ever explicitly stated as such in the original literature
(rather than simply "uncountable"), but "cardinality
continuum" is immediate from the fact that all the sets
involved are Borel (at most level 2, in fact).
Another of my pet peeves, now that I'm on a roll, is
when people do not clearly distinguishing between second
category (or non-meager) and residual (or co-meager).
I realize that some authors at the turn of the century
used "second category" to mean "complement of first category",
but I'm not talking about them. I'm talking about people
in much more recent times. Another thing is a lack of
awareness of different notions of density, at least in
regard to ordinary density and c-dense in the reals
(cardinality c in every open interval). Sometimes I'll
see a theorem stated where something is shown to be
dense and uncountable, when almost no additional work
is needed to get c-dense in the reals, which is a
strengthening in both aspects ("c-dense" strengthens
"dense" and "c-dense" strengthens "uncountable").
Krishna M. Garg wrote several papers in the 1960's
on various precise notions of non-monotonic behavior
(among other things), and he's an excellent example of
someone who was extremely careful in stating maximally
strong statements, whether his own or those of others,
all the while carefully distinguishing between what
an earlier author stated in a theorem, what an earlier
author clearly _proved_ when proving the theorem, what
an earlier author "essentially proved" (almost no extra
work needed), and the like.
Dave L. Renfro
Seán O'Leathlóbhair
> Seán O'Leathlóbhair wrote:
> > I have been trying to remember when I learnt quite
> > how many infinities there were and whether I ever
> > wondered whether or not there were countably many.
> > I have failed. I can remember first learning that
> > N, Z, Q etc were the same size, that R was bigger,
> > that there was no biggest, but I cannot remember
> > when I learnt more or what I thought about the
> > subject in the meantime.
>
> For me this was an obvious question from the two
> main (only?) treatments of cardinal numbers that
> I knew about and had read when I first learned
> about the idea (6'th or 7'th grade, back around
> 1970 or 1971).
I am not quite sure how your grades relate to our school years but it
seems that I am a little older than you but you discovered this
subject at a rather younger age than me. Even though my knowledge of
math was fairly advanced in school (e.g. calculus and group theory),
my knowledge of infinity was still quite naive until I went to
university at 18. Even then, I learnt about infinite cardinals quite
some time before I learnt much about infinite ordinals. Not knowing
of uncountable ordinals, may be why I did not wonder if the sequence
aleph_0, aleph_1 etc reached aleph_uncountable.
> There were two paperback books that
> I had gotten around this time, during one of my
> family's shopping trips to Charlotte, NC (about 40
> miles away from where I then lived), that had a few
> pages about cardinal numbers in them. [The books were
> purchased at something called a "Candy Kitchen",
> a combination book/magazine/coffee/breakfast/candy
> store, a type of store that disappeared from the
> entire region within the next 5 to 8 years.]
I have never heard of a Candy Kitchen before, not even in novels set
in the US. I don't recall anything quite like that here (UK) but
maybe they existed in some odd corner that I have not visited. The
closest that we have (had) is probably "The Corner Shop". These are
so called since they are often in towns at the end of a row of houses
and hence at a corner (intersection). However, they are usually
called "corner shops" even if they are not at a corner. They are
typically small family run businesses in ordinary houses. They
concentrate on food but often have a diverse range of other goods.
Newspapers and magazines are common but I don't think that books are.
A few of these corner shops remain but they are dying rapidly in an
era of large out of town stores and high car ownership.
> One was Isaac Asimov's "Realm of Numbers". The other
> was George Gamov's "One, Two, Three, ..., Infinity".
>
> I still have both of them, too. Asimov's book is
> the first one that I got and read.
I have heard of that Gamov book but not read it. I have read a few of
Asimov's non-fiction books but I did not know that he wrote any on
mathematics.
Thanks very much for all that. I will need to do a little revision to
fully understand the last paragraph.
--
Seán O'Leathlóbhair
Dave L. Renfro
> I have heard of that Gamov book but not read it.
> I have read a few of Asimov's non-fiction books
> but I did not know that he wrote any on mathematics.
Asimov wrote several books on (elementary) mathematics,
even a book on using the slide rule, I believe. In general,
if a topic is generally known to the public (slide rule,
the planet Jupiter, limericks, the bible, etc.), there's
a pretty good chance that Asimov has written a book about it.
Dave L. Renfro
Dave L. Renfro
> What I mean is that their construction provably gives continuum
> many (regardless of the status of CH), unlike some constructions
> that are provably aleph_1 many. For example, the number of alternate
> iterations of the operations "complement" and "countable union"
> needed to generate the smallest sigma-algebra from an arbitrarily
> specified collection of subsets of the reals (take unions at the
> limit ordinals) is provably aleph_1. Also, the number of iterations
> of the pointwise limit operation (take pointwise limits of the
> union of the previous collections at limit ordinals) needed to
> obtain the Baire functions (smallest collection of functions
> f:R --> R closed under pointwise limits and containing the
> continuous functions) from the continuous functions is provably
> aleph_1. Or, every analytic set is the union of some collection
> of at most aleph_1 many Borel sets. In this last case, if
> aleph_1 = c, then the result is trivial, since any subset of
> the reals is a union of at most c many singleton sets.
I recently came across another example that dates from the
turn of the century (19'th to 20'th), one that fits with the
topics of several recent sci.math threads. There are aleph_1
many countable well order types [Cantor, early 1880's], but
c many countable linear order types [Cantor, 1884 or 1885 for
"at least c many"; F. Bernstein, 1901 Dissertation under Hilbert
and (essentially, I think) Cantor) for "at most c many"].
Dave L. Renfro