Russell-like paradoxes
Witt
The assumption of the existence of non-existent things leads to
contradiction!
Neither the 'Russell Class' nor the 'Barber of Seville' exist!
The essence of these contradictions resides in the notion that:
~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)).
Both are valid.
That the x's and the y's are not of the same type is not relevant.
Russell's theory of types does not resolve the paradox of the Barber of
Seville!
Although, his 'Types' does eliminate the possibility of expressing (x e x).
(ix: (x e y) <->x ~(x e x )) is just as contradictory as is (ix: x=y <->x
~(x=x)).
~E!(iy: (x e y <->x ~(x e x))
~E!(iy: x=y <->x ~(x=x))
In general: ~E!(iy: xRy <->x ~(xRx)), and,
~E!(yRx <->x ~(xRx)) for every R.
It is the case that: Ax(xRy <-> ~(xRx)) and Ax(yRx <-> ~(xRx)) are both
contradictions.
for all R's.
Witt
Charlie-Boo
> The Russell's paradox and the Barber of Seville share a comminality.
>
> The assumption of the existence of non-existent things leads to
> contradiction!
>
> Neither the 'Russell Class' nor the 'Barber of Seville' exist!
>
> The essence of these contradictions resides in the notion that:
>
> ~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)).
> Witt
Witt,
While the above is true (although it looks like you have an extra x in
~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)) after each ">") one can
understand Russell's Paradox better if, rather than thinking about
what it is analogous to, develop a general theorem for which each is
an instance. You will see that the following are instances of a more
general theorem:
1. The Liar Paradox
2. Russell's Paradox
3. Tarski's Theorem on the undefinability of truth.
4. The set of Turing Machines that do not halt yes on themselves is
not recursively enumerable.
5. Various aspects of axiomized set theory (e.g. ZFC).
In fact, most every incompleteness theorem of Computer Science
(Recursion Theory, Theory of Computation), Logic and Mathematics, such
as those of Godel and Turing, are formally derivable from this theorem
- that is, are also instances of a more general theorem that results
from applying additional logic.
Let N#P(x)[Q(a,b)] mean that P(a)=Q(N,a) (P(a) means set P and Q(N,a)
means the set defined by fixing the first component of Q at N), and
P(x) mean there exists an N such that N#P(x), where [Q(a,b)] stated
separately allows us to abbreviate all references P(x)[Q] as P(x).
This is a generalization of the notion of a set P being recursively
enumerable, where N is the program that lists P. When Q(a.b) is
"Turing Machine a halts yes on input b." then P(x)[Q(a,b)] is the
standard definition of r.e. Let -P(x)[Q(a,b)] mean that there is no N
such that N#P(x)[Q(a,b)].
The general theorem is:
-~P(x,x)[P(a,b)]
Some researchers have considered certain values of Q, such as the
above example or Q(a,b) = "Wff a with input b is provable" or "... is
refutable" (called "contrarepresentable", I believe - too lazy to
check). But we need to make the "domain" Q a variable to see the
general case.
All of this depends on the the "rules of inference" that apply to Q.
For example, does P(x) => ~P(x)? For some values of Q this holds.
The paradoxes occur when there are 3 certain rules in effect for Q,
which lead to the contradiction that ~P(x,x)[P(a,b)]. The various
attempts to "resolve" these paradoxes (e.g. Russell's Type Theory and
disallowing a universal set) are generally disallowing some particular
step in the formal proof.
When applied to English, this actually produces the indeterminable
sentence:
"'It is not true of itself.' is true of itself."
Certain additional formal rules of English can then be applied to show
this is equivalent to:
"This is false."
Charlie Volkstorf
Cambridge, MA
http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1
http://www.arxiv.org/html/cs.lo/0003071
PS If people like you-know-who tell you "it's been done before", be
sure to very politely respond, "Thanks for the reference, but, gee, I
just don't see that result in the 400 page book you referred me to.
Can you tell me what page it's on or give me a brief summary/example
of how it is handled?"
G. Frege
> >
> > Neither the 'Russell Class' nor the 'Barber of Seville' exist!
> >
> > The essence of these contradictions resides in the notion that:
> >
> > ~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)).
> >
>
> ...it looks like you have an extra x in ~ER(xRy <->x ~(xRx)) or
> ~ER(yRx <->x ~(xRx)) after each ">")
>
This is just Russell's shorthand notation for
Ax(... <-> ...),
originally introduced by Peano, afaik. Actually, the formulas should
read (in 2nd order logic):
~EREyAx(xRy <-> ~(xRx)) / ~EREyAx(yRx <-> ~(xRx))
or
AR~EyAx(xRy <-> ~(xRx)) / AR~EyAx(yRx <-> ~(xRx)).
George Greene:
"Russell's Paradox is WELL-expressed as the following
2nd-order tautology:
AR[~Er[Ax[(xRr) <-> ~(xRx)]]]
Short & natural: 'For any R, r does not exist'."
>
> [Indeed] one can understand Russell's Paradox better if, rather than
> thinking about what it is analogous to, develop a general theorem for
> which each is an instance.
>
Indeed. This theorem is called Thomson's theorem:
"Let S be any set and R any relation defined at least on S.
Then no element of S has R to all and only those elements
in S which do not R to themselves."
(J. F. Thomson, "On Some Paradoxes" in Analytical Philosophy,
ed. R. J. Butler, New York: Barnes & Noble, 1962, p. 104-119.)
>
> Thanks for the reference, but, gee, I just don't see that result
> in the 400 page book you referred me to. Can you tell me what page
> it's on or give me a brief summary/example of how it is handled?
>
???
F.
P.S.
You might also be interested in the /short/ paper 'Is Russell's Paradox
Deep?' by Prof. Laurence Goldstein:
Witt
"Charlie-Boo" <ch...@aol.com> wrote in message
news:3df1e59f.03121...@posting.google.com...
> "Witt" <oori...@yahoo.com> wrote
> > The Russell's paradox and the Barber of Seville share a commonality.
> >
> > The assumption of the existence of non-existent things leads to
> > contradiction!
> >
> > Neither the 'Russell Class' nor the 'Barber of Seville' exist!
> >
> > The essence of these contradictions resides in the notion that:
> >
> > ~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)).
> . . .
> > Witt
>
> Witt,
>
> While the above is true (although it looks like you have an extra x in
> ~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)) after each ">")
(xRy <->x ~(xRx)) means Ax(xRy <-> ~(xRx)).
1.Ax(xRy <-> ~(xRx)) is contradictory for all y's and for all R's.
2. ~Ax(xRy <-> ~(xRx))
3. ~EyAx(xRy <-> ~(xRx))
4. ~ER(EyAx(xRy <-> ~(xRx))).
2. ~Ax(xRy <-> ~(xRx))
Proof:
2a. Ax(xRy <-> ~(xRx)) -> (yRy <-> ~(yRy))
By, AxFx -> Fy.
2b. ~(yRy <-> ~(yRy))
By, ~(p <-> ~p).
2c. Ax(xRy <-> ~(xRx)) -> (contradiction)
By: 2a, 2b, (p <-> ~p) <-> (p & ~p) <-> (contradiction).
2d. ~Ax(xRy <-> ~(xRx))
By: 2c, (p -> (contradiction)) <-> ~p.
Every instance of R (in Ax(xRy <-> ~(xRx))) implies a contradiction.
(the illusion of paradox lies in contradiction ...implying a logical misuse
of language,
i.e. there are no 'paradoxes' or 'contradictions' in reality.)
A. Ax(x e y <-> ~(x e x)) -> (y e y <-> ~(y e y)).
B. Ax(y shaves x <-> ~(x shaves x)) -> (y shaves y <-> ~(y shaves y)).
C. Ax(y shampoos x <-> ~(x shamoos x)) -> (y shampoos x <-> ~(x shamoos x))
D. Ax(x=y <-> ~(x=x)) -> (y=y <-> ~(y=y))
Etc.
A1. ~E!(the y: Ax(x e y <-> ~(x e x)) [ i.e. ~E!{x:~(x e x)}]
B1. ~E!(iy: Ax(y shaves x <-> ~(x shaves x))
C1. ~E!(iy: Ax(y shampoos x <-> ~(x shamoos x))
D1. ~E!(iy:Ax(x=y <-> ~(x=x)))
That which is not equal to itself does not exist, whether identity is
reflexive or not.
(P(x) -> ~P(x)) <-> ~P(x), for every P.
(P(x) -> ~P(x)) <-> (P(x) -> (contradiction))
(P(x) -> ~P(x)) <-> (P(x) <-> (contradiction))
I will study your remarks about a general solution to paradoxes and
reply later.
I don't think that all of the 'paradoxes' that you mention have a common
form, so I need to consider your position.
>
> The paradoxes occur when there are 3 certain rules in effect for Q,
> which lead to the contradiction that ~P(x,x)[P(a,b)]. The various
> attempts to "resolve" these paradoxes (e.g. Russell's Type Theory and
> disallowing a universal set) are generally disallowing some particular
> step in the formal proof.
None of the attempted solutions to the Russell Paradoxes
(e.g. Russell's Type Theory and disallowing a universal set)
work for the Barber Paradox.
Russell's Types only apply to the membership predicate.
Zermelo, von Neumann and Quine, etc., want to restrict the range of the
individual/class
variable to include existent individuals/classes only.
Z: EyAx(x e y <-> (x e z & Fx))
N: EyAx(x e y <-> (Ez(x e z) & Fx))
Q: EyAx(x e y <-> ((x e V) & Fx))
Z: If z=V or z={z}then Ey(x e y <-> (x e z & Fx)) reduces to EyAx(x e y <->
Fx)),
which leads to the Russell antinomy.
They want to avoid the contradictions by restricting 'y' in
Ax(x e y <-> ~(x e x)) -> (y e y <-> ~(y e y)). (since it cannot exist).
But, the problem is not resolved in general by their approach.
The predicate F is what needs restricting, not y.
EyAx(x e y <-> (EzAx(x e z <-> Fx) & Fx)) does the job for set
theory...without types.
The predicate ~(x e x) has no extension.
The individual (ix:~(x=x)) has no extension.
The individual (ix: x=x) has no extension.
The x such that Ay(x shaves y <-> ~(y shaves y)) has no extension.
The Barber of Seville has no extension.
>
> When applied to English, this actually produces the indeterminable
> sentence:
>
> "'It is not true of itself.' is true of itself."
>
> Certain additional formal rules of English can then be applied to show
> this is equivalent to:
>
> "This is false."
>
> Charlie Volkstorf
> Cambridge, MA
> http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1
> http://www.arxiv.org/html/cs.lo/0003071
>
> PS If people like you-know-who tell you "it's been done before", be
> sure to very politely respond, "Thanks for the reference, but, gee, I
> just don't see that result in the 400 page book you referred me to.
> Can you tell me what page it's on or give me a brief summary/example
> of how it is handled?"
Some posters, unfortunately, prefer to insult others or to simly be profane
because they can. ???
