Some basic set theory questions
gratis mezn
I am sure this has been asked many times but couldn't find
exactly what I am looking for.
Basically, I am trying to understand what sets are (as opposed to
classes). As I understand :
1) If you can define it by a formula its a class
2) its a set only if it is in the cumulative hierarchy.
Now, how do we know that there exists atleast one set in the
cumulative hierarchy. How do we establish the existence of the null
set.
For instance, {x : x \neq x} would be empty. How do we know its a set.
Which axiom says it is a set?.
We cant use separation here(???). I cant think how this becomes a set.
William Elliot
> Basically, I am trying to understand what sets are (as opposed to
> classes). As I understand :
>
> 1) If you can define it by a formula its a class
> 2) its a set only if it is in the cumulative hierarchy.
>
Where did you get that understanding?
What a set is depends upon what set theory you are using.
In ZFC, every thing is a set and there are no classes.
In NF, no distinction is made.
In NBG, x is a set when some y with x in y.
> I am sure this has been asked many times but couldn't find
> exactly what I am looking for.
> Now, how do we know that there exists atleast one set in the
> cumulative hierarchy. How do we establish the existence of the null
> set.
>
Just what is the cumulative hierarchy? The constructible universe?
> For instance, {x : x \neq x} would be empty. How do we know its a set.
> Which axiom says it is a set?.
>
What set theory are you using?
It exists in ZFC by use of existence and separation,
in NF because x /= x is stratified
and is a set in NBG by comprehension.
David C. Ullrich
<zero...@googlemail.com> wrote:
>Hello all,
> I am sure this has been asked many times but couldn't find
>exactly what I am looking for.
>
>Basically, I am trying to understand what sets are (as opposed to
>classes). As I understand :
>
>1) If you can define it by a formula its a class
>
>2) its a set only if it is in the cumulative hierarchy.
>
>Now, how do we know that there exists atleast one set in the
>cumulative hierarchy. How do we establish the existence of the null
>set.
Can you give us a list of the axioms?
(The list I might give you could be different from your
list; these things get formulated in different equivalent
ways. In every list of the axioms of ZFC I've ever seen
there's one that makes this clear...)
>For instance, {x : x \neq x} would be empty. How do we know its a set.
>Which axiom says it is a set?.
>
>We cant use separation here(???). I cant think how this becomes a set.
You can use separation _if_ you know that at least one set exists...
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Frederick Williams
>
> On Wed, 1 Jul 2009, gratis mezn wrote:
>
> > Basically, I am trying to understand what sets are (as opposed to
> > classes). As I understand :
> >
> > 1) If you can define it by a formula its a class
> > 2) its a set only if it is in the cumulative hierarchy.
> >
> Where did you get that understanding?
>
> What a set is depends upon what set theory you are using.
>
> In ZFC, every thing is a set and there are no classes.
In ZFC one can "identify" classes with formulae. See Takeuti and Zaring
for details.
> >
> Just what is the cumulative hierarchy? The constructible universe?
No.
R(0) = 0,
R(alpha + 1) = R(alpha) union powerset(R(alpha))
R(limit) = union_{beta < limit} R(beta).
The collection of the R's is the cumulative hierarchy.
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
gratis mezn
> On Wed, 1 Jul 2009 02:47:39 -0700 (PDT), gratis mezn
>
> >Hello all,
> > I am sure this has been asked many times but couldn't find
> >exactly what I am looking for.
>
> >Basically, I am trying to understand what sets are (as opposed to
> >classes). As I understand :
>
> >1) If you can define it by a formula its a class
>
> >2) its a set only if it is in the cumulative hierarchy.
>
> >Now, how do we know that there exists atleast one set in the
> >cumulative hierarchy. How do we establish the existence of the null
> >set.
>
> Can you give us a list of the axioms?
>
I work in ZF. By cumulative hierarchy I mean V.
list -
Axiom of extensionality,
foundation,
separation ,
pairing,
union,
collection,
infinity,
powerset
> (The list I might give you could be different from your
> list; these things get formulated in different equivalent
> ways. In every list of the axioms of ZFC I've ever seen
> there's one that makes this clear...)
>
> >For instance, {x : x \neq x} would be empty. How do we know its a set.
> >Which axiom says it is a set?.
>
> >We cant use separation here(???). I cant think how this becomes a set.
>
> You can use separation _if_ you know that at least one set exists...
>
How do we know atleast one set exists?. For `infinity' to exist, we
need null set. How does null set exist?.
Han de Bruijn
For what you will find it's worth:
http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf
Han de Bruijn
Aatu Koskensilta
> On Wed, 1 Jul 2009, gratis mezn wrote:
>
>> Basically, I am trying to understand what sets are (as opposed to
>> classes). As I understand :
>>
>> 1) If you can define it by a formula its a class
>> 2) its a set only if it is in the cumulative hierarchy.
>
> Where did you get that understanding?
It is the standard understanding, in context of ZFC.
> Just what is the cumulative hierarchy?
Just consult any standard text on the subject.
> The constructible universe?
No.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Stephen J. Herschkorn
>> Basically, I am trying to understand what sets are (as opposed to
>>
>>>classes). As I understand :
>>>
>>>
>>>1) If you can define it by a formula its a class
>>>
>>>
>>>2) its a set only if it is in the cumulative hierarchy.
>>>
>>>
>>> <>Now, how do we know that there exists at least one set in the
>>> cumulative hierarchy. How do we establish the existence of the null
>>> set.
>>
>
>
>I work in ZF. By cumulative hierarchy I mean V.
>
>list -
>Axiom of extensionality,
>foundation,
>separation ,
>pairing,
>union,
>collection,
>infinity,
>powerset
>
>
>
>How do we know atleast one set exists?. For `infinity' to exist, we
>need null set. How does null set exist?.
