Some while ago I made this inquiry, but I don’t recall it produced much
action, so I’ll try again.
The usual stated motivation for accepting AC, (which doesn’t seem to
help in deciding other matters like CH, though), is that of the so-called
"combinatorial notion" of sets. Godel seems to have been the first to
put this in words. Roughly, he said…
"We adopt the combinatorial notion of sets, whereby it is taken that
the collection of all subsets is unrestricted by any notion of rule
definability, but may range over all imaginable subsets."
[Apologies if I've got it badly wrong.] Now this sounds very fine, being
unrestrictive and so forth; till one takes a closer look at it. Then it
appears as though subsets are to be "things which might be there, even
though we can’t think straight away what might be a way of deciding
what members they may have". There is a clear suggestion that subsets
are in some sense, "indefinitely extendible", and that elements may be
added *at will*, i.e. without any pre-assigned method.
This notion appears quite clearly in many AC-using proofs, e.g. the proof
of the existence of a partition of R^3 into non-parallel lines.
Now it seems to me there is a definite "psychological" element in this
notion of subset. It’s not usually noticed, but it smells *very* strongly
of the intuition behind Brouwer’s notion of "Choice Sequence". For those,
one is to conceive of an indefinitely extendible, but never complete,
sequence of (say) integers. Now most mathematicians, including ZFC
platonists, reject choice sequences as a true mathematical idea, while
perhaps allowing that their theory may be a way of doing some odd form
of applied math – "psychological math". Most would not give ChSeqs even
that much time. I myself don’t like them because of the way they seem
to "build subjectivity into the system", something I often rail against,
whether in intuitionism, Bayesian statistics, or quantum theory.
Now most mathematicians reject choice sequences on these psychological
grounds, without noticing the very similar psychological nature of
the combinatorial notion of set; in particular as it applies to AC.
For example, in the partition proof referred to above, one observes that
*one can* at each stage add in another line that doesn’t intersect
the previous ones; and this is true, for various (discernable) stages.
But it then glibly assumes this can proceed indefinitely, so that new
lines can be added in *by free choice*, without any defining rule,
until a whole c-ordinal of them have been so added.
IMHO this indefinitely-extendible-choosing is almost identical to the
choice function notion, which most have completely rejected.
I would like to hear some defense of their view, and its apparent
inconsistency, by vigorous AC-supporters.
——————————————————————————-
Bill Taylor W.Tay…@math.canterbury.ac.nz
——————————————————————————-
Set theory is a shotgun marriage – between well-ordering and power-set.
The two parties get along OK; but they hardly seem made for each other.
——————————————————————————-
Bill Taylor <math…@math.canterbury.ac.nz> wrote in message
news:8pkir0$9n1$1@cantuc.canterbury.ac.nz…
| Some while ago I made this inquiry, but I don’t recall it produced much
| action, so I’ll try again.
|
<snip>
| put this in words. Roughly, he said…
|
<snip>
| [Apologies if I've got it badly wrong.]
Excuse me Bill,
but this is not a good start for a post, you did have
quite a while to think about (according to your first
sentence).
Apology would be given readily, if you had thought of
that all of a sudden. Like a flash everybody should share.
Sorry – no apology.
Rainer
———————————-
0000000100000000000
- Hide quoted text — Show quoted text -
Bill Taylor wrote:
> Some while ago I made this inquiry, but I don’t recall it produced much
> action, so I’ll try again.
> The usual stated motivation for accepting AC, (which doesn’t seem to
> help in deciding other matters like CH, though), is that of the so-called
> "combinatorial notion" of sets. Godel seems to have been the first to
> put this in words. Roughly, he said…
> "We adopt the combinatorial notion of sets, whereby it is taken that
> the collection of all subsets is unrestricted by any notion of rule
> definability, but may range over all imaginable subsets."
> [Apologies if I've got it badly wrong.] Now this sounds very fine, being
> unrestrictive and so forth; till one takes a closer look at it. Then it
> appears as though subsets are to be "things which might be there, even
> though we can’t think straight away what might be a way of deciding
> what members they may have". There is a clear suggestion that subsets
> are in some sense, "indefinitely extendible", and that elements may be
> added *at will*, i.e. without any pre-assigned method.
