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| Bill Taylor... |
 Posted: Fri Oct 30, 2009 7:15 pm |
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stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]People assume the axiom of choice because reasoning without
is an incredible pain.
[/quote]
That is a very wise observation! And you examples are all
well-chosen. One might add that it's very slickly comforting
to know that all vector spaces have bases, and many other
similar uniformity results.
But I note you have slipped into "ease and elegance"
mode of reasoning, or rather of apologetics, as opposed to
"truth and philosophical clarity" mode, which we started with.
[quote]For example, without choice, we can't reason
about cardinality based on bigger/smaller. There are many different
notions of cardinality that are *inequivalent*: X < Y can be defined
by the existence of an injection from X to Y, or the existence of a
surjection from Y to X, and the two definitions are inequivalent.
[/quote]
As you say, these things are all a terrible pain for the mainstream
mathie doing mainstream math. OTOH, they are a fascinating
bywater tributary to the main stream, for those inclined to dawdle
there!
[quote]and seemingly not in an interesting way.
[/quote]
Not interesting to the mainstream sculler.
But fascinating to the dawdler.
-- Backwater Bill |
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| Bill Taylor... |
| Posted: Fri Oct 30, 2009 7:21 pm |
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On Oct 29, 12:02 pm, Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
[quote]I don't see why you would believe it.
I can't say it better, at this stage, than i already did. Sorry.
(Can you /explain/ /why/ you think the Jordan curve
should definitely be true? I can't.)
[/quote]
Now THAT is a wonderful question, and I would dearly love
to hear a response from some of the people it is directed toward!
-- Wide-eyed William
* Changing your mind because of emotion - that is faith.
* Changing your mind because of thinking - that is philosophy.
* Changing your mind because of facts - that is science. |
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| Nam Nguyen... |
| Posted: Fri Oct 30, 2009 11:44 pm |
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I'm a bit off topic here but since it's still about
(counter)intuition so I'm going to ask.
Let 2ZFC be a formal system written in a language
L with 2 epsilon relations: L = L(e,e'). The axioms
of 2ZFC are the following:
- A set of axioms that has formulas using only e and
all together would be a ZFC axioms (for e).
- A set of axioms that has formulas using only e' and
all together would be a ZFC axioms (for e').
- An additional axiom:
Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)]
(This says the 2 empty sets be identical.)
How would intuition perceive such a multiple epsilon relation
sets? How would AC's and AD's connotations be impacted by 2ZFC? |
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| Nam Nguyen... |
| Posted: Sat Oct 31, 2009 12:40 am |
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Nam Nguyen wrote:
[quote]I'm a bit off topic here but since it's still about
(counter)intuition so I'm going to ask.
Let 2ZFC be a formal system written in a language
L with 2 epsilon relations: L = L(e,e'). The axioms
of 2ZFC are the following:
- A set of axioms that has formulas using only e and
all together would be a ZFC axioms (for e).
- A set of axioms that has formulas using only e' and
all together would be a ZFC axioms (for e').
- An additional axiom:
Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)]
(This says the 2 empty sets be identical.)
How would intuition perceive such a multiple epsilon relation
sets? How would AC's and AD's connotations be impacted by 2ZFC?
[/quote]
The way my intuition is thinking is that we'd probably add
more axioms to make the sets isomorphically equals between
the 2 epsilons: from pairing, unions, intersection, power-sets,
etc... But then when AC is required, we'd make the 2 choice
functions *different* all together could be used to formulate
a sort of "Axiom of Destiny" in 2ZFC. Something like that.
Would you see it the same way? |
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| Herman Jurjus... |
| Posted: Sat Oct 31, 2009 2:35 am |
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Bill Taylor wrote:
[quote]On Oct 29, 12:02 pm, Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
I don't see why you would believe it.
I can't say it better, at this stage, than i already did. Sorry.
(Can you /explain/ /why/ you think the Jordan curve
should definitely be true? I can't.)
Now THAT is a wonderful question, and I would dearly love
to hear a response from some of the people it is directed toward!
-- Wide-eyed William
* Changing your mind because of emotion - that is faith.
* Changing your mind because of thinking - that is philosophy.
* Changing your mind because of facts - that is science.
[/quote]
May i ask one question to you, too?