Regards,
Owen
Charlie-Boo
> On 13 Dec 2003 15:30:36 -0800, ch...@aol.com (Charlie-Boo) wrote:
> > ...it looks like you have an extra x in ~ER(xRy <->x ~(xRx)) or
> > ~ER(yRx <->x ~(xRx)) after each ">")
> >
> This is just Russell's shorthand notation for
>
> Ax(... <-> ...),
It seems strange to refer to a variable and only later in the
expression indicate that it is universally quantified. (I assumed the
common convention of universally quantifying otherwise unquantified
variables, so that each such "x" is superfluous.)
But as far as that goes, if the x means ...<->... then you have "<->"
twice in each expression. Is that what you really mean? I assume you
mean that x represents Ax, but, again, why have it in the MIDDLE of
the references to x?
> > [Indeed] one can understand Russell's Paradox better if, rather than
> > thinking about what it is analogous to, develop a general theorem for
> > which each is an instance.
> >
> Indeed. This theorem is called Thomson's theorem:
>
> "Let S be any set and R any relation defined at least on S.
> Then no element of S has R to all and only those elements
> in S which do not R to themselves."
>
> (J. F. Thomson, "On Some Paradoxes" in Analytical Philosophy,
> ed. R. J. Butler, New York: Barnes & Noble, 1962, p. 104-119.)
Yes, that is the instance based on sets. But I am talking about a
more general theorem that can also be used to formally derive the
various incompleteness theorems that I mentioned, from the Theory of
Computation (Turing), Proof Theory (Godel), English (the Liar Paradox)
and other domains.
(Thomson's Theorem is the special case where Q(a,b)="Set a contains
element b." and no additional rules of inference are applied. Thus
there is no generalization of the domain, Q, which is what is needed
to derive these other theorems, as I mentioned in my first post.)
It is also necessary to apply addition rules of inference to derive
these and their variations. My general theorem, -~P(x,x)[P(a,b)],
which I call the Axiom of Incompleteness (at a lower level of
abstraction it is a theorem, at the higher level at which I prefer to
work it is an axiom), is the starting point in each case, which per se
actually produces:
1. Theory of Computation: "The set of Turing Machines that don't halt
yes on themselves is not recursively enumerable." (Q(a,b) = "Turing
Machine a halts yes on input b.")
2. Proof Theory: "There is no wff that is provable iff its free
variable is replaced by the Godel number of a wff that is not provable
when its free variable is replaced by its Godel number (this predicate
is not 'representable')." (Q(a,b)="Wff a with b substituted for its
free variable is provable.")
3. Set Theory: There is no set of those sets that do not contain
themselves. (Q(a,b)="Set a contains element b.")
4. English: " 選t is not true of itself.' is true of itself." (Q(a,b)
= "English sentence a with its pronouns replaced by noun b is true.")
Additional rules are needed to derive variations such as:
1. Turing's unsolvability of the Halting Problem.
2. Godel's 1st Incompleteness Theorem (both versions given by Godel in
1931 - not to be confused with his 2nd Incompleteness Theorem) and
Smullyan's Dual Form theorem.
3. The (original) Liar Paradox: "This is false."
(See section VI of my arxiv paper for the formal derivation of
Turing's result and variations.)
(Dozens of variations of Godel's 1st Incompleteness Theorem are
generated, many of which are described, more informally, by Smullyan.
I recently posted some corrections to Smullyan's work that I uncovered
while formally deriving his theorems, programming my Theorem Generator
on a PC. Compare that to published "Theorem Provers" that generate
only 1 theorem, Godel's! I also posted 500 theorems generated by my
software.)
Note that the notion that Russell's Paradox, the Liar Paradox, and
Godel and Turing's results are analogous is wrong. Merely think about
variations of these and you realize that they are just the first of
many possible related theorems. The theorems from the first list 1-4
above are the true siblings. By formalizing (automating) the
derivation of these results we can see exactly what each theorem is
saying and why it is so.
In fact, my formalization clarifies a number of misconceptions
concerning paradoxes and incompleteness. The theorem described in the
introduction of Godel's 1931 article is NOT the same as the one given
in detail. Turing's result is NOT Godel's result applied to Turing
Machines (as Gregory Chaitin maintains.) My rigorous formalization
shows exactly which theorems are siblings. (For example, the sibling
to Turing's result is the fact that, assuming consistency, the set of
provable wffs is not recursive.)
With the hindsight of recursive computer programming, as well as the
Recursion Theory result that there is a program that outputs only
itself, can we believe in Russell's Vicious Circle Principle that "no
totality can contain members defined in terms of itself"?
And, believe it or not, the belief that not every predicate has an
associated set of the values for which it holds (which prompted ZF Set
Theory) is also wrong. For, think of predicates and sets as being
different ways of looking at the same thing: limiting our attention to
a part of our universe. Why can't we define predicates in parallel to
sets? The answer is, we can (and in fact do, intuitively.) But what
about the predicate "x is a set that is not a member of itself"? Keep
in mind that predicate = set and see if you can tell what the real
problem is. (I'll give the answer upon request.)
The moral of the story is, if you really want to understand something,
write a program to generate it. You'll see exactly how it works - and
lots of new results will pop out as well.
> F.
>
> P.S.
> You might also be interested in the /short/ paper 'Is Russell's Paradox
> Deep?' by Prof. Laurence Goldstein:
>
> http://www.hku.hk/philodep/dept/lg/docs/IRPD.pdf
Thanks for the reference, but gee, I don't see where he (Goldstein
describing Thomson's Theorem) addresses the general case, including
the Liar Paradox and Godel and Turing's theorems. Can you tell me
what page it's on or give me a brief summary/example of how they're
handled? (BTW: Goldstein repeats many of the misconceptions that I
mention above.)
Charlie-Boo
> Hello Charlie-Boo,
>
> (P(x) -> ~P(x)) <-> ~P(x), for every P.
> (P(x) -> ~P(x)) <-> (P(x) -> (contradiction))
> (P(x) -> ~P(x)) <-> (P(x) <-> (contradiction))
I was afraid of this. My bad. I should've explained further. P(x)
of course represents any wff that is "recursively enumerable" ("r.e.")
in the more general sense that I defined. The logical connective =>
is NOT used in wffs, but has as arguments wffs. Then P(x) => ~P(x) is
not a single wff, but rather the assertion that if a given set P is
"r. e." then ~P is "r. e.".
This is certainly not the case if Q(a,b) = "Turing Machine a halts yes
on input b." (thus P(x) is the actual definition of r.e.) For
example, the set of programs that halt yes on themselves is r.e. but
its complement is not. However, if Q(a,b) = "Set a contains element
b.", then P(x) means that P is a predicate which has a corresponding
set, and P(x) => ~P(x) does hold. Every set has a complement
(although I guess you really should check the axioms of the set theory
that you are using.)
P(x) => ~P(x) is one of the 3 rules of inference that together create
a contradiction. When Q(a,b) = "Set a contains element b.", this
contradiction is the Russell Paradox.
> > The paradoxes occur when there are 3 certain rules in effect for Q,
> > which lead to the contradiction that ~P(x,x)[P(a,b)]. The various
> > attempts to "resolve" these paradoxes (e.g. Russell's Type Theory and
> > disallowing a universal set) are generally disallowing some particular
> > step in the formal proof.
>
> None of the attempted solutions to the Russell Paradoxes
> (e.g. Russell's Type Theory and disallowing a universal set)
> work for the Barber Paradox.
The Theory of Types disallows a set from being a member of itself.
Applied to barbers, it disallows a barber from shaving himself.
> The predicate ~(x e x) has no extension.
That's what THEY think. Aren't predicates and sets intuitively the
same thing? The problem isn't that a predicate doesn't have a
corresponding set. The problem is that there is no predicate "x is a
set that does not contain itself", because that would be the predicate
"x is a predicate that does not hold for itself", which doesn't exist.
> Some posters, unfortunately, prefer to insult others or to simly be profane
> because they can. ???
"He who establishes his argument by noise and command,
shows that his reason is weak."
Michel de Montaigne
> Regards,
>
> Owen
G. Frege
... <->x ...
> >
> > is just Russell's shorthand notation for
> >
> > (x)(... <-> ...),
>
> It seems strange to refer to a variable and only later in the
> expression indicate that it is universally quantified.
>
Huh?
Actually, there's NOTHING "strange" concerning that notation. It's just
not common these days any more. That's all. (Hint: it's just an "infix
notation".)
Look, man, things would be simpler if you weren't such an ignorant
bonehead.
F.
Witt
Not so. x shaves x, is not dependent on any theory of types.
In, x shaves y, x and y are of the same type. x shaves x, has the same
sense,
with or without the theory of types.
Ax(y shaves x <-> ~(x saves x)) -> (y shaves y <-> ~(y shaves y)).
(y shaves y <-> ~(y shaves y)) is paradoxical in that it cannot be true,
it is a contradiction!
The y such that Ax(yRx <-> ~(xRx)) does not exist, with or without Types,
for all R's.
Types are relevant wrt the membership predicate.
~(x e x) violates the theory of types, but, ~(x shaves x) does not.
>
> > The predicate ~(x e x) has no extension.
>
> That's what THEY think. Aren't predicates and sets intuitively the
> same thing?
No. It depends on the intuitive assumptions of the structure of the language
in use.
For a language that imposes the theory of types, they are equal. See:
Russell, Carnap.etc.
In those systems, F = {x:Fx}, Exists{x:Fx} for all F's, and y e {x:Fx} <->
Fy, are true.
These systems deny the existence of troublesome predicates such as: x e x,
~ (x e x),
x e y & y e z & x e z, x = {x}, etc.
For systems that do not adopt a theory of types, eg: Zermelo, von Neumann,
Quine, etc.,
other methods of avoiding the 'paradoxes' are required. For example,
Quine: EyAx(x e y <->Fx) only for stratified F, or, EyAx(x e y <-> (x e V &
Fx)).
Zermelo: EyAx(x e y <-> (x e z & Fx)).
Me: EyAx(x e y <-> (EzAx(x e z <-> Fx) & Fx)).
etc.
> The problem isn't that a predicate doesn't have a
> corresponding set. The problem is that there is no predicate "x is a
> set that does not contain itself", because that would be the predicate
> "x is a predicate that does not hold for itself", which doesn't exist.
Not so. ~(x e x) exists for Quine and others, but {x:~(x e x)} does not
exist. (NF)
Some writers claim {x:~(x e x)}exists as a 'proper' class but not as a set
???
Note: If we define {x:Fx} as (the y such that: Ax(x e y <-> Fx)), (see
Quine,
Methods of Logic, page 300) then, G{x:Fx} <-> Ey(Ax(x e y <-> Fx) & Gy).
i.e. G{x:~(x e x)} <-> Ey(Ax(x e y <-> ~(x e x) & Gy), but, Ax(x e y <-> ~(x
e x))
is contradictory, therefore, ~EG(G{x:~(x e x)}), ie. {x:~(x e x)} does not
exist!
Also, y e {x:Fx} <-> (EzAx(x e y <-> Fx) & Fx) is a theorem here.