> This notion appears quite clearly in many AC-using proofs, e.g. the proof
> of the existence of a partition of R^3 into non-parallel lines.
> Now it seems to me there is a definite "psychological" element in this
> notion of subset. It’s not usually noticed, but it smells *very* strongly
> of the intuition behind Brouwer’s notion of "Choice Sequence". For those,
> one is to conceive of an indefinitely extendible, but never complete,
> sequence of (say) integers. Now most mathematicians, including ZFC
> platonists, reject choice sequences as a true mathematical idea, while
> perhaps allowing that their theory may be a way of doing some odd form
> of applied math – "psychological math". Most would not give ChSeqs even
> that much time. I myself don’t like them because of the way they seem
> to "build subjectivity into the system", something I often rail against,
> whether in intuitionism, Bayesian statistics, or quantum theory.
That doesn’t make much of a difference though. Logicians and Popes
have always assumed to no avail that what they rail for or against has some
correlation to existence.
math…@math.canterbury.ac.nz (Bill Taylor) writes:
>Some while ago I made this inquiry, but I don’t recall it produced much
>action, so I’ll try again.
>The usual stated motivation for accepting AC, (which doesn’t seem to
>help in deciding other matters like CH, though), is that of the so-called
>"combinatorial notion" of sets. Godel seems to have been the first to
>put this in words. Roughly, he said…
>"We adopt the combinatorial notion of sets, whereby it is taken that
> the collection of all subsets is unrestricted by any notion of rule
> definability, but may range over all imaginable subsets."
I would have that that the class of all imaginable subsets is
countable. The reason people argue about AC, is that it allows one
to have unimaginable sets.
>[Apologies if I've got it badly wrong.] Now this sounds very fine, being
>unrestrictive and so forth; till one takes a closer look at it. Then it
>appears as though subsets are to be "things which might be there, even
>though we can’t think straight away what might be a way of deciding
>what members they may have". There is a clear suggestion that subsets
>are in some sense, "indefinitely extendible", and that elements may be
>added *at will*, i.e. without any pre-assigned method.
>This notion appears quite clearly in many AC-using proofs, e.g. the proof
>of the existence of a partition of R^3 into non-parallel lines.
>Now it seems to me there is a definite "psychological" element in this
>notion of subset.
I think you talked yourself into this view by taking "all imaginable
subsets" too literally.
>notion of subset. It’s not usually noticed, but it smells *very* strongly
>of the intuition behind Brouwer’s notion of "Choice Sequence". For those,
>one is to conceive of an indefinitely extendible, but never complete,
>sequence of (say) integers. Now most mathematicians, including ZFC
>platonists, reject choice sequences as a true mathematical idea, while
>perhaps allowing that their theory may be a way of doing some odd form
>of applied math – "psychological math". Most would not give ChSeqs even
>that much time. I myself don’t like them because of the way they seem
>to "build subjectivity into the system", something I often rail against,
>whether in intuitionism, Bayesian statistics, or quantum theory.
>Now most mathematicians reject choice sequences on these psychological
>grounds, without noticing the very similar psychological nature of
>the combinatorial notion of set; in particular as it applies to AC.
Brouwer’s methods call on intuition. Proofs with AC are often
counter-intuitive. It is a little odd that you would see
similarity.
> This notion appears quite clearly in many AC-using proofs, e.g. the proof
> of the existence of a partition of R^3 into non-parallel lines.
Wow! Could you describe the proof? Or a reference?
Thanks,
Dan Goodman
Neil W Rickert <rickert…@cs.niu.edu> writes:
> I would have that that the class of all imaginable subsets is
> countable. The reason people argue about AC, is that it allows one
> to have unimaginable sets.
Can you imagine that there are sets that no-one can imagine?
–
Alan Smaill email: A.Sma…@ed.ac.uk
Division of Informatics tel: 44-131-650-2710
Edinburgh University
On Wed, 13 Sep 2000, it was written:
> > This notion appears quite clearly in many AC-using proofs, e.g. the proof
> > of the existence of a partition of R^3 into non-parallel lines.