What are your /reasons/ for rejecting AC, other than your acceptance of
AD and the fact that ZFC+AD is inconsistent?
--
Cheers,
Herman Jurjus |
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| Frederick Williams... |
| Posted: Sat Oct 31, 2009 6:46 am |
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Nam Nguyen wrote:
[quote]
I'm a bit off topic here but since it's still about
(counter)intuition so I'm going to ask.
Let 2ZFC be a formal system written in a language
L with 2 epsilon relations: L = L(e,e').
[/quote]
Assuming that e and e' have intended interpretations, what are they?
[quote]The axioms
of 2ZFC are the following:
- A set of axioms that has formulas using only e and
all together would be a ZFC axioms (for e).
- A set of axioms that has formulas using only e' and
all together would be a ZFC axioms (for e').
- An additional axiom:
Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)]
(This says the 2 empty sets be identical.)
How would intuition perceive such a multiple epsilon relation
sets? How would AC's and AD's connotations be impacted by 2ZFC?
[/quote]
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested. |
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| Daryl McCullough... |
| Posted: Sat Oct 31, 2009 7:55 am |
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Bill Taylor says...
[quote]
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
You gave the answer yourself,in terms of infinte-depth quantifiers
But I really don't have an intuition about what infinitely many
alternations of quantifiers *mean*,
Of course you do!
[/quote]
No, I don't. It's a limit of finite quantifiers, but in what
sense of "limit"? I can guess at several *different* things
that the infinite quantifier expression *might* mean, but
the possibilities don't seem equivalent. So the whole concept
seems insufficiently "tied down".
For example, to say
Ex_1 Ax_2 Ex_3 ... Phi([x_1, x_2, ...])
Might mean the infinite conjunction of the following
statements:
1. Ex_1 [x_1} can be extended to an infinite sequence s such that Phi(s)
2. Ex_1 Ax_2 [x_1,x_2] can be extended to a sequence s such that Phi(s)
etc.
In terms of games, this would be formalized by "there is a strategy
on the part of the first player such that following that strategy,
he never loses at any finite stage. Following that strategy, he could
still lose, but you can only know that by looking at the whole
infinitely many moves played by both players.
Another meaning is given in terms of strategy functions. If we
define play(f,g) to be the infinite sequence resulting from
the first player following strategy f, and the second player
following strategy g, then the infinite quantifier expression
Ex_1 Ax_2, ... Phi([x_1,x_2,...])
might mean
Ef Ag Phi(play(f,g))
It could *also* mean the quantifier-switched version:
Af Eg Phi(play(f,g))
We can also formulate "strategy" in terms of sets, instead
of functions. Those are called "quasistrategies". When a
player follows a quasistrategy, the quasistrategy doesn't
give a unique move at each stage, but gives a set of possible
moves, and the player just picks one. Then in terms of
quasistrategies, Ex_1, A_x_2, ... Phi([x_1, x_2, ...])
could be formalizes as:
Exists quasistrategy Q1, Forall quasistrategies Q2, forall
infinite sequences s: s is consistent with the first player
following quasistrategy Q1, and is consistent with the second
player following quasistrategy Q2, and Phi(s).
It could also be formalized as the quantifier-switched
version of the above:
Forall Q2, Exists Q1, blah, blah (same as above).
[quote](Else I have a quite different understanding of the words you use.)
The mere fact that you *came up* with the idea, independently of me,
shows you have a fair idea of what it means.
[/quote]
The germ of the idea is: generalize quantifiers to the case of
infinitely many alternations of quantifiers. That doesn't uniquely
pin down what it is we are talking about.
[quote]It's not *obvious* that chess or checkers has a
winning strategy;
It IS.(That is, a winning strategy or a drawing strategy for both)
It was obvious to me even before I started high school.
I'm sure you never played infinite games before high school.
I's talking about CHESS, you goose, when I mentioned high school! 
[/quote]
My point is that you simultaneously developed two different intuitions
about chess (or whatever): (1) Eventually, the game always ends.
(2) There is a winning strategy for one player or the player.
So I don't see how chess gives you any intuitions about games
that *don't* end.