Witt
Charlie-Boo
> "Charlie-Boo" <ch...@aol.com> wrote
> > The Theory of Types disallows a set from being a member of itself.
> > Applied to barbers, it disallows a barber from shaving himself.
>
> Not so. x shaves x, is not dependent on any theory of types.
The Theory of Types is an arbitrary prohibition meant to avoid
Russell's Paradox. As noted by all, there is a parallel between
barbers shaving and sets containing, i.e., Russell's Paradox and the
Barber Paradox. If we continue this analogy, then the Theory of Types
translates into a prohibition against barbers shaving themselves.
If you are saying that "barbers can shave themselves", then one could
say "sets can contain themselves" and deny that the Theory of Types
applies to sets either.
Whether you believe that sets or barbers can be self-applied, the same
set of 3 rules of inference are in effect in both cases, and that
combination, as I have said, is inconsistent. The Theory of Types is
merely arbitrarily prohibiting one of these 3 rules of inference. The
point is that you can't have those 3 rules simultaneously, whatever
system (sets, Turing Machines, Logic, English) you are using.
> > > The predicate ~(x e x) has no extension.
> >
> > That's what THEY think. Aren't predicates and sets intuitively the
> > same thing?
>
> No. It depends on the intuitive assumptions of the structure of the language
> in use. For a language that imposes the theory of types, they are equal. See:
> Russell, Carnap.etc.
This is merely adding an arbitrary restriction, making their
definition of set and predicate different from the intuitive
definition. I am saying that is not necessary. One can keep the
definition of a set and predicate as being essentially the same,
rather than altering these concepts to the point of contradicting the
intuitive notions.
The problem isn't that sets and predicates can't be the same thing.
This just leads to more problems, besides the fact that the system
denies that sets and predicates are synonymous. Define a "tet" to be
the same thing as a "set". Then there is no tet of all tets that
don't contain themselves, but there is a tet that contains all sets
that don't contain themselves. But then, a tet is a set, so we have
another contradiction. Solution: There is no tet of sets that don't
contain themselves, because that is equivalent to the tets that don't
contain themselves. Likewise, there is no predicate "x is a set that
does not contain itself". (This is just applying one more step in my
formal proof of inconsistency.)
SETS = CLASSES = PREDICATES. That is true intuitively and it is not
necessary to contradict that fact.
> For systems that do not adopt a theory of types, eg: Zermelo, von Neumann,
> Quine, etc., other methods of avoiding the 'paradoxes' are required.
Right. They have to avoid one of the three rules that I formalized.
But they don't have to deny that sets, classes and predicates are all
the same thing. I KNOW they say that in their systems. But they're
wrong. Sorry.
> > The problem isn't that a predicate doesn't have a
> > corresponding set. The problem is that there is no predicate "x is a
> > set that does not contain itself", because that would be the predicate
> > "x is a predicate that does not hold for itself", which doesn't exist.
>
> Not so. ~(x e x) exists for Quine and others, but {x:~(x e x)} does not
> exist. (NF)
> Some writers claim {x:~(x e x)}exists as a 'proper' class but not as a set
> ???
I'm really not talking about what some people write, believe and work
with. They can have their theories, but they are unnecessarily
abandoning the real, intuitive notion of a set being another name for
a predicate.
Question: Would you prefer ("all else considered equal") a set theory
in which sets and predicates are the same thing, or one in which they
are different?
> Witt
KRamsay
(Charlie-Boo) writes:
>The Theory of Types is an arbitrary prohibition meant to avoid
>Russell's Paradox.
I don't think it's so arbitrary, since it has some philosophical
motivation. One analysis of the paradoxes goes sort of like
this:
When we consider the sets of X's for some previously
known type of thing X, such as "sets of birds", we are performing
an act of linguistic construction. We are creating a set of
conventions concerning what phrases referring to sets of birds
mean, which explain the meaning of those phrases in terms
of terms we already know ("bird", especially).
In Russell's paradox, he deals with arbitrary sets. On the one
hand, these alleged arbitrary sets are allowed to contain any
kind of arbitrary set. So the type of thing that they are permitted
to contain is "arbitrary set". On the other hand, we haven't
justified yet the construction of such a notion as "arbitrary
set", so we don't have a pre-existing type of thing to base our
abstract construction on. This (by this story) is why his notion
fails to cohere.
In a type theory, one describes the construction of certain
specific types of set, and the constructions are individually
justified by deriving references to sets of one level from
references to sets of the previous level.
So if it's meant as a cure, it is at least not a completely
arbitrary cure. It's a cure motivated by a certain understanding
of what makes references to sets of things okay, and a belief
that the paradox is a result of having wandered away from
such an approach.
If you want a different idea of what needs to be fixed, take a
look at Weyl's _The Continuum_.
I'll agree that some of the later "adjustments" to the theory
of types are not so well motivated, and were motivated more
by a practical desire to make the axiom system one in which
more ordinary mathematics could be formalized. Probably
it makes more sense to go either the one direction or the
other direction whole hog, either to construct a theory based
on a philosophical motivation, or to construct a theory which
is intended purely to formalize "wild" unformalized mathematics,
like ZFC. I understand the axioms of ZFC where chosen in order
to formalize a proof of the well-ordering theorem.
Keith Ramsay
Thomas Bushnell, BSG
There is no paradox in the "barber who shaves each man that does not
shave himself."
The barber is a woman. Shame on Russell. :)
Thomas
Witt
"Thomas Bushnell, BSG" <tb+u...@becket.net> wrote in message
news:87r7z5k...@becket.becket.net...
Shame on you, the barber cannot exist.
Therefore it cannot be male or female.
There is no thing that it is.
Those who do not shave are not excluded from the argument.
Thomas Bushnell, BSG
Huh? "The barber who shaves each *man* that does not shave himself."
I haven't got Russell's book at hand to check the wording, but I
believe this is what he actually said.
Thomas
Thomas Bushnell, BSG
I believe the original is a sign in a barber shop that says "I shave
all those men, and only those men, who do not shave themselves."
The sign can be consistently true, provided the barber is not a man.
I wonder if Russell actually saw such a sign before he exposed this
paradox in naive set theory.
Thomas
Witt
"Thomas Bushnell, BSG" <tb+u...@becket.net> wrote in message
news:874qw1k...@becket.becket.net...
>
> I believe the original is a sign in a barber shop that says "I shave
> all those men, and only those men, who do not shave themselves."
>
> The sign can be consistently true, provided the barber is not a man.
No it cannot. Ax(y shaves x <-> ~(x shaves x)) is a contradiction.
There cannot be a thing (male or female) that satisfies this condition.
Thomas Bushnell, BSG
> "Thomas Bushnell, BSG" <tb+u...@becket.net> wrote in message
> news:874qw1k...@becket.becket.net...
> >
> > I believe the original is a sign in a barber shop that says "I shave
> > all those men, and only those men, who do not shave themselves."
> >
> > The sign can be consistently true, provided the barber is not a man.
>
> No it cannot. Ax(y shaves x <-> ~(x shaves x)) is a contradiction.
> There cannot be a thing (male or female) that satisfies this condition.
Ax(S(y,x) <-> ~S(x,x)) is a contradiction.
But the quoted sentence is not of that form. It is actually
ambiguous, and could be either of the following:
Ax(M(x) -> (~S(x,x) <-> S(y,x)))
Ax((M(x) & ~S(x,x)) <-> S(y,x))
From either of those, one can conclude ~M(x).
Thomas
Charlie-Boo
> On 14 Dec 2003 12:37:00 -0800, ch...@aol.com (Charlie-Boo) wrote:
>
> ... <->x ...
>
> > > is just Russell's shorthand notation for
> > >
> > > (x)(... <-> ...),
> >
> > It seems strange to refer to a variable and only later in the
> > expression indicate that it is universally quantified.
> >
> Huh?
>
> Actually, there's NOTHING "strange" concerning that notation. It's just
> not common these days any more. That's all.
strange [straengz]: adjective: "not common"
> (Hint: it's just an "infix notation".)
Infix notation refers to functions (often called "operators" in this
context), not quantifiers.
Predicate Calculus puts quantifiers before any reference to the
variable quantified, so that you know when you get to it (without
lookahead) - sort of like how one would naturally say it, "For all x .
. ."
> Look, man, things would be simpler if you weren't such an ignorant
> bonehead.
Whether someone considers something strange or not is a funnction of
their value system, not their intellect.
> F.
Torkel Franzen
> I wonder if Russell actually saw such a sign before he exposed this
> paradox in naive set theory.
You mean in Frege's highly non-naive system - not in naive set
theory.
G. Frege
>
> Infix notation refers to functions (often called "operators" in this
> context), not quantifiers.
>
Go away, idiot.
G. Frege
> >
> > There is no paradox in the "barber who shaves each man that does not
> > shave himself."
> >
> > The barber is a woman. Shame on Russell. :)
> >
>
> Shame on you, the barber cannot exist.
>
You are wrong (this time), Owen, IF we phrase the "paradox" as mentioned
above. There really IS NO paradox. The barber may be any being _except a
man_.
Only ONE thing is sure (i.e. can be derived from the statement above):
>
> it [the barber] cannot be male.
>
Of course, with the presupposition that ONLY man can be barbers
[something that certainly w a s true when Russell came up with his
statement] we get the conclusion:
>
> the barber cannot exist.
>
F.
The formalization of the "paradox" would be now:
Ax(men x -> b shaves x <-> ~(x shaves x))
Now *assume*
men b.
Then we would (immediately) get
b shaves b <-> ~(b shaves b)
and hence a contradiction. Thus
~men b.
Thomas Bushnell, BSG
Sorry, yes, I should have been more precise.
Naive set theory is an underspecified beast, so whether the
Burali-Forti paradox or the Russell paradox really affect it is hard
to answer clearly.
Thomas
G. Frege
> >
> > I believe the original is a sign in a barber shop that says "I shave
> > all those men, and only those men, who do not shave themselves."
> >
> > The sign can be consistently true, provided the barber is not a man.
>
> No it cannot.
>
Of course, it can.
>
> Ax(y shaves x <-> ~(x shaves x)) is a contradiction.
>
Right. But your statement is NOT an appropriate translation of
"y shaves all those men, and only those men, who do not shave
themselves",
for you dropped the condition "men" (!). Hence we have to use the
translation:
Ax(men x -> (y shaves x <-> ~(x shaves x))). (*)
And actually there CAN be a female barber, say Barbara (b), which shaves
all men [of Seville] who do not shave themselves.
BTW: Actually the statement (*) is much better than the one without the
condition on "man". Since in the latter case the barber would be
condemned to shave ANYTHING which does not shave itself... well...
actually a "supertask" NO MAN can perform... :-)
F.
Well, of course, the "solution" to this conundrum is that our "universe
of discourse" is [silently] restricted just to _the man of Seville_.
Then we may ask:
EyAx(y shaves x <-> ~(x shaves x))?
And the answer certainly will be:
~EyAx(y shaves x <-> ~(x shaves x)).
G. Frege
BSG) wrote:
> From ...