>
[The quote is much edited.]
There are no classes in ZF. Formally, one treats classes as logical
formulae. See, for example, Kunen.
The axiom of infiinty implies there exists a set which has the empty set
as an element. Let x be such a set. Let y = {z in x: z is empty},
which exists by separation. Let z = Uy, which exists by the axiom of
union. Then z is the empty set.
Better yet: Let x be a set containing the empty set; x exists by
the axiom of inifinity. The There exists y in x suxh that y is
empty. This is just shorhand for "There exists y such y in x and
y is empty." By logicla inference, this implies that there exists y
such that y is empty.
Or here's another. Let x be a set satisfying the axiom of inifinity.
Let y = {z in x: z != z}, which exists by separation. y is the
empty set.
By the way, is your axiom of collection the same thing as the axiom of
replacement?
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey
Aatu Koskensilta
> The axiom of infiinty implies there exists a set which has the empty
> set as an element. Let x be such a set. Let y = {z in x: z is
> empty}, which exists by separation. Let z = Uy, which exists by
> the axiom of union. Then z is the empty set.
This is a baffling proof. If there is something that has the empty set
as an element clearly the empty set exists.
> Or here's another. Let x be a set satisfying the axiom of
> inifinity. Let y = {z in x: z != z}, which exists by separation.
> y is the empty set.
There's no need for x to be a set satisfying the axiom of infinity.
> By the way, is your axiom of collection the same thing as the axiom of
> replacement?
No, but the two are (classically) equivalent.
MoeBlee
> In ZFC, every thing is a set and there are no classes.
In ZFC (without urelements), every object is a set and a class, but
there are no PROPER classes.
> Just what is the cumulative hierarchy? The constructible universe?
No, the constructible universe is a proper subclass of the cumulative
hierarchy.
> It exists in ZFC by use of existence and separation,
"use of existence"; That's built into the first order logic. The
existence of an empty set follows in Z set theory from separation (as
you mentioned), and uniqueness from extensionality.
MoeBlee
MoeBlee
> How do we know atleast one set exists?.
It's built into the first order logic we use.
Using the axiom schema of separation with first order logic is enough
to prove the existence of an object that has no members.
MoeBlee
Aatu Koskensilta
> No, the constructible universe is a proper subclass of the cumulative
> hierarchy.
Indeed it is -- but not provably so in ZFC.
MoeBlee
wrote:
> There are no classes in ZF.
Yes there are. There are no PROPER classes with ZF.
> The axiom of infiinty implies there exists a set which has the
> empty set
> as an element.
We don't need the axiom of infinity to prove that there exists a set
with no members. The axiom schema of separation is sufficient for
that.
MoeBlee
gratis mezn
How ? could you give me the formula?. you already need a set to use
seeparation?. no?
> MoeBlee
Dave
What does the axiom of infinity say, exactly?
MoeBlee
> > We don't need the axiom of infinity to prove that there exists a set
> > with no members. The axiom schema of separation is sufficient for
> > that.
>
> How ? could you give me the formula?. you already need a set to use
> seeparation?. no?
That there exists an object is given to us just by our first order
logic. Take any logically true predicate P(x), then we have this
theorem of first order logic:
ExP(x)
But we don't even need to deploy that specifically.
A derivation of the existence of an empty set from the axiom schema of
separation is simple ['e' stands for the epsilon membership symbol]:
AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
schema of separation
EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
Ay(yeb <-> (yez & (yey & ~yey)) ... existential instantiation
yeb <-> (yez & (yey & ~yey)) ... universal instantiation
yeb <-> (yey & ~yey) ... sentential logic
yeb ... supposition toward a contradiction
yey & ~yey ... sentential logic
~yeb ... by contradiction
Ay ~yeb ... universal generalization
EbAy ~yeb .... existential generalization
Then uniqueness comes from the axiom of extensionality:
Ay ~yeb & Ay ~yec ... supposition
Ay(yeb <-> yec) ... easy predicate logic
b=c ... from axiom of extensionality
E!bAy ~yeb ... from above
So with the existence and uniqueness theorem, we define:
0=b <-> Ay ~yeb
MoeBlee
MoeBlee
> What does the axiom of infinity say, exactly?
It depends on the exact formulation. But the most ordinary formulation
says:
There exists a set w such that the empty set is a member of w and w is
closed under the successor operation.
(The successor operation takes x to x union {x}.)
MoeBlee
MoeBlee
> On Jul 1, 12:19 pm, gratis mezn <zeroni...@googlemail.com> wrote:
>
> > > We don't need the axiom of infinity to prove that there exists a set
> > > with no members. The axiom schema of separation is sufficient for
> > > that.
>
> > How ? could you give me the formula?. you already need a set to use
> > seeparation?. no?
>
> That there exists an object is given to us just by our first order
> logic. Take any logically true predicate P(x), then we have this
> theorem of first order logic:
>
> ExP(x)
>
> But we don't even need to deploy that specifically.
>
> A derivation of the existence of an empty set from the axiom schema of
> separation is simple ['e' stands for the epsilon membership symbol]:
>
> AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
> schema of separation
> EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
> Ay(yeb <-> (yez & (yey & ~yey)) ... existential instantiation
Oops, I didn't mean to change 'x' to 'z' there. So, it's 'x' where 'z'
appears here and for the next line.
MoeBlee
MoeBlee
> On Jul 1, 3:21 am, William Elliot <ma...@rdrop.remove.com> wrote:
>
> > In ZFC, every thing is a set and there are no classes.
>
> In ZFC (without urelements), every object is a set and a class, but
> there are no PROPER classes.
>
> > Just