I am curious, though, about the *form* that your intuition
about the existence of a winning strategy took. You really
thought in terms of *functions* from board positions to
your next move? That's very remarkable to me, if that's
the case. When I played chess, I would pick a certain number
of moves to "think ahead" (and my pitiful brain could never
think more than two or three). On the basis of looking ahead,
I could decide that certain moves are losers. If a win
was sufficiently close at hand, then I could decide that
certain moves were winners. But in the general case, the
best I could say was that certain moves were neither
obviously winners, nor obviously losers.
I'm curious as to how the idea that there *must* be a winning
move occurred to you. Presumably, if the win was too many
moves in the future, then you didn't actually *find* this
winning move. So how did you figure that it must exist?
[quote]I don't agree. Once you introduce infinite games, the intuition
disappears completely for me.
Well, I could say exactly the same thing about choice,
which you seem to find no problems with.
How are we to resolve this quagmire!?
[/quote]
Well, I would *not* say that my intuition about choice
has anything to do with generalization from finite
sets. The size of the set has nothing to do with it,
as far as I can see. My intuition about sets is that
they are like boxes, which can contain things (possibly
other boxes). If a box is nonempty, then I can reach
in and pull something out of the box. It doesn't matter
how many things are in the box, and it doesn't matter
whether those things have unique names, or whether they
can be well-ordered or whatever.
Now, suppose I have a box that contains a bunch of other
nonempty boxes (it doesn't matter how many). I can imagine
doing the following: for each box, I assign a helper
(that might mean infinitely many helpers, but if I'm
going to imagine infinitely many boxes, I can imagine
infinitely many helpers). On my signal, each of my
helpers reaches into his corresponding box and pulls
something out. I put this collection together into
one box. In terms of sets, I conclude:
If there is a set of nonempty sets, then it is possible
to get a new set that contains one element from each set.
There is nothing special about finiteness of the sets
in this intuition.
[quote]Using infinite quantifiers, we can't (as far as I know) make
the distinction between "Player 1 has no winning strategy" and
"Player 2 has a winning strategy".
OCN! Because there isn't one, (or so I claim), for tie-free games.
[/quote]
You don't see that what your saying is circular? There are two
different claims being made: (1) The infinite quantifier notation
is adequate to capture all distinctions that are relevant.
(2) There is no distinction between "Player 1 has no winning
strategy" and "Player 2 has a winning strategy".
I agree that *if* you accept (1), then (2) follows, and that
*if* you accept (2), then (1) follows. But you can't use
(1) as an argument for the truth of (2), and then turn around
and use (2) as an argument for the truth of (1).
I would think that you need an *independent* argument for (2)
(one that doesn't involve infinite quantifiers) in order to
justify the use of infinite quantifiers in the first place.
--
Daryl McCullough
Ithaca, NY |
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| Daryl McCullough... |
| Posted: Sat Oct 31, 2009 8:14 am |
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Bill Taylor says...
[quote]
On Oct 29, 12:02 pm, Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
I don't see why you would believe it.
I can't say it better, at this stage, than i already did. Sorry.
(Can you /explain/ /why/ you think the Jordan curve
should definitely be true? I can't.)
Now THAT is a wonderful question, and I would dearly love
to hear a response from some of the people it is directed toward!
[/quote]
Here's my way of thinking: We have a rubber band (infinitely stretchable
and bendable) that is initially a circle. We place the rubber band
on the floor, and place a tiny ant inside the circle made by the rubber
band. Is there any way for the ant to get out of the region enclosed by
the rubber band without breaking the rubber band? (We want this problem
to be two-dimensional, so we ignore the possibility of it climbing over
or under the rubber band) Clearly not.
Now, imagine the ant pushing or pulling spots on the rubber, distorting
it away from a circle. Does that help? No. Imagine painting the inner
surface of the rubber band red and the outer surface green. No matter
how the ant pushes or pulls the rubber band, no matter how he distorts
its shape away from a circle, nothing he can do can ever change the
fact that he only has access to the red surface, and to be out, he
must be able to touch the green surface.
Now, to mathematize this picture, we let the rubber shrink to zero
thickness and height, so that it can be represented by a closed curve,
and let the ant be a mathematical point, and the ant's trajectory in
trying to get becomes another curve. For any closed curve that is
obtainable by stretching and bending from a circle, there is no
nonintersecting curve connecting the inside to the outside.