>
> Ax(M(x) -> (~S(x,x) <-> S(y,x)))
>
> ...one can conclude ~M(y).
>
Right. Probably one actually should interpret M(x) with "x is a men of
Seville". This way the /pseudoparadox/ would take a quite reasonable
form.
("Curry uses the term pseudoparadox to describe an apparent paradox,
such as the catalogue paradox, for which there is no underlying actual
contradiction." --mathworld.wolfram.com)
F.
Witt
"Charlie-Boo" <ch...@aol.com> wrote in message
news:3df1e59f.03121...@posting.google.com...
> "Witt" <oori...@yahoo.com> wrote
> > "Charlie-Boo" <ch...@aol.com> wrote
>
> > > The Theory of Types disallows a set from being a member of itself.
> > > Applied to barbers, it disallows a barber from shaving himself.
> >
> > Not so. x shaves x, is not dependent on any theory of types.
>
> The Theory of Types is an arbitrary prohibition meant to avoid
> Russell's Paradox. As noted by all, there is a parallel between
> barbers shaving and sets containing, i.e., Russell's Paradox and the
> Barber Paradox.
Yes, they are both instances of Ax(yRx <-> ~(xRx)), ie. they have R in
common.
There is no y of any type such that Ax(yRx <-> ~(xRx)).
> If we continue this analogy, then the Theory of Types
> translates into a prohibition against barbers shaving themselves.
No it does not! We cannot continue this analogy.
The theory of types does deny ~(x e x), but, it does not deny ~(x shaves x).
>
> If you are saying that "barbers can shave themselves", then one could
> say "sets can contain themselves" and deny that the Theory of Types
> applies to sets either.
Of course, many languages deny the theory of types.
>
> Whether you believe that sets or barbers can be self-applied, the same
> set of 3 rules of inference are in effect in both cases, and that
> combination, as I have said, is inconsistent. The Theory of Types is
> merely arbitrarily prohibiting one of these 3 rules of inference. The
> point is that you can't have those 3 rules simultaneously, whatever
> system (sets, Turing Machines, Logic, English) you are using.
What 3 rules are you talking about?
>
> > > > The predicate ~(x e x) has no extension.
> > >
> > > That's what THEY think. Aren't predicates and sets intuitively the
> > > same thing?
> >
> > No. It depends on the intuitive assumptions of the structure of the
language
> > in use. For a language that imposes the theory of types, they are equal.
See:
> > Russell, Carnap.etc.
>
> This is merely adding an arbitrary restriction, making their
> definition of set and predicate different from the intuitive
> definition. I am saying that is not necessary. One can keep the
> definition of a set and predicate as being essentially the same,
> rather than altering these concepts to the point of contradicting the
> intuitive notions.
You are correct only if you include some 'Type' theory in your assumed
intuitive theory.
>
> The problem isn't that sets and predicates can't be the same thing.
> This just leads to more problems, besides the fact that the system
> denies that sets and predicates are synonymous. Define a "tet" to be
> the same thing as a "set". Then there is no tet of all tets that
> don't contain themselves, but there is a tet that contains all sets
> that don't contain themselves. But then, a tet is a set, so we have
> another contradiction. Solution: There is no tet of sets that don't
> contain themselves, because that is equivalent to the tets that don't
> contain themselves. Likewise, there is no predicate "x is a set that
> does not contain itself". (This is just applying one more step in my
> formal proof of inconsistency.)
>
> SETS = CLASSES = PREDICATES. That is true intuitively and it is not
> necessary to contradict that fact.
Only if you include some 'Type' theory in your assumed intuitive theory.
>
> > For systems that do not adopt a theory of types, eg: Zermelo, von
Neumann,
> > Quine, etc., other methods of avoiding the 'paradoxes' are required.
>
> Right. They have to avoid one of the three rules that I formalized.
> But they don't have to deny that sets, classes and predicates are all
> the same thing. I KNOW they say that in their systems. But they're
> wrong. Sorry.
They are not wrong within their intuitive set theory.
They are wrong within your intuitive set theory.
Why do you think that your way is the only correct way?
>
> > > The problem isn't that a predicate doesn't have a
> > > corresponding set. The problem is that there is no predicate "x is a
> > > set that does not contain itself", because that would be the predicate
> > > "x is a predicate that does not hold for itself", which doesn't exist.
> >
> > Not so. ~(x e x) exists for Quine and others, but {x:~(x e x)} does not
> > exist. (NF)
> > Some writers claim {x:~(x e x)}exists as a 'proper' class but not as a
set
> > ???
>
> I'm really not talking about what some people write, believe and work
> with. They can have their theories, but they are unnecessarily
> abandoning the real, intuitive notion of a set being another name for
> a predicate.
>
> Question: Would you prefer ("all else considered equal") a set theory
> in which sets and predicates are the same thing, or one in which they
> are different?
I admit that set theories with a type theory are easier to deal with, but,
I prefer a set theory without the very awkward theory of types.
y e {x:Fx} <-> Fy is valid for you, and, E!{x:Fx} ->. y e {x:Fx} <-> Fy
is valid for me.
Your assumption that every predicate determines a class is false without
the assumption of 'types'.
George Greene
You mean you're a tendentious jackass as usual, Torkel.
The reason WHY Russell's paradox occurs in Frege's system
IS BECAUSE it occurs in naive set theory. The axiom in Frege's
system that produced this particular paradox is itself DEFINITIVE
of naive set theory.
--
---
"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
George Greene
: Torkel Franzen <tor...@sm.luth.se> writes:
:
: > tb+u...@becket.net (Thomas Bushnell, BSG) writes:
: >
: > > I wonder if Russell actually saw such a sign before he exposed this
: > > paradox in naive set theory.
: >
: > You mean in Frege's highly non-naive system - not in naive set
: > theory.
:
: Sorry, yes, I should have been more precise.
No, really, you shouldn't have.
Torkel should learn to wait until he has more to say than
irrelevant 1-liners before posting.
: Naive set theory is an underspecified beast,
Maybe, but that's not the point. The point is that
unrestricted comprehension is dangerous.
: so whether the
: Burali-Forti paradox or the Russell paradox really affect it is hard
: to answer clearly.
No, it isn't. Even though "naive set theory is under-
specified", it is ALWAYS specified well enough for people to
know that it includes unrestricted comprehension. That is enough
for Russell's paradox.
Thomas Bushnell, BSG
> : Burali-Forti paradox or the Russell paradox really affect it is hard
> : to answer clearly.
>
> No, it isn't. Even though "naive set theory is under-
> specified", it is ALWAYS specified well enough for people to
> know that it includes unrestricted comprehension. That is enough
> for Russell's paradox.
Well, the reason for my semi-retraction is twofold:
To give a good example to James Harris, and
At least one naive set theory text on my bookshelf doesn't have
unrestricted comprehension, but instead waves about with "this is a
dangerous area" when it gets near the universal set, or the largest
ordinal, and whatnot., and says "you can't just comprehend anything
without problems".
Thomas
Torkel Franzen
> At least one naive set theory text on my bookshelf doesn't have
> unrestricted comprehension,
In what "naive set theory" do you in fact find unrestricted
comprehension?
George Greene
: unrestricted comprehension,
I'm sorry, I don't believe you.
: but instead waves about with "this is a
: dangerous area"
Well, WHY is it dangerous, if it restricts comprehension?
If you restrict comprehension, you can eliminate the danger; that
is the whole reason why you accept to the restriction! It would certainly
be BAD to accept restrictions and STILL be in danger, would it not??
: when it gets near the universal set, or the largest
: ordinal, and whatnot.,
Well, at this point, the antecedent of "it" needs clarifying.
I'll not wax so pedantic as to demand that you post the book's
actual particular axiomatization, but my point is, its framework
either calls a universal set into existence or it doesn't, and if
it does, it either suffers from Russell's paradox or it doesn't.
If it does, then saying it lacks unrestricted comprehension
is almost irrelevant: it's got something provably just as dangerous.
: and says "you can't just comprehend anything
: without problems".
But to be *aware* of PRECISELY *this* is PRECISELY what it
means NOT to be *naive*, in the relevant sense.
More to the point, why would it *need* to warn, "you can't just
comprehend anything" IF it was not (on the theoretical basis
of what was presented before the warning) in fact about to
ALLOW you to comprehend "anything"? To say "you can't just
comprehend anything" IS, dictionarially, restricting comprehension.
If the theory-as-presented actually incorporated this natural-language
prohibition mathematically, into its axioms, then it was not "naive"
in the relevant sense. If it didn't, then that's what it means for
its comprehension to be "unrestricted".
Thomas Bushnell, BSG
As I said, "naive set theory" as a term is underspecified, and
different texts give different accounts.
Barwise and Etchemendy's "Language, Proof, & Logic", pp 405-441, in
describing naive set theory, then builds up to Russell's paradox, and
then shows a ZFC axiom. On page 408, in the section titled "Naive Set
Theory":
"The second principle of naive set theory is the so-called
Unrestricted Comprehension Axiom. It states, roughly, that every
determinate property determines a set. That is, given any
determinate property P, there is a set of all objects that have this
property.... This way of talking about the Axiom of Comprehension
has a certain problem, namely it talks about properties. We don't
want to get into the business of having to axiomatize properties as
well as sets. To get around this, we use formulas of first-order
logic. Thus, for each formula P(x) of FOL, we take as a basic axiom
the following: EaVx[x in a <-> P(x)]."
By contrast, the textbook Shen and Vereshchagin "Basic Set Theory",
takes the approach of saying that there are "danger areas", which they
warn about, and describe ZFC as giving "safety rules" to keep one out
of the "danger areas".
This difference is of course related to the difference in the point of
these two books. LPL is concerned with showing the value and
importance of FOL axiomatizations, whereas Basic Set Theory is
concerned with introducing potential working mathematicians to the
necessary basics of set theory.
Thomas
George Greene
:
: > At least one naive set theory text on my bookshelf doesn't have
: > unrestricted comprehension,
Torkel Franzen <tor...@sm.luth.se> writes:
: In what "naive set theory" do you in fact find unrestricted
: comprehension?
All of them, by definition.
Torkel Franzen
> Barwise and Etchemendy's "Language, Proof, & Logic", pp 405-441, in
> describing naive set theory, then builds up to Russell's paradox, and
> then shows a ZFC axiom. On page 408, in the section titled "Naive Set
> Theory":
>
> "The second principle of naive set theory is the so-called
> Unrestricted Comprehension Axiom.
But where is this naive set theory to be found? Has anybody ever
used naive set theory?
Thomas Bushnell, BSG
What I said was that the term "naive set theory" is underdetermined,
and that some texts on my shelf say it has unrestricted comprehension,
and some instead say it doesn't, but are vague (and non-axiomatic, and
non-rigorous) about what the restrictions are.
From two sides, I was disbelieved. I've now posted quotes from two
texts. (Halmos's "Naive Set Theory" isn't at my home, so I can't
check it as easily, or I would have.)
Now you shift, onto a question of "where is this to be found". Well,
one place it is to be found is in Barwise and Etchemendy's LPL.