> Wow! Could you describe the proof? Or a reference?
There’s not much too it if you’re familiar with transfinite induction.
Let c be the least ordinal such that {n: n < c} has the same
cardinality as R, which is also the cardinality of R^3. List all the
points of R^3 in a transfinite sequence (P_n: n < c}. Define a
transfinite sequence of lines (L_n: n < c) as follows. Suppose that
L_i has already been defined for all i < n. If P_n is not in the union
of the family {L_i: i < n}, let P = P_n; otherwise, let P be any point
of R^3 not in that union. Choose a line L through P which is distinct
from all the lines L_i, i < n. Each of those lines L_i lies on at most
one of the (continuum many) planes containing L, so we can choose a
plane H containing L which contains none of the lines L_i with i < n.
For each i < n, the plane determined by P and L_i intersects the plane
H in a line M_i. Let L_n be some line through P which lies in H and is
distinct from all of the lines M_i (i < n). In this way, we get a
family of noncoplanar lines L_n (n < c) covering every point of R^3.
–
"Any clod can have the facts, but having opinions is an art."–McCabe
Alan Smaill <sma…@dai.ed.ac.uk> writes:
>Neil W Rickert <rickert…@cs.niu.edu> writes:
>> I would have that that the class of all imaginable subsets is
>> countable. The reason people argue about AC, is that it allows one
>> to have unimaginable sets.
>Can you imagine that there are sets that no-one can imagine?
Imagining that there are such sets is not the same as imagining
the sets themselves.
> There’s not much too it if you’re familiar with transfinite induction.
I’m not (although I can sort of guess what it is based on the proof you
gave), but the proof is interesting, thanks.
Dan Goodman
math…@math.canterbury.ac.nz (Bill Taylor) writes:
: "We adopt the combinatorial notion of sets, whereby it is taken that
: the collection of all subsets is unrestricted by any notion of rule
: definability, but may range over all imaginable subsets."
:
: [Apologies if I've got it badly wrong.] Now this sounds very fine, being
: unrestrictive and so forth; till one takes a closer look at it. Then it
: appears as though subsets are to be "things which might be there, even
: though we can’t think straight away what might be a way of deciding
: what members they may have".
The operative phrase here is : "a way".
WHAT IS "a way"? Since the Church-Turing thesis, we
have an answer to this question, but it is only ONE possible
answer, and the biggest problem with the Church-Turing answer is that
it requires each individual way to be finitely specificable
(#of memory-states and #of program-instructions in all TMs are
always finite). As a result of this constraint,
there are only denumberably many ways.
Since there are (obviously) more than denumerably many subsets
of a denumerable set, the question of whether
<something that TAKES a "bit-string of length omega" TO specify>
CAN be <a "way"-thing>
is the one that needs answering first.
: There is a clear suggestion that subsets
: are in some sense, "indefinitely extendible", and that elements may be
: added *at will*, i.e. without any pre-assigned method.
You have to know what "method" means before you can say this.
And the problem is, We Just Don’t Know. The only generally
accepted notion of method that we have is finitistic.
For dealing with "ALL subsets" of an infinite set, that
is arguably just too restrictive a notion. Worse, "extendible"
is metaphorically incorrect. It connotes actually DOING something,
actually constructing something. The issue is not whether all
these subsets are going to be constructible (of course they
won’t). The issue is whether they EXIST. To even CALL them
non-constructible is therefore almost to concede the argument
in advance.
: This notion appears quite clearly in many AC-using proofs, e.g. the proof
: of the existence of a partition of R^3 into non-parallel lines.
:
: Now it seems to me there is a definite "psychological" element in this
: notion of subset. It’s not usually noticed, but it smells *very* strongly
: of the intuition behind Brouwer’s notion of "Choice Sequence". For those,
: one is to conceive of an indefinitely extendible, but never complete,
: sequence of (say) integers.
This "never complete" constraint is just ridiculous.