--
Daryl McCullough
Ithaca, NY |
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| Nam Nguyen... |
| Posted: Sat Oct 31, 2009 9:27 am |
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Frederick Williams wrote:
[quote]Nam Nguyen wrote:
I'm a bit off topic here but since it's still about
(counter)intuition so I'm going to ask.
Let 2ZFC be a formal system written in a language
L with 2 epsilon relations: L = L(e,e').
Assuming that e and e' have intended interpretations, what are they?
[/quote]
They each would have the same usual set-membership interpretation: but same
interpretation of 2 different kinds of mereological concepts. Together
they'd make 2ZFC a 2-dimensional set theory. (So potentially we could have
n-ZFC).
An "rudimentary" example would be in the area of mind-vs-body. A physical
body part could be said to be a member of another body part, thus one
epsilon symbol. But a thought could be said to be part of (or reflected by)
another thought, thus another epsilon symbol.
But that's just one intuition. There might be more from others.
[quote]
The axioms
of 2ZFC are the following:
- A set of axioms that has formulas using only e and
all together would be a ZFC axioms (for e).
- A set of axioms that has formulas using only e' and
all together would be a ZFC axioms (for e').
- An additional axiom:
Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)]
(This says the 2 empty sets be identical.)
How would intuition perceive such a multiple epsilon relation
sets? How would AC's and AD's connotations be impacted by 2ZFC?
[/quote] |
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| Nam Nguyen... |
| Posted: Sat Oct 31, 2009 9:37 am |
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Nam Nguyen wrote:
[quote]Frederick Williams wrote:
Nam Nguyen wrote:
I'm a bit off topic here but since it's still about
(counter)intuition so I'm going to ask.
Let 2ZFC be a formal system written in a language
L with 2 epsilon relations: L = L(e,e').
Assuming that e and e' have intended interpretations, what are they?
They each would have the same usual set-membership interpretation: but same
interpretation of 2 different kinds of mereological concepts. Together
they'd make 2ZFC a 2-dimensional set theory. (So potentially we could have
n-ZFC).
An "rudimentary" example would be in the area of mind-vs-body. A physical
body part could be said to be a member of another body part, thus one
epsilon symbol. But a thought could be said to be part of (or reflected by)
another thought, thus another epsilon symbol.
But that's just one intuition. There might be more from others.
[/quote]
Another example would be SR but I'd think in this case instead of having
just n epsilon symbols, we might need countably infinite of them. (But
haven't given this "example" too much thought though).
[quote]
The axioms
of 2ZFC are the following:
- A set of axioms that has formulas using only e and
all together would be a ZFC axioms (for e).
- A set of axioms that has formulas using only e' and
all together would be a ZFC axioms (for e').
- An additional axiom:
Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)]
(This says the 2 empty sets be identical.)
How would intuition perceive such a multiple epsilon relation
sets? How would AC's and AD's connotations be impacted by 2ZFC?
[/quote] |
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| Bill Taylor... |
| Posted: Sat Oct 31, 2009 10:24 pm |
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Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
[quote]What are your /reasons/ for rejecting AC,
[/quote]
Well! That would be a whole thread in itself.
And I don't wish to bore the old hands YET AGAIN.
So I'll try to be brief.
AC produces sets, by fiat of existence, which simply DON'T EXIST!
Non-measurable sets in R,free ultrafilters on N, partitions of R^3
into non-parallel lines - all these things simply don't exist,
in that no explicit set can be named, with those properties.
Now "explicit" and "naming" are very vague terms, so we must
determine our philosophy of sets, to try and illuminate them.
And mine is - the whole IDEA of a set, is that one can say
WHAT THE MEMBERS ARE. Now, I do not mean by this, that one
must be able to prove whether or not some particular
putative element is in it or out of it. But at least
*a condition* for membership should be available. i.e.
{ x | phi(x) }
should be taken as a model or paradigm for what a set actually IS.
So, to be a set, IMHO, means there is (in principle, implicitly,
or whatever other caveat one cares about) a determining property
for what constitutes an element or not.
Now, we all know, there are many ways of doing this, not all
of them very explicit at all. But at least, in "normal" math,
they exist in the background.