"Naive set theory" just isn't a single well-defined thing. That's my
point. Your insistence that it *must* be, thus defining out of
existence the texts which take an opposing view on what it is, does
not demonstrate that, in *fact*, the term "naive set theory" refers to
a variety of different things, and simply has no one universal and
well understood definition.
Thomas
G. Frege
BSG) wrote:
>
> Halmos's "Naive Set Theory" isn't at my home, so I can't
> check it as easily, or I would have.
>
Halmos' book is -despite it's title- NOT about "naive set theory". :-)
(He concedes that in the preface of the book. :-)
Actually, he just describes good old ZFC in the book.
F.
George Greene
:
: > Barwise and Etchemendy's "Language, Proof, & Logic", pp 405-441, in
: > describing naive set theory, then builds up to Russell's paradox, and
: > then shows a ZFC axiom. On page 408, in the section titled "Naive Set
: > Theory":
: > "The second principle of naive set theory is the so-called
: > Unrestricted Comprehension Axiom.
Torkel Franzen <tor...@sm.luth.se> writes:
: But where is this naive set theory to be found?
Well, last&least, on pp.405-441 of this book.
: Has anybody ever used naive set theory?
Well, since it is inconsistent, maybe not.
"Used" is too strong a term. People have attempted
to axiomatize set theory and possibly stumbled across it
along the way. People (specifically Frege) have assumed that
unrestricted comprehension was legitimate and have used it.
There was a lot of it going on before it was FIGURED OUT that
comprehension needed to be limited. Specifically, Russell
wrote Frege about the paradox in 1902, and the axiomatization
of set theory that cured it was published by Zermelo in 1908.
All I'm saying is that people were forming sets without being
careful about how, for several years prior to 1903.
Since you already knew this, the motivation for your
question remains tragically obscure, but while you may
have just barely avoided actual impropriety here, you have
certainly not avoided the appearance of impropriety.
George Greene
: point.
But it is well-enough defined to make its comprehension, if not
"un"restricted, at least overbroad enough to get you into trouble.
Naive already has a dictionary meaning in natural language, before
set theory comes along. Even if "naive set theory" is not well
defined, all of the various things it could be defined as STILL
have to merit being called naive about something. The importance of
not waxing overbroad in comprehension is usually that something.
: Your insistence that it *must* be, thus defining out of
: existence the texts which take an opposing view on what it is,
The text you cite DOES NOT "take an opposing view on what it is".
You cannot hope to demonstrate that the text you cited thinks that
it can restrict comprehension and still call itself "naive". That text
does not in fact do that.
: does
: not demonstrate that, in *fact*, the term "naive set theory" refers to
: a variety of different things, and simply has no one universal and
: well understood definition.
It is universally well understood that it is about avoiding things
like Russell's paradox that can arise from unrestricted comprehension.
The set theory that you were trying to allege was naive but lacked
unrestricted comprehension does in fact restrict comprehension, but
it is NOT naive. That's not due to some choice made by Torkel Franzen;
it's due to the community's linguistic practice generally, which,
around this particular term and issue, is in fact more homogeneous
than you are giving it credit for.
But Torkel is still abusing you.
Thomas Bushnell, BSG
> : existence the texts which take an opposing view on what it is,
>
> The text you cite DOES NOT "take an opposing view on what it is".
> You cannot hope to demonstrate that the text you cited thinks that
> it can restrict comprehension and still call itself "naive". That text
> does not in fact do that.
Huh? The back cover identifies it as a presentation of "naive set
theory"; the text is a translation from Russian, so perhaps it's a
poor guide about usage.
At best, what you are saying is that the authors of the book, or the
translator, or the AMS editors who put the back cover together, are
using the term wrongly.
Ok--then fine, but they are still using it in a certain way, to refer
to a vaguely and suggestively limited comprehension.
> It is universally well understood that it is about avoiding things
> like Russell's paradox that can arise from unrestricted comprehension.
> The set theory that you were trying to allege was naive but lacked
> unrestricted comprehension does in fact restrict comprehension, but
> it is NOT naive.
I'm not "trying to allege it's naive". I'm saying that it is in print
labelled as naive. That labelling may well be wrong, but it is there,
nontheless.
I think I would generally agree that the usage of naive to mean
vaguely limited comprehension is a disappointing usage. I would
prefer to restrict it just as you do, and in my own usage, I tend to.
One also hears of "informal set theory" to be the vaguely limited
comprehension version; that's probably the best way to describe the
book I refer to as well as Halmos.
Wikipedia thinks that the name "Naive set theory" may well have
originated with Halmos' book, which is odd, given his preface.
> But Torkel is still abusing you.
Well, I made a mistake here not too long ago, which I corrected after
about two posts. He apparently thinks that if a person makes a
mistake, it's fair game from then on to take random pot shots without
limitation.
Thomas
Thomas Bushnell, BSG
Here's a lecture note page I ran across, as evidence for my claim that
the phrase "naive set theory" often refers to a system with vaguely
limited comprehension:
http://www.cs.odu.edu/~toida/nerzic/content/set/intr_to_set.html:
Though the concept of set is fundamental to mathematics, it is not
going to be defined rigorously here. Instead we rely on everyone's
notion of "set" as a collection of objects or a container of
objects. In that sense "set" is an undefined concept here. Similarly
we say an object "belongs to " or "is a member of" a set without
rigorously defining what it means. This approach to set theory is
called "naive set theory" as opposed to "axiomatic set theory". The
naive set theory produces paradoxes such as Russell's paradox, hence
it is not consistent, meaning that a statement which should be true
may not be proven true following the naive set theory. However, it is
simpler and practically all the results we need can be derived within
the naive set theory. Thus we shall be following this naive set theory
in this course.
So the course will use "naive set theory", which "produces paradoxes",
but at the same time since "practically all the results we need can be
derived within the naive set theory."
Now if "naive set theory" produces real antinomies, then of course all
the results we need can be derived in it. And a lot more results that
we don't want either.
So the authors of that paragraph are dancing a fine line, and are
problably saying something strictly incoherent. They are saying that
naive set theory is inconsistent, and they are saying that it matters
what results it proves.
But there is another interpretation, in which they mean to say it
provise "practically all the results we need", and then they say in
the back of their head "and we will only use methods that we know can
be repeated in a proper axiomatized [ZFC, GBN, etc] set theory".
Imagine if a bright student comes up and says "hey, if it's
inconsistent, then what does it matter if a result can be proved in
it---*anything* can be proved in it?!" The instructor would answer by
saying... that uses of comprehension will be limited in certain
ways...which don't produce the paradoxes...
Now if you want "naive set theory" to refer to a single, well-defined
thing, then sure, it probably must refer to the unrestricted
comprehension system. If there is an axiom system for naive set
theory, it must be Frege's or one like it.
But my whole point is that people very frequently use "naive set
theory" in a vague hand-wavy sort of way, which does not conform to
the expectation that it be a single well-defined thing.
Thomas
Thomas Bushnell, BSG
The following page describes naive set theory implying limited
comprehension:
http://www.math.niu.edu/~rusin/known-math/index/03EXX.html
Naive set theory considers elementary properties of the union and
intersection operators -- Venn diagrams, the DeMorgan laws,
elementary counting techniques such as the inclusion-exclusion
principle, partially ordered sets, and so on. This is perhaps as
much of set theory as the typical mathematician uses. Indeed, one
may "construct" the natural numbers, real numbers, and so on in this
framework. However, situations such as Russell's paradox show that
some care must be taken to define what, precisely, is a set.
So here "naive set theory" means some kind of "subset" of axiomatic
set theory. Lest one get a paradox, vaguely defined "some care" must
be taken. Moreover, it is clear that this "naive set theory" is
really used.
Thomas
Thomas Bushnell, BSG
The following article from the FOM list, by Soren Riis:
http://www.cs.nyu.edu/pipermail/fom/1998-September/002167.html
(which gives an interesting variation on Freiling's argument, I
thought), describes there as being mathematicians who "accept naive
set-theory". Now perhaps Soren Riis thinks that these mathematicians
really accept a system which is inconsistent. But I don't think
that's what he means.
I think he means "naive set theory" to include a vaguely restricted
comprehension, without committing to any particular restriction or any
particular axiomatization.
Thomas
Torkel Franzen
> From two sides, I was disbelieved. I've now posted quotes from two
> texts. (Halmos's "Naive Set Theory" isn't at my home, so I can't
> check it as easily, or I would have.)
That book presents ZFC.
> Now you shift, onto a question of "where is this to be found". Well,
> one place it is to be found is in Barwise and Etchemendy's LPL.
You mean that the term "naive set theory" is to be found
there. Sure. But if you say that Russell showed "naive set theory" to
be inconsistent, there is a strong suggestion that somebody had
actually formulated, explicitly or implicitly, such a theory. Is this
the case?
Thomas Bushnell, BSG
> > Now you shift, onto a question of "where is this to be found". Well,
> > one place it is to be found is in Barwise and Etchemendy's LPL.
>
> You mean that the term "naive set theory" is to be found
> there. Sure. But if you say that Russell showed "naive set theory" to
> be inconsistent, there is a strong suggestion that somebody had
> actually formulated, explicitly or implicitly, such a theory. Is this
> the case?
No, I mean that Barwise and Etchemendy explicitly state that what they
present is "naive set theory". This does not prove what naive set
theory means. But it *does* prove that the term is used, in print, to
refer to a system with unlimited comprehension.
My point was that there are divergent uses of the term in print. If
you want to pick one and say "that's the right one", go ahead. Just
be careful, because there are plenty out there in print saying the
other one--for whichever you print.
Since the term "Naive Set Theory" may only go back to Halmas's book,
which he titles "Naive Set Theory", and then says isn't about naive
set theory, there is something amusing in
1) Insisting that the term is relevant to what happened a hundred
years ago, and
2) Insisting that it must have a sure rigid meaning.
Thomas
Torkel Franzen
> 1) Insisting that the term is relevant to what happened a hundred
> years ago, and
Given that the term is irrelevant to what happened a hundred years
ago (e.g. in set theory), why did you wonder "if Russell actually saw
such a sign before he exposed this paradox in naive set theory"?
Thomas Bushnell, BSG
Good grief, you are an annoying little pedant, aren't you?
So far you have screwed up several times, but I'm sure that won't stop
you in your continual attempt to make me look bad.
Yes, people did formulate a family of theories which *today* we call
naive set theory; at Russell's time, it was not called "naive set
theory".
Indeed, Frege's theory is rightly called naive *today*--and is, in
print--because it is unaware of the paradoxes and naively proceeds as
if they won't occur.
Did Russell call Frege's theory naive? No.
Is the *term* relevant to what happened a hundred years ago? No.
Does that mean that the *concept* is not important? No. The
*concept* is different from the *term*.
Perhaps the problem is that your English isn't so good. Regardless,
please take a big step back and stop the stupid attempts to score
points. Is that really all that fun?
In any case, I hereby award you a zillion points, thus saving you from
any further need to score any.