Any denumerable subset is "complete", in the sense that it must
by definition contain denumerably many members, AT ALL TIMES,
even if it is not recursive and even if the TM by which you are
representing it is still running and has not "yet" enumerated all of
it (and never will).
: Now most mathematicians reject choice sequences on these psychological
: grounds, without noticing the very similar psychological nature of
: the combinatorial notion of set; in particular as it applies to AC.
: For example, in the partition proof referred to above, one observes that
: *one can* at each stage add in another line that doesn’t intersect
: the previous ones; and this is true, for various (discernable) stages.
But you’re conflating the doing and the being again.
Yes, one observes that one [mathematician] can. But that is NOT the point.
One FIRST observes that one [line] EXISTS. THAT is the point.
: But it then glibly assumes this can proceed indefinitely, so that new
: lines can be added in *by free choice*, without any defining rule,
: until a whole c-ordinal of them have been so added.
This is not an assumption. It is a definitional consequence.
:
: IMHO this indefinitely-extendible-choosing is almost identical to the
: choice function notion, which most have completely rejected.
One does not have to extend or construct the relevant set in order
to prove that it exists. That ALL the subsets exist has already
BEEN presumed, even if actually stipulating it is problematic.
: I would like to hear some defense of their view, and its apparent
: inconsistency, by vigorous AC-supporters.
I would agree that we deprecate choice-sequences.
I would then insist that our choosing to define an
object in an ordinal way does not impute any sort of
choice-sequentiality or incompleteness to the object
thereby defined. The fact that the object in question
doesn’t have a finitistic definition in our underlying
framework doesn’t make it in any sense 2nd-class.
It was always existing even we hadn’t yet made the
infinity of dependent choices necessary to define it.
Getting well and properly described (or even pointed at)
by us is NOT a prerequisite for the existence of any
of these subsets.
If you believe that ALL the subsets exist, then
OF COURSE you are a strong supporter of AC since
all the choice sets or choice-functions are themselves
subsets of some relevant larger set. More to the point, a lot
of equivalents of AC are a lot more intuitively bulletproof.
Consider the Cartesian product of two sets. Now, consider
the cartesian product of a set of sets. E.g., S={B,C,D},
c(S) = c({B,C,D}) = B x C x D = { (b,c,d) | beB & ceC & deD }.
Under what circumstances, if S is not empty, could c(S)
be empty? Well, obviously, if this is being written
with a finite number of letters, c(S) is empty if and
only if some element of S is empty. But in the case
where S is infinite, is it even meaningful to speak of
B x C x D x … x Y x Z x …. x W x W+1 x W+2 … etc?
Do our usual associative laws for products even have
MEANING for infinitely long products? Well, maybe they
don’t, but the point is, IF they do, one needs to remember
that the cardinality of B x C x D is |B|*|C|*|D| if they
are disjoint, and is never less than n! even when they
are identical (with n elements each) unless n is 0.
In other words, c(S) HAS to get BIGGER as S does,
in the finite realm.
So, WHY, then, should it EVER be possible, by making S INFINITELY
large, to collapse c(S) to EMPTY?
Well, fundamentally, it ISN’T possible.
But to allege that the cartesian product of an infinite
set can be empty only if one of its elements is empty is
to allege the axiom of choice (Russell called this his
"multiplicative axiom" and it is equivalent to the axiom
of choice).
The bottom line is, even if you are not willing to assume
the existence of ALL non-constructible subsets, denying EVEN
the mere choice-subsets is just absurd. They are doing nothing
more or less than demanding that you "honor" all claims that
some allegedly non-empty sets (to be chosen from) are in fact
non-empty. That algorithmic confirmation of this may be
unavailable is the algorithmic framework’s problem, not
the AC-proponent’s.
—
—
"It’s difficult … you need to be united to have any
strength, but internal issues have to be addressed."
— E. Ray Lewis, on liberalism in America
Alan Smaill <sma…@dai.ed.ac.uk> writes:
>Neil W Rickert <rickert…@cs.niu.edu> writes:
>> Alan Smaill <sma…@dai.ed.ac.uk> writes:
>> >Neil W Rickert <rickert…@cs.niu.edu> writes:
>> >> I would have that that the class of all imaginable subsets is
>> >> countable. The reason people argue about AC, is that it allows one
>> >> to have unimaginable sets.