But for the sets declared/created by AC, this informal criterion
is cast to the winds. There is no longer the slightest hint,
pretense or intuition of what is a member and what isn't.
Sure, there are plenty of intuitions etc as to what PROPERTIES
the set itself may have - but that isn't enough for existence,
IMHO, because a set with those properties may or may not actually
exist. But until one has a handle on the members, the set
does not EXIST. No set-hood without element-hood.
Interestingly, this idea, that elements must be (in some sense)
logically prior to the set, is at the heart of almost all the
methods of addressing "the paradoxes", which are easily seen
to be caused by attempts to put a set on the same footing as
its members.
Sadly, the math world drops this principle like a hot potato
once it has done its minimal work. I feel this is bad.
Please note:- I'm not against set theorists using whatever
bunch of (consistent) axioms and principles and ideas they want,
for abstract set theory, just please stop pretending it has any
bearing on the existence or otherwise of basic concerns about
N, R and their immediate neighborhood.
Rant over. But see my sig...
====== W.Taylor ======
** No set-hood without elementhood!
** No set-hood without elementhood!
** No set-hood without elementhood! |
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| Daryl McCullough... |
| Posted: Sun Nov 01, 2009 8:36 am |
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[quote]AC produces sets, by fiat of existence, which simply DON'T EXIST!
Non-measurable sets in R,free ultrafilters on N, partitions of R^3
into non-parallel lines - all these things simply don't exist,
in that no explicit set can be named, with those properties.
[/quote]
The reason I have little sympathy for this point of view is
that I don't see that traditional mathematics ever makes the
assumption that everything has an explicit name or description.
Yes, some things do---namely naturals, but points in a
topological space? The *names* of those points are completely
irrelevant.
It's an amazing accomplishment of ZF and the concept of the
cumulative hierarchy that we can come so close to an explicit
taxonomy of all the objects that will ever come into play in
any mathematical argument, but I don't see that the existence
of such a standard model has any relevance to what mathematicians
*do* with sets. Instead, they start with certain basic sets
that are *not* defined, except implicitly by axioms governing
them, and then they apply set-theoretic operations on *those*
sets to get new sets.
To me, the astounding claim about set theory is that, whatever
your basic objects are --- paths of particles, wavefunctions,
games, strategies, probabilities, trees, computations --- we
can describe and reason about them using set theory. Having
explicit names for the basic objects is just *not* necessary
in order to use the techniques of set theory.
[quote]Now, I do not mean by this, that one
must be able to prove whether or not some particular
putative element is in it or out of it. But at least
*a condition* for membership should be available. i.e.
{ x | phi(x) }
should be taken as a model or paradigm for what a set actually IS.
[/quote]
I sort of agree with this, but I don't feel that it is
appropriate to assume that the only possible conditions
phi are ones that are definable using pure set theory.
For example, if we are trying to formulate Newtonian
physics using set theory, then we can talk about such
sets as "the set of all past locations of the particle".
That is a set of triples of real numbers, and it has an
associated condition, but the condition is not given by
*set* theory, it is given by the premise of the problem
being analyzed using set theory.
If we are talking about sets of naturals, then "the set
of phone numbers of mathematicians" is a perfectly good
set of naturals, even though the condition defining that
set isn't a set-theoretic condition. Now, in the case of
finite sets of naturals, we can always *find* a
set-theoretic condition that is extensionally equivalent.
We can cook up a formula Phi(x) expressed in the language
of arithmetic, such that Phi(x) holds only of those x-s
such that x is a phone number of a mathematician. But
such an exercise would be pointless---the existence of
such a Phi would be completely irrelevant to anything
we want to do with such a set.
In a sense, the insistence that all mathematical objects
have an explicit description is more radical than
constructivism. Constructivists take the principle that
everything should be constructable as an obligation on
the part of someone trying to *prove* something. It only
counts as a proof if everything can be made explicit and
computable. But constructive mathematics does *not* assume
that nothing *exists* unless it is constructable. Some
constructive mathematicians may believe that, but such an
assumption is not itself something that can be used in
a constructive proof. The claim: "every real number is
computable" is not constructively provable, so it can't
be assumed in a constructive proof.