Thomas
Torkel Franzen
> Yes, people did formulate a family of theories which *today* we call
> naive set theory; at Russell's time, it was not called "naive set
> theory".
Who formulated such a theory? Cantor certainly didn't, and it's far
from clear who else you might have in mind.
Witt
"Torkel Franzen" <tor...@sm.luth.se> wrote in message
news:vcbfzfk...@beta13.sm.luth.se...
Hi Torkel,
Yes, it is the case that, for example, Frege's set theory is naive.
It is naive to assume that all predicates have an extension.
It is naive to assume that EyAx(x e y <-> Fx) is true for all predicates F.
~EyAx(x e y <-> ~(x e x)) is a theorem.
Frege's axiom V: Ax(Fx <-> Gx) -> {x:Fx}={x:Gx}, is naive, because it is not
valid.
It fails if either {x:Fx} or {x:Gx} do not exist.
Ax(~(x e x) <-> ~(x e x)) -> {x:~(x e x)}={x:~(x e x)} fails. Because,
Ax(~(x e x) <-> ~(x e x)) is tautologous and {x:~(x e x)}={x:~(x e x)}is
contradictory.
It is naive to assume: y e {x:Fx} <-> Fy is true for all F's. Because it
fails if {x:Fx} does not exist.
~Ey(y e {x:~(x e x)}) is a theorem that conflicts with the naive set
theories of: Frege, Cantor , Quine, etc..
Witt
Thomas Bushnell, BSG
Have you heard of Frege? His set theory certainly naive.
And Cantor certainly did, to the extent he is rightly the first
serious discoverer of Set Theory.
But the funny thing is that you are trying to distract attention from
*your* screw up, which was your silly claim implication nobody thinks naive
set theory includes unrestricted comprehension.
When I make mistakes, I have the decency and honesty to admit them.
Do you?
Thomas
Torkel Franzen
> Have you heard of Frege? His set theory certainly naive.
I don't think you want to describe Frege's Grundgesetze system as
"naive set theory" in the sense of Barwise and Etchmendy.
> And Cantor certainly did, to the extent he is rightly the first
> serious discoverer of Set Theory.
Cantor did indeed create set theory. Since he did not introduce any
unrestricted comprehension axiom there is no basis for the idea that
he used or introduced "naive set theory" in your technical sense.
Alan Smaill
arguably, naive set theory is not a theory at all, in the current
logical sense;
(but that doesn't stop the term from being meaningful,
IMHO).
--
Alan Smaill email: A.Sm...@ed.ac.uk
School of Informatics tel: 44-131-650-2710
University of Edinburgh
Torkel Franzen
> (but that doesn't stop the term from being meaningful,
> IMHO).
As a technical term, it's perfectly meaningful. In the context of
discussions of Russell's paradox, it is too often used as though
referring to some set theory used at the time, e.g. by Cantor.
G. Frege
wrote:
> >
> > Has anybody ever used naive set theory?
> >
Actually, that question is already moot.
"6.5 For an answer which cannot be expressed the question
too cannot be expressed."
(L. Wittgenstein, TLP)
As Thomas Bushnell said: "'naive set theory' as a term is
underspecified".
"6.53
The right method of philosophy would be this: To say nothing
except what can be said, i.e. the propositions of natural
science, i.e. something that has nothing to do with philosophy:
and then always, when someone else wished to say something
metaphysical, to demonstrate to him that he had given no
meaning to certain signs in his propositions. This method would
be unsatisfying to the other -- he would not have the feeling
that we were teaching him philosophy -- but it would be the
only strictly correct method."
In this case the "meaning of the signs"
naive set theory
in Torkels question [i.e. in its context], is "underspecified".
Still, we might consider the following to be a reasonable answer:
>
> Well, since it is inconsistent, maybe not.
>
But...
>
> "Used" is too strong a term. People have attempted
> to axiomatize set theory and possibly stumbled across it
> along the way. People (specifically Frege) have assumed that
> unrestricted comprehension was legitimate and have used it.
>
Right.
>
> There was a lot of it going on before it was FIGURED OUT that
> comprehension needed to be limited [somehow].
>
Well. Actually, CANTOR *himself* knew about that! But for many others it
w a s a discovery.
>
> Specifically, Russell wrote Frege about the paradox in 1902, and the
> axiomatization of set theory that cured it was published by Zermelo
> in 1908.
>
Right. Zermelo actually STATES in his paper that he considers his theory
to be a cure of (for?) "set theory".
>
> All I'm saying is that [many] people were forming sets without being
> careful about how, for several years prior to 1903.
>
Right.
>
> Since you already knew this, the motivation for your
> question remains tragically obscure [...]
>
Not really, I guess. :-)
Probably his point is that CANTOR never (actually) used "naive set
theory" (i.e. allowed for unrestricted comprehension). And he is right.
On the other hand..., it's ALSO true, that Cantor never published
ANYTHING that could be considered a comprehensive description of his
theory that actually forms a _Set Theory_ in which the more obvious
antinomies cannot arise. If Torkel thinks otherwise it would be NICE if
he could describe the (this) PRINCIPLES [axioms] of "Cantor's theory".
F.
Daryl McCullough
>arguably, naive set theory is not a theory at all, in the current
>logical sense;
>(but that doesn't stop the term from being meaningful,
>IMHO).
The way I understood it, "using naive set theory" means using definitions
such as "the set of all x such that Phi(x)" without worrying too much
about the sorts of formulas Phi(x) for which this definition makes
sense.
--
Daryl McCullough
Ithaca, NY
George Greene
: The way I understood it, "using naive set theory" means using definitions
: such as "the set of all x such that Phi(x)" without worrying too much
: about the sorts of formulas Phi(x) for which this definition makes
: sense.
It always makes "sense";
it just doesn't always have "reference".
It causes whatever trouble it causes in virtue of the
PARTICULAR "sense" that is made by the inconvenient applications.
It is not the case that P&~P "does not make sense" or is meaningless --
it is contradictory precisely BECAUSE it has the particular meaning that it has.
Thomas Bushnell, BSG
> > And Cantor certainly did, to the extent he is rightly the first
> > serious discoverer of Set Theory.
>
> Cantor did indeed create set theory. Since he did not introduce any
> unrestricted comprehension axiom there is no basis for the idea that
> he used or introduced "naive set theory" in your technical sense.
Is there any suggestion anywhere in his work of restrictions on
comprehension?
Thomas Bushnell, BSG
You don't seem to be very good at keeping track.
As I said, the term is used to refer to many different things, and
doesn't have a single well-specified reference in the literature.
Now you seem to agree: "it is too often used ...". That's exactly my
point: it is used in a variety of different ways, with somewhat
divergent meanings.
Thomas
Torkel Franzen
> Is there any suggestion anywhere in his work of restrictions on
> comprehension?
Certainly. Furthermore, the work in set theory by Cantor was and remains
perfectly good mathematics.
You can save yourself some time by looking up earlier exchanges in
the group on this topic.
Torkel Franzen
> Now you seem to agree: "it is too often used ...". That's exactly my
> point: it is used in a variety of different ways, with somewhat
> divergent meanings.
If we use the phrase in the sense of "non-axiomatized set theory",
Russell's paradox did not show the "naive set theory" of Cantor or
Dedekind to be in the least inconsistent. If we use the phrase in the
sense of "the first order theory with extensionality and unlimited
comprehension as its only axioms", we need to recognize that this is
not a description of any set theory proposed by Cantor or Dedekind or
any other pioneer of the subject.
George Greene
:
: > Is there any suggestion anywhere in his work of restrictions on
: > comprehension?
Torkel Franzen <tor...@sm.luth.se> writes:
: Certainly.
"Suggestion" will NOT cut it.
: Furthermore, the work in set theory by Cantor
"furthermore"? Why is set theory "more" an "further"
than "restrictions on comprehension" and the paradoxes
that can arise if you don't restrict? Surely that is a
KEY aspect of set theory.
: was and remains perfectly good mathematics.
"the work"? That's a ridiculously ambiguous locution.
What EXACTLY is the extension of that predicate? EVERY LAST piece
of the work? If the WHOLE work was PERFECTLY good then that would imply
he had NEVER made ANY mistake -- which is inherently ridiculous, but
it's what you said.
: You can save yourself some time by looking up earlier exchanges in
: the group on this topic.
No, he can't. That will waste his time.
Cantor alluded to his awareness of "absolutely infinite multiplicities"
but he did not come up with any formal machinery for explaining why
they weren't sets. He noticed that calling them sets would lead to
contradictions but he could not explain formally why any of his earlier
set-formation techniques could avoid them.
Cantor wrote this in a letter to Dedekind, for which I need a date
but don't have one:
On the one hand a multiplicity can be such that
the assumption that all of its elements "are together"
leads to a contradiction, so that it is impossible to
conceive of the multiplicity as a unity, as "one finished
thing". Such multiplicities I call "absolutely infinite"
or "inconsistent" multiplicities.
As one easily sees, "the totality of everything thinkable",
for example, is such a multiplicity; later still other examples
will present themselves. When, on the other hand, the totality
of elements of a multiplicity can be thought of without contradiction
as "being together", so that their collection into "one thing" is
possible, I call it a consistent multiplicity or a set.
But the point is, THIS is NOT set THEORY.
Canotr himself didn't publish these "other examples", and
in any case, it is NOT inconceivable for them to exist as one
thing -- it is just contradictory to CALL that thing a set --
if you call it a class, the contradiction goes away.
Of course, it is re-constructible in analogous terms at
the class level, but that just means it is re-avoidable
by one more semantic ascent.
In addition to not being aware of any of this, Cantor also
did not have a formalism for it, and while the intuitions
may well have constituted VERY good mathematics, Torkel Franzen
is just flat out LYING if he alleges that this take on "inconsitent
multiplicities" was PERFECTLY "good" mathematics: good mathematics
knows what axioms it's using.
George Greene
: > Now you seem to agree: "it is too often used ...". That's exactly my
: > point: it is used in a variety of different ways, with somewhat
: > divergent meanings.
Torkel Franzen <tor...@sm.luth.se> writes:
: If we use the phrase in the sense of "non-axiomatized set theory",
That is NOT the definition. And you knew that, Torkel, even if TB/BSG
didn't.
: Russell's paradox did not show the "naive set theory" of Cantor or
: Dedekind to be in the least inconsistent.
"non-axiomatized set theory" is a contradiction in terms.
If they didn't know what their axioms where then whatEVER it
was, it WASN'T a theory. But that is not even the point: OF COURSE
they knew what they considered axiomatic, EVEN if they didn't
have a systematic notation for it yet.
: If we use the phrase in the
: sense of "the first order theory with extensionality and unlimited
: comprehension as its only axioms", we need to recognize that this is
: not a description of any set theory proposed by Cantor or Dedekind or
: any other pioneer of the subject.
That's ridiculous: Frege is a pioneer of the subject.
Thomas Bushnell, BSG
However, the phrase *is* used in print to refer to both those things
(and a variety of other shades of meaning as well). Which is what I
said, if you recall, when you decided to try and score points.