>> >Can you imagine that there are sets that no-one can imagine?
>> Imagining that there are such sets is not the same as imagining
>> the sets themselves.
>Agreed;
>but what is your answer to the question?
I would have thought that my answer was implicit in my
earlier statement — the one quoted with ">> >>" above.
Neil W Rickert wrote:
> Alan Smaill <sma…@dai.ed.ac.uk> writes:
> >Neil W Rickert <rickert…@cs.niu.edu> writes:
> >> I would have that that the class of all imaginable subsets is
> >> countable. The reason people argue about AC, is that it allows
> >> one to have unimaginable sets.
> >Can you imagine that there are sets that no-one can imagine?
> Imagining that there are such sets is not the same as imagining
> the sets themselves.
This all sounds very familiar, with associations to:
How many angels can split a hair in how many pieces,
simultaneously and/or in sequence, while standing
(or hanging onto) the head of a non-existing needle ?-)
ACDC — Don’t you _love_ ‘math’ (intuively or counter_intuitively) ??
Neil W Rickert <rickert…@cs.niu.edu> writes:
> Alan Smaill <sma…@dai.ed.ac.uk> writes:
> >Neil W Rickert <rickert…@cs.niu.edu> writes:
> >> I would have that that the class of all imaginable subsets is
> >> countable. The reason people argue about AC, is that it allows one
> >> to have unimaginable sets.
> >Can you imagine that there are sets that no-one can imagine?
> Imagining that there are such sets is not the same as imagining
> the sets themselves.
Agreed;
but what is your answer to the question?
–
Alan Smaill email: A.Sma…@ed.ac.uk
Division of Informatics tel: 44-131-650-2710
Edinburgh University
Bill Taylor <math…@math.canterbury.ac.nz> wrote in message
news:8pkir0$9n1$1@cantuc.canterbury.ac.nz…
- Hide quoted text — Show quoted text -
> Some while ago I made this inquiry, but I don’t recall it produced much
> action, so I’ll try again.
> The usual stated motivation for accepting AC, (which doesn’t seem to
> help in deciding other matters like CH, though), is that of the so-called
> "combinatorial notion" of sets. Godel seems to have been the first to
> put this in words. Roughly, he said…
> "We adopt the combinatorial notion of sets, whereby it is taken that
> the collection of all subsets is unrestricted by any notion of rule
> definability, but may range over all imaginable subsets."
> [Apologies if I've got it badly wrong.] Now this sounds very fine, being
> unrestrictive and so forth; till one takes a closer look at it. Then it
> appears as though subsets are to be "things which might be there, even
> though we can’t think straight away what might be a way of deciding
> what members they may have". There is a clear suggestion that subsets
I think Goedel’s statement is misleading. There are two ways to interprete
it; either you take imaginable literally, and get something quite absurd, or
you take it metaphorically, i.e. as saying simply that "all" subsets are to
be included, no matter their definability or whatnot.
> are in some sense, "indefinitely extendible", and that elements may be
> added *at will*, i.e. without any pre-assigned method.
Not really. What it seems to say is that even though we can’t think of
the set in any "quasi-concrete" way it still very well exists. AC is a
statement
to the effect that the set theoretical universe is "well-behaved", not that
some thinking subject goes about adding things to it at will.
> This notion appears quite clearly in many AC-using proofs, e.g. the proof
> of the existence of a partition of R^3 into non-parallel lines.
> Now it seems to me there is a definite "psychological" element in this
> notion of subset. It’s not usually noticed, but it smells *very* strongly
> of the intuition behind Brouwer’s notion of "Choice Sequence". For those,
> one is to conceive of an indefinitely extendible, but never complete,
> sequence of (say) integers. Now most mathematicians, including ZFC
> platonists, reject choice sequences as a true mathematical idea, while
> perhaps allowing that their theory may be a way of doing some odd form
> of applied math – "psychological math". Most would not give ChSeqs even
> that much time. I myself don’t like them because of the way they seem
> to "build subjectivity into the system", something I often rail against,
> whether in intuitionism, Bayesian statistics, or quantum theory.