--
Daryl McCullough
Ithaca, NY |
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| Daryl McCullough... |
| Posted: Sun Nov 01, 2009 11:02 am |
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Bill Taylor says...
[quote]
Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
What are your /reasons/ for rejecting AC,
Well! That would be a whole thread in itself.
And I don't wish to bore the old hands YET AGAIN.
So I'll try to be brief.
AC produces sets, by fiat of existence, which simply DON'T EXIST!
Non-measurable sets in R,free ultrafilters on N, partitions of R^3
into non-parallel lines - all these things simply don't exist,
in that no explicit set can be named, with those properties.
[/quote]
There is something a little strange about this philosophical
position: If you assume that all sets have an explicit
"membership criterion" Phi(x), then choice is *automatically*
true. You can well-order sets by their defining formulas.
The only way to get a nonempty set of reals *without* a
choice function is if contains some undefinable reals.
--
Daryl McCullough
Ithaca, NY |
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| Bill Taylor... |
| Posted: Sun Nov 01, 2009 8:29 pm |
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stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]The reason I have little sympathy for this point of view is
that I don't see that traditional mathematics ever makes the
assumption that everything has an explicit name or description.
[/quote]
OC it doesn't. It never needed to. Before Cantor everything DID.
[quote]It's an amazing accomplishment of ZF and the concept of the
cumulative hierarchy that we can come so close to an explicit
taxonomy of all the objects that will ever come into play in
any mathematical argument, but I don't see that the existence
of such a standard model has any relevance to what mathematicians
*do* with sets.
[/quote]
I don't follow this sentence at all, sorry.
[quote]Instead, they start with certain basic sets
that are *not* defined,
[/quote]
In pure set theory they all are. They are the empty set.
[quote]To me, the astounding claim about set theory is that, whatever
your basic objects are --- paths of particles, wavefunctions,
[/quote]
That is all applied mat, applied set theory.
We are (I thought) speaking of pure set theory, as applied to
(at most) other pure math objects like N & R.
[quote]{ x | phi(x) }
should be taken as a model or paradigm for what a set actually IS.
I sort of agree with this,
[/quote]
Well! There's something.
[quote]but I don't feel that it is
appropriate to assume that the only possible conditions
phi are ones that are definable using pure set theory.
[/quote]
Set theory as applied to physical objects is completely trivial.
I don't think it has any place in this debate.
[quote]If we are talking about sets of naturals, then "the set
of phone numbers of mathematicians" is a perfectly good
[/quote]
No, it's perfectly bad!
[quote]In a sense, the insistence that all mathematical objects
have an explicit description is more radical than constructivism.
[/quote]
It's NOT an "insistence", as you call it, but merely an observation
of standard (pre-Cantor) math. And I feel that set theory should
be conducted along the lines of those traditions.
At least, if it is to make traditional sense.
Anyway, I was asked for my view, and I gave it.
-- Bill of basics |
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| Bill Taylor... |
| Posted: Sun Nov 01, 2009 9:02 pm |
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stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]AC produces sets, by fiat of existence, ...
... - all these things simply don't exist,
in that no explicit set can be named, with those properties.
There is something a little strange about this philosophical
position: If you assume that all sets have an explicit
"membership criterion" Phi(x), then choice is *automatically*
true. You can well-order sets by their defining formulas.
[/quote]
AHA! A very astute observation. This is the kind of thing
that had me bothered for a long time. And it is all bound up
with the UNDEFINABILITY of "DEFINABILITY",
as I have noted before.
All sets will be definable, but this notion cannot be delimited
in advance. So e.g. once you have a bounded set of reals,
it must have a least upper bound, BUT this lub will not be
definable at the same level - if the reals in the set are
definable at level "a" and below, then the lub will be most
likely be at level a+1. There is no end to the levels, they
can be notated exactly as any initial segment of the recursive
ordinals. But not of the whole set itself - that would be
declaring existences BY FIAT again. omega_1^CK itself is
purely a creature of ZF, with its all-encompassing power set
operation, (without which it cannot be defined.)
So the levels of definibility proceed endlessly onward,
without (sans Cantor) a definite or upper bound.
This sort of property is called "extensible" in
philosophical logic circles, and is the topic
of the paper I alluded to earlier.
-- Well-ordered William
** Definibility is itself undefinable! |
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