I wish you had the decency to say "sorry, I was wrong".
Thomas
G. Frege
BSG) wrote:
>
> I wish you had the decency to say "sorry, I was wrong".
>
Now way. :-(
F.
Torkel Franzen
>However, the phrase *is* used in print to refer to both those things
>(and a variety of other shades of meaning as well).
Really? Anyway, my present comments concerned the use of the phrase
to suggest that Russell's paradox revealed an inconsistency in
19th century set theory.
Thomas Bushnell, BSG
> Really? Anyway, my present comments concerned the use of the phrase
> to suggest that Russell's paradox revealed an inconsistency in
> 19th century set theory.
In <vcbk74x...@beta13.sm.luth.se>, you said:
"You mean in Frege's highly non-naive system - not in naive set
theory".
In <87oeu9r...@becket.becket.net>, I said:
"Naive set theory is an underspecified beast, so whether the
Burali-Forti paradox or the Russell paradox clearly affect it is hard
to answer clearly".
Where here do I "suggest that Russell's paradox revealed an
inconsistency in 19th century set theory"? Rather, I express the view
that there is no good answer to the question of whether Rulless's
paradox clearly affects "naive set theory", because different people
use the words "naive set theory" to refer to different things.
I do not have much hope, however, that you will admit your error.
George Green took issue with my statement that it was underspecified,
and I replied, in <87ekv43...@becket.becket.net> that:
"At least one naive set theory text on my bookshelf doesn't have
unrestricted comprehension"
And you asked, in <vcbvfog...@beta13.sm.luth.se>:
"In what 'naive set theory' do you in fact find unrestricted
comprehension?"
Now your errors are compounded, for I posted a reference which
explicitly lists unrestricted comprehension as an axiom of what those
authors term "naive set theory". Nor is Language, Proof, and Logic a
textbook written by ignoramus wackos.
Now, perhaps you define "naive set theory" as "19th century set
theory". That would be one more definition of the term, further
demonstrating my point that "naive set theory" has a variety of uses
and definitions, in sufficient disagreement that, as I said, it is
hard to answer clearly whether Russell's paradox clearly affects
"naive set theory".
Thomas
Thomas Bushnell, BSG
> tb+u...@becket.net (Thomas Bushnell, BSG) writes:
>
> > Where here do I "suggest that Russell's paradox revealed an
> > inconsistency in 19th century set theory"?
>
> I thought you did so in your earlier remark about Russell exposing a
> paradox in naive set theory. In this, I was most likely influenced by
> your earlier comment about cranks who hear a little voice saying
> "maybe I'll get the fame Bertrand Russell got for his proof that the
> old set theory was inconsistent". But I see now that I misread this
> comment! After all, this is a crank's inner voice speaking, and such
> inner voices are frequently mistaken on historical matters.
The old set theory, including at least Frege's theory, was in fact
torn down. It is hardly a settled and clear matter just what
axiomatization Cantor would have given for his theory, if he had given
one.
> But I'm not quite clear about what tearing down you have in mind in
> your further comment, not attributed to the crank, that "Russell's
> greatest achievements were constructive enterprises, extending
> knowledge, rather than tearing down mistakes of the past."
Frege's mistake, at least, in proposing (in ignorance) an inconsistent
theory.
You seem to want to categorize things into "wonderful" and "useless",
in which the wonderful don't make any blunders. I think Cantor made
lots of blunders (including the failure to adequately prove the
Schroeder-Bernstein theorem, for example), and was also wonderful.
Likewise Frege was wonderful, who also made a blunder.
Without that blunder, however, much important progress might not have
been made.
Thomas
Thomas Bushnell, BSG
> tb+u...@becket.net (Thomas Bushnell, BSG) writes:
>
> > The old set theory, including at least Frege's theory, was in fact
> > torn down.
>
> So what would you include in "the old set theory" apart from Frege's
> Grundgesetze system?
I would include Cantor, at least. Many other folks worked in the same
general framework that Cantor described, without destinguishing
themselves. I tend to conceive of it as one large project, which
Frege ended up axiomatizing, and whose axiomatization turned out to be
inconsistent.
You find the occasional recognition that certain things "can't" be
taken as sets, without any statement why, beyond "it leads to an
inconsistency". But that isn't axiomatizable: "Predicate P defines a
set, provided it doesn't result in an inconsistency"? With an axiom
like that, you can't prove the existence of...anything?
Nor could they have articulated such an axiom, depending as it does on
proof-theoretic notions that only came about later, in part, as a
response to Russell. (I take Goedel, for example, to be in part a
response to Russell.)
So there are, it seems to me, three possible ways to understand
comprehension for the workers in early set theory:
1) There is a set for any predicate. That's Frege's axiom, and it
doesn't work.
2) There is a set for any predicate, provided it doesn't produce an
inconsistency. This allows you to prove there is no set of all
sets, and thus deal with paradoxes nicely, but is a positive
detraction if you want to prove sets exist.
3) For any predicate, go ahead and assume a corresponding set exists,
until you can prove it doesn't. This just isn't an "axiom of
comprehension" in any sense, however. It's research advice.
Cantor seems to have worked along number (3). Number (3) is a fine
way to make some progress in a new area, but it's not axiomatizable.
I suspect it's not really a good idea to even ask what kind of "axiom
of comprehension" Cantor used. He pretty much did not use any,
because he wasn't working in an axiomatic system.
Now it seems to me that you can either take either of two tacks:
1) Russell's paradox only really hampers Frege's system, and not
anyone else's, or
2) Russell's paradox hampers the old set theory in general.
I understand why you might prefer (1). It means you can continue to
have a "hero/idiot" dichotomy, and put Cantor on the hero side.
But it's just not right. If Russell's result was so obviously a
consequence of unrestricted comprehension, such that every early
worker in set theory knew that you couldn't have unrestricted
comprehension for Russell's-paradox-reasons, then---why did Frege ever
bother to write down such an axiom?
I think that Russell's paper came as a suprise to an awful lot of
people, just as Goedel's came as a surprise to Russell. (Of course,
Goedel's was a whole lot more clever!) But the reaction to Russell
was not "oh, sure, Frege's project is hurt by that, but not anything
else." If Frege's system was a sideline, and everything else wasn't
really affected--then why was there an urgent desire on many parts to
come up with an axiomatization that could work?
The answer, it seems to me, is obvious: if you asked people to write
down axioms before Frege, most of them would have produced something
much like Frege's. In their rough intuitions, they looked at Frege's
system, and said "yes, that's the right axiomatization". If Cantor
and all the rest already knew the danger of unrestricted
comprehension, then why did Frege ever bother writing it down?
And finally, a meta-comment. Do a favor, and spend some time writing
something substantial. One-line pot shots are quite annoying. If you
can't be bothered to put your thoughts to words, then they really
aren't worth much at all. So please, if you want to have a
conversation, make it something worthwhile instead of pointless
oneliners.
Thomas
Torkel Franzen
>I would include Cantor, at least.
So you have the idea that Cantor's set theory was torn down by
Russell's paradox? This is a misconception. Nothing in Cantor's set
theory was invalidated by Russell's paradox.
> Many other folks worked in the same
> general framework that Cantor described, without destinguishing
> themselves. I tend to conceive of it as one large project, which
> Frege ended up axiomatizing, and whose axiomatization turned out to be
> inconsistent.
This again is a misconception (and an odd one). I hardly know what
to say about it except to suggest that you read up on the history of
set theory. There's a readable BSL article by Kanamori, "The mathematical
development of set theory from Cantor to Cohen" available on the net
at http://www.math.ucla.edu/%7Easl/bsl/0201-toc.htm
Thomas Bushnell, BSG
> tb+u...@becket.net (Thomas Bushnell, BSG) writes:
>
> >I would include Cantor, at least.
>
> So you have the idea that Cantor's set theory was torn down by
> Russell's paradox? This is a misconception. Nothing in Cantor's set
> theory was invalidated by Russell's paradox.
It's like you didn't mother reading what I said. Go back and re-read
it. I wrote more words, precisely because you have a penchant for
snipping isolated sentences, and misreading them. It's clear you read
the first sentence. It's also clear you didn't bother reading more
than that.
> This again is a misconception (and an odd one). I hardly know what
> to say about it except to suggest that you read up on the history of
> set theory. There's a readable BSL article by Kanamori, "The mathematical
> development of set theory from Cantor to Cohen" available on the net
> at http://www.math.ucla.edu/%7Easl/bsl/0201-toc.htm
Perhaps I shall. Regardless, your words will get very little respect
from me if you can't be bothered to respond to the points I make, the
questions I ask, or what I actually say.
So far you have so frequently misread what I said, and gone into
pointless stupid distractions in an apparent attempt to score pathetic
little debating points. Frankly, I think you are probably doing the
same here, which means I won't bother taking what you say very
seriously, unless you start to show signs of reading a little more
carefully and considering what I actually say instead of the first
sentence of each post.
Thomas
Torkel Franzen
> Perhaps I shall. Regardless, your words will get very little respect
> from me if you can't be bothered to respond to the points I make, the
> questions I ask, or what I actually say.
Then I suggest you just set my words aside, and read Kanamori's
article instead.
Thomas Bushnell, BSG
> > Many other folks worked in the same general framework that Cantor
> > described, without destinguishing themselves. I tend to conceive
> > of it as one large project, which Frege ended up axiomatizing, and
> > whose axiomatization turned out to be inconsistent.
>
> This again is a misconception (and an odd one). I hardly know what
> to say about it except to suggest that you read up on the history of
> set theory. There's a readable BSL article by Kanamori, "The mathematical
> development of set theory from Cantor to Cohen" available on the net
> at http://www.math.ucla.edu/%7Easl/bsl/0201-toc.htm
So the paper is quite interesting. Perhaps my word "distinguishing"
is open to two contrasting understandings. By no mean do I suggest
that early set theorists were not distinguished, in the sense of being
important or seminal.
Rather, I meant that they did not attempt to articulate a new
framework for set theory as such, in the way that is frequently
described today when in distinguishing "naive set theory" from
"axiomatic set theory". The paper tends to blur that distinction; if
the blurring is correct, then that only proves my point all the more,
that the term "naive set theory" has a variety of meanings, without
any single one having claim to fame.
(I would note in passing that "naive set theory" need not be taken as
a *historical* term at all, in the sense that it is a perfectly
sensible [if blurry] concept, whether or not it had any particular
simply described historic existence. In this, it is similar to
"Euclidean geometry".)
The paper suffers from an understandable flaw, in its trumpeting of
Cohen's work. Without doubt, Cohen's work is every bit as important
as the paper describes. And since the paper is billed as "Cantor to
Cohen" (though it does go a bit further, in giving a feel for some of
the post-Cohen fallout), it is natural for Cohen to be the pinnacle of
the presentation.
But I fear that the focus on Cohen's brilliance has tended to mark him
as the one great dividing line in set theory. I believe a very
similar story could be told in which Goedel, not Cohen, is the Big
Hero, and Cohen is a postscript to Goedel. With a bit of a wink,
Skolem could be given the same status.