The theory of choice sequences is imho quite mathematical, and need not
be plaqued by "psychologism". Of course the motivation was originally
to represent the "free construction of mathematical entities" by the
intuitionist mathematical subject, but that’s quite irrelevant.
Turing Machines are no less mathematical just because Turing invented
them as a model of what a human computer does when computing
something.
I too dislike subjective elements, but I fail to see how a mathematical
model of someone’s idea what the subjective elements are like is, in
itself, subjective. And not being competent, I won’t say anything
about my dislike of the orthodox interpretation of QM
> Now most mathematicians reject choice sequences on these psychological
> grounds, without noticing the very similar psychological nature of
> the combinatorial notion of set; in particular as it applies to AC.
> For example, in the partition proof referred to above, one observes that
> *one can* at each stage add in another line that doesn’t intersect
> the previous ones; and this is true, for various (discernable) stages.
> But it then glibly assumes this can proceed indefinitely, so that new
> lines can be added in *by free choice*, without any defining rule,
> until a whole c-ordinal of them have been so added.
> IMHO this indefinitely-extendible-choosing is almost identical to the
> choice function notion, which most have completely rejected.
If I recall correctly, your point has historical roots. Originally AC was
objected because it was thought, like you seem to think, that AC involves
the notion of "someone making a choice" for an inifnite number of sets.
Gradually this conception was replaced with the now accepted one
of AC as asserting the existence of a set satisfying certain properties.
As such, it is no more psychological than the powerset axiom.
> I would like to hear some defense of their view, and its apparent
> inconsistency, by vigorous AC-supporters.
I’m not a vigorous AC-supporter, in fact I have no strong feelings either
way.
> ————————————————————————–
—–
> Bill Taylor W.Tay…@math.canterbury.ac.nz
> ————————————————————————–
—–
> Set theory is a shotgun marriage – between well-ordering and
power-set.
> The two parties get along OK; but they hardly seem made for each
other.
> ————————————————————————–
—–
: > Neil W Rickert <rickert…@cs.niu.edu> writes:
: > I would have that
: > that the class of all imaginable subsets is
: > countable.
"Imaginable" is a contested, colloquial, conversational term.
This whole thing will go better if you make it a defined term.
Intuitively, the reason why the class of "all things that can be
imagined by people" is countable is supposed to be that people have
finite brains, wherefore "whatever" is "in" their brain constituting
their "image" of any thing they "imagine" must also be in some deep
sense finitistic (else it wouldn’t "fit"). There are only
a countable number of finite "things", assuming that the letters
out of which we build strings are themselves also no more complex
than countable.
: > The reason people argue about AC, is that it allows one
: > to have unimaginable sets.
But one can trivially "imagine" a choice set.
One can trivially IMAGINE a denumerable collection of pairs
of socks. One can trivially IMAGINE some other collection
consisting of one from each pair of the first; or, equivalently,
imagine partitioning the
infinite
set of
pairs of
socks
into a
pair of
infinite
sets of
socks ,
where the two new sets each contain one sock from each of
the original sock-pairs, and neither new set ever contains
0 or both socks from the same original pair. IMAGINING
this is trivial. The fact that you can’t imagine DOING
it in finite time is just irrelevant.
More to the point, if it were an infinite number of pairs of
SHOES, the issue wouldn’t even come up; we can OBVIOUSLY
trivially IMAGINE a set of all the left shoes and a set of
all the right shoes, given the original set of pairs of shoes.
The claim that the partition of the set of socks is harder to
IMAGINE than the partition of the set of shoes is has the property
that, well, it is hard to IMAGINE how that could be relevant —
to the imagination, anyway. Obviously it is relevant
to constructibility and computability, but TMs are notoriously
lacking in imagination.
Alan Smaill <sma…@dai.ed.ac.uk> writes:
: Can you imagine that there are sets that no-one can imagine?
> Imagining that there are such sets is not the same as imagining
> the sets themselves.
: Agreed;
I get your point but I still say we need better words.
"Imagine" is too overloaded here.
The verb I personally favor for the sense of "imagine"
occurring in "imagining the sets themselves" above is
"represent". The point being that some sets are not
finitely representable, EVEN after you cheat by
allowing TMs to run arbitrarily long to generate them.
There then needs to be a complementary verb for
the other sense of "imagine", the one that connotes
pointing at something far away, and "knowing what"
we are [or intend to be] pointing at, while not
yet being close enough to see "what’s inside" it,
i.e., how to represent it or what its representation
is. I can know that I mean "that Swedish car over there"
without yet knowing whether it is a Volvo or a Saab.
For a better analogy with AC, I can know I want "one of
those two Swedish cars over there" without knowing OR CARING
whether I get the Volvo or the Saab.
I can know that I mean "the shoe Chosen from the 4303809th
pair" without knowing (or, more to the point, as always
happens we actually need AC, without CARING) whether the
choice function named the left shoe or the right one Chosen.
My personal choice for this verb is "define", although
constructivists will insist that that choice begs the question.
: but what is your answer to the question?
I would answer it with another question:
How is a finite brain supposed to imagine an infinite set?
You can imagine a finite TM that enumerates it, but why
should that COUNT as "imagining" the set itself?
The usual answer to THAT question is that "for any INDIVIDUAL
potential element of the set, we CAN answer in recursive
time whether it is or isn’t in", if the infinite set is
recursive. If it is r.e. but not recursive, there still
IS a fact of the matter as to whether its membership gets
confirmed,
The question then becomes, "Can we define sets we can’t
represent?" And one answer is, "Yes, and every r.e. set
that is not recursive is such a set." I was trying
to suggest above why it is reasonable to allow its TM
to count as a definition, but not a representation,
of such a set.
Some people will insist that that is splitting the
hair too fine, that all r.e. sets ought to be
considered represented as well as defined by their
TMs. THEN the question of whether we can imagine
a set we can’t represent devolves into the question
of whether any set could ever be more or less "imaginable"
(or definable) than its complement. We know for a fact that
some sets are more (or less) representable than their
complements: r.e. sets that are not recursive have the
property that their complements are not r.e..
Can it EVER make sense to say that we can "imagine"
a set without simultaneously being able to IMAGINE
its complement??
—
—
"It’s difficult … you need to be united to have any
strength, but internal issues have to be addressed."
— E. Ray Lewis, on liberalism in America
- Hide quoted text — Show quoted text -
George Greene <gree…@eagle.cs.unc.edu> writes:
> : > Neil W Rickert <rickert…@cs.niu.edu> writes:
> : > I would have that
> : > that the class of all imaginable subsets is
> : > countable.
>"Imaginable" is a contested, colloquial, conversational term.
>This whole thing will go better if you make it a defined term.
>Intuitively, the reason why the class of "all things that can be
>imagined by people" is countable is supposed to be that people have
>finite brains, wherefore "whatever" is "in" their brain constituting
>their "image" of any thing they "imagine" must also be in some deep
>sense finitistic (else it wouldn’t "fit"). There are only
>a countable number of finite "things", assuming that the letters
>out of which we build strings are themselves also no more complex
>than countable.
> : > The reason people argue about AC, is that it allows one
> : > to have unimaginable sets.
>But one can trivially "imagine" a choice set.
With a different sense of "imagine".
We have to remember the context, where Bill Taylor was attempting to
make the case that AC is psychological. You can only make that case
with a psychological sense of "imagine."
You perhaps took me as disagreeing with the Goedel statement, but my
disagreement was actually with Bill’s interpretation.
In sci.logic George Greene <gree…@eagle.cs.unc.edu> wrote:
…
> their "image" of any thing they "imagine" must also be in some deep
> sense finitistic (else it wouldn’t "fit"). There are only
> a countable number of finite "things", assuming that the letters
> out of which we build strings are themselves also no more complex
> than countable.
…
It’s perhaps not a good assumption, even supposing that what we
can imagine is limited to what can be spelled in letters.
–
Greg Lee <l…@hawaii.edu>
Neil W Rickert wrote:
> Alan Smaill <sma…@dai.ed.ac.uk> writes:
> >Neil W Rickert <rickert…@cs.niu.edu> writes:
> >> I would have that that the class of all imaginable subsets is
> >> countable. The reason people argue about AC, is that it allows one
> >> to have unimaginable sets.
> >Can you imagine that there are sets that no-one can imagine?
> Imagining that there are such sets is not the same as imagining
> the sets themselves.
I’m very interested in the implications of even this super short
statement.
It seems to lean on a traditional HABIT of thought that is ASSUMED, that
the reader ‘understands’.
But without a rigorous definition of ‘imagining’ any ‘rigorous’
construction based on it is always suspect.
To start, "All words cannot be defined by only words." It’s a circular
or infinite definition to try.
All of us got our first meanings of words by childhood EXPERIENCE.
Forgetting or never stating that experience needed for definition does
not create ‘rigor’.
SOME improvement in the definitions one uses may be made by trying to
remake ONE’S definitions so they are consistent and general. BUT, how
can one be sure, it is not rigor to simply say "I’m sure".
As long as the exact detail of a brain’s response to a word is not
rigorously and practically known, no word can be rigorously and
practically defined for others or oneself. I don’t think it ever will be
in human history – the brain is too imperfect, forgetful, biased.
SOME improvement for humans can be made by simply providing a simple
STANDARD interactive experience on the internet for basic logic
definitions. My site provides such:
http://www.clickenfind.com/2d_logic/2d_frames3.asp?group=LHS&to_=Babb...
The ultimate solution for humans is their creation of a greatly improved
DNA generated species or AI.
–
Rigorous understanding requires a rigorous definition of thought.
http://www.clickenfind.com/pidmass/bob.html
rlmassey <rlmas…@mho.net> writes:
>Neil W Rickert wrote:
>> Alan Smaill <sma…@dai.ed.ac.uk> writes:
>> >Neil W Rickert <rickert…@cs.niu.edu> writes:
>> >> I would have that that the class of all imaginable subsets is
>> >> countable. The reason people argue about AC, is that it allows one
>> >> to have unimaginable sets.
>> >Can you imagine that there are sets that no-one can imagine?
>> Imagining that there are such sets is not the same as imagining
>> the sets themselves.
>I’m very interested in the implications of even this super short
>statement.
>It seems to lean on a traditional HABIT of thought that is ASSUMED, that
>the reader ‘understands’.
All effective communication ultimately depends on some degree of
shared understanding.
>But without a rigorous definition of ‘imagining’ any ‘rigorous’
>construction based on it is always suspect.
I never claimed that my comment was a rigorous construction.
"Imaginable set" is, after all, an intuitive notion, particularly as
Bill Taylor used it when starting this thread.
>To start, "All words cannot be defined by only words." It’s a circular
>or infinite definition to try.
Agreed, although it seem you are changing subjects.
>All of us got our first meanings of words by childhood EXPERIENCE.
>Forgetting or never stating that experience needed for definition does
>not create ‘rigor’.
>SOME improvement in the definitions one uses may be made by trying to
>remake ONE’S definitions so they are consistent and general. BUT, how
>can one be sure, it is not rigor to simply say "I’m sure".
Ordinary speech is not, and could not be rigorous in the same sense that we
try to make mathematics rigorous.
>As long as the exact detail of a brain’s response to a word is not
>rigorously and practically known, no word can be rigorously and
>practically defined for others or oneself. I don’t think it ever will be
>in human history – the brain is too imperfect, forgetful, biased.
Whether the brain is imperfect, forgetful or biased would seem to be
irrelevant to the point.
>SOME improvement for humans can be made by simply providing a simple
>STANDARD interactive experience on the internet for basic logic
>definitions. My site provides such:
>http://www.clickenfind.com/2d_logic/2d_frames3.asp?group=LHS&to_=Babb...
>The ultimate solution for humans is their creation of a greatly improved
>DNA generated species or AI.
Okay, I get it. You are a net kook, pushing exotic ideas.