The paper focuses on "set theory as branch of mathematics" instead of
"set theory as foundation for mathematics". Itself blameless, that
focus has the effect of sliding a little too quickly over Frege and
Russell's work; Russell is treated as trying to develop a weird (to
modern eyes) set theory, and Frege is a more-or-less sideline
distraction. For the "set theory as branch of mathematics" set, this
is harmless. But "set theory as foundation for mathematics" requires
a different story, one in which the key work is Russell, Frege,
Skolem, Goedel, and in which Cantor (!), Zermelo, and Cohen take a
back seat.
The author's desire to put Cohen in big sharp terms as the Big Hero,
as I said, gives an entirely earned distinction to Cohen. But I
believe that it is also a bit of foreshortening. In the sequence 1,
2, 4, 16, ..., it is certainly true that the jump from 1024 to 2048 is
bigger than any preceding, but there is also an important sense in
which it is not the arithmetical difference between terms that
expresses how "important" a number is in that sequence, but rather the
geometric ratio between successive terms.
Cohen might therefore be a damn big advance over previous work, but
that previous work was--relatively speaking--an equally big advance,
if you consider constance of "ratio" instead of constancy of
"increment". Which is to say that there is a very important sense in
which many earlier people are just as important--relative to their
context--as Cohen, and the singular praise handed out to Cohen, in
that sense, should not have been reserved for him alone.
And this is relevant to the present topic. The paper, by desiring to
laud Cohen in the strongest possible terms, presents all that preceded
as a basically simple progression, with a stunning discontinuity at
the moment of Cohen's work. This, as a result, tends to make Cantor,
Russell, Zermelo, Skolem, and the rest, as more or less participating
in one single forward going "thing", with no clear discontuities, and
then--poof--Cohen comes along. And so, for someone who sees important
discontinuities earlier as well, I find the paper to be lacking in
that regard.
The paper also contributes to that (erroneous, in my view) vision, by
its order of presentation. It seems chronological, but only very
roughly. Each section tends to begin fifteen years before the
preceding one ended; the "feel" is as if the results were reached in
the order described, but in fact, it's more like:a zigzag ascent: up
some, down some, up some more. But the order makes it seem like "they
solved this, then they solved that, ..." when in fact both were being
done at once.
And as a result, when a key result happened in many different branches
at once, it doesn't stand out in the paper, because it's buried in the
presentation of each branch on its own. As a result, discontinuities
and sea changes (before Cohen) can't be detected in the history it
presents.
None of this is intended, I hasten to say, that the paper is *wrong*,
just that it isn't the only word; and likewise, not at all do I intend
to derogate from the high praise the paper gives Cohen, which is, I
believe, entirely deserved.
I suspect that Torkel will never read this last paragraph. I wonder
if he will prove me wrong.
Thomas
Thomas Bushnell, BSG
As I did. Now, will you do me the respect of reading what I write?
Torkel Franzen
> The paper focuses on "set theory as branch of mathematics" instead of
> "set theory as foundation for mathematics". Itself blameless, that
> focus has the effect of sliding a little too quickly over Frege and
> Russell's work; Russell is treated as trying to develop a weird (to
> modern eyes) set theory, and Frege is a more-or-less sideline
> distraction.
It's unclear what passage in Kanamori's paper you have in mind here,
since he does not describe Russell's system as a set theory. It would
be odd to do so in view of the fact that sets were "logical fictions"
to Russell. But what I'm chiefly wondering is whether, after reading
this paper, you still hold to the view that Frege axiomatized a
set-theoretical project to which Cantor and many other folks
contributed? If so, I can only admit to bafflement.
Thomas Bushnell, BSG
> It's unclear what passage in Kanamori's paper you have in mind here,
> since he does not describe Russell's system as a set theory. It would
> be odd to do so in view of the fact that sets were "logical fictions"
> to Russell. But what I'm chiefly wondering is whether, after reading
> this paper, you still hold to the view that Frege axiomatized a
> set-theoretical project to which Cantor and many other folks
> contributed? If so, I can only admit to bafflement.
I'm sorry, but you don't deserve a reply until you read my entire
post, and not just the first two paragraphs.
Thomas
G. Frege
"...among all the mathematical theories,
it is just the theory of sets that requires
clarification more than any other."
(Mostowski 1979)
The following is a quote from:
Issues in commonsense set theory
by Müjdat Pakkan and Varol Akman
----------------------------------------------------
Earliest Developments
G. Cantor's work on the theory of infinite series and related topics
should be considered as the foundation of the research in set theory.
In Cantor's conception, a /set/, or /aggregate/, is a collection into a
whole of definite, distinct objects of our perception or our thought,
called the /elements/ of the set (Cantor 1883). This property of
/definiteness/ implies that given a set and an object, it is possible to
determine if the object is a member of that set [...].
In the earlier stages of his research, Cantor did not work from axioms
(Suppes 1972). However, all of his theorems can be derived from three
axioms: /Extensionality/ which states that two sets are identical if
they have the same members, /Abstraction/ which states that for any
given property there is a set whose members are just those entities
having that property, and /Choice/ which states that if A is a set, all
of whose elements are non-empty sets no two of which have any elements
in common, then there is a set B which has precisely one element in
common with each element of A.
The theory was soon threatened by the introduction of some paradoxes
which led to its evolution. In 1902, Russell found a contradiction in
Frege's foundational system (Frege 1893) which was developed on Cantor's
naive set conception (van Heijenhoort 1967). Frege's reaction to this
can be found in the appendix to the second volume of his famous
/Grundgesetze der Arithmetik/: "Hardly anything more unfortunate can
befall a scientific writer than to have one of the foundations of his
edifice shaken after the work is finished. This was the position I was
placed in by a letter of Mr. Bertrand Russell." This contradiction could
be derived from the Axiom of Abstraction (which was named Axiom V in
Frege's system) by considering "the set of all things which have the
property of not being members of themselves". This property can be
denoted as ~(x e x) in the language of first-order logic. (~(x e x) will
be denoted as x !e x from now on.)
The Axiom of Abstraction can be formulated as
EyAx[x e y <-> phi(x)],
where phi(x) is a formula in which x is free. In the case of Russell's
Paradox phi(x) = x !e x and we have: EyAx[x e y <-> x !e x]. [And hence
Ax[x e y <-> x !e x] for a certain y --ff]. Substituting y for x, we
reach y e y <-> y !e y. The problematic thing here is the set y with the
property [x e y iff x !e x --ff].
Another antinomy occurred with the conception of the "set of all sets",
V = {x : x = x}. The well-known Cantor's Theorem states that the power
set (set of all subsets) of V has a greater cardinality than V itself.
This is obviously paradoxical since V by definition is the most
inclusive set. This is the so-called Cantor's Paradox (Cantor 1932) and
led to discussions on the sizes of comprehensible sets. Strictly
speaking, it was Frege's foundational system that was overthrown by
Russell's Paradox, not Cantor's naive set theory. The latter came to
grief precisely because of the preceding "limitation of size"
constraint. Later, von Neumann would clarify this problem of size by
stating that (Goldblatt 1984) "Some predicates have extensions that are
too large to be successfully encompassed as a whole and treated as a
mathematical object."
Such paradoxes shook the theory to its foundations and were instrumental
in new axiomatizations of the set theory or in alternate approaches.
However, it is believed that axiomatic set theory would still have
evolved in the absence of paradoxes because of the continuous search for
foundational principles. Axiomatization of a theory is important since
it provides a concise formulation of the principles of the theory and
allows fundamental notions like completeness and consistency to be
discussed in a precise way; these would be formulated in an imprecise
manner (e.g., in natural language) otherwise.
------------------------------
References:
Cantor, G. (1883). Fondaments d'une théorie générale des ensembles. Acta
Mathematica 2:381--408.
Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts (ed. by E. Zermelo). Springer-Verlag, Berlin.
Frege, G. (1893). Grundgesetze der Arithmetik, begriffsschriftlich
abgeleitet (Volume I). Jena.
van Heijenhoort, J. (1967). From Frege to Gödel. Harvard University
Press, Cambridge, MA.
Goldblatt, R. (1984). Topoi: The Categorial Analysis of Logic.
North-Holland, Amsterdam.
Mostowski, A. (1979). Thirty Years of Foundational Studies. In
Kuratowski, K. et al. (eds.) Andrzej Mostowski: Foundational Studies,
Selected Works. North-Holland, Amsterdam.
von Neumann, J. (1925). Eine Axiomatisierung der Mengenlehre. J. für
Math. 154:219--240. (Corrections: ibid., 155:128, 1926.)
Suppes, P. (1972). Axiomatic Set Theory. Dover, New York.
F.
G. Frege
>
> So you have the idea that Cantor's set theory was torn down by
> Russell's paradox? This is a misconception. Nothing in Cantor's set
> theory was invalidated by Russell's paradox.
>
Now let's talk about "unrestricted comprehension", and Frege's vs.
Cantor's approach concerning this principle of "naive set theory".
Quoting BURGESS:
"Frege really did assume that every condition determines a set, or more
precisely, determines a "concept" and thereby an "extension", and so
fell into paradox. Even before the discovery of any such paradoxes,
however, Cantor [1885] had rejected Frege's assumptions in a review of
Frege's Grundlagen:
The author's own attempt to give a strict foundation to the
number-concept seems to me less successful. Specifically, the
author has the unfortunate idea ... to take as the foundation
for the number-concept what in Scholastic logic is called the
"extension of a concept". He completely overlooks the fact that
in general the "extension of a concept" is quantitatively wholly
indeterminate. Only in certain cases is the extension
quantitatively determinate, in which cases it can then of course
be assigned a definite number, if it is finite, or power, in
case it is infinite. But for this sort of quantitative
determination we must already possess the concepts of "number"
and "power", and it is getting things backwards to try to found
these latter concepts on the concept "extension of a concept".
Exactly what Cantor meant by "quantitatively indeterminate" [quantitativ
unbestimmt] is not entirely clear, but he seems to be alluding to the
kind of distinction within the "actually infinite" that he makes
elsewhere, between merely "transfinite" or "consistent multiplicities",
which do form sets, and the "absolutely infinite" or "inconsistent
multiplicities", which do not. The Russell paradox does not arise for
Cantor because he never assumes that every condition determines a set,
but only those conditions that do not hold of too many objects, an
assumption known as the principle of limitation of size."
---
F.
G. Frege
>
> ...what I'm chiefly wondering is whether, after reading
> this paper, you still hold to the view that Frege axiomatized
> [Cantor's] set-theoretical project.
>
Actually, this is a widely held believe (among non-scholars).
But it's -obviously- wrong.
Though I didn't study Cantor's view in detail [read: at all], I've come
to the 'conclusion' that MK is a nice "approximation" of whatever theory
Cantor himself had in mind. Just read his "multiplicities" as "classes".
Then his "consistent multiplicities" are /sets/, and his "inconsistent
multiplicities" are /proper classes/ or /non-sets/ (in MK).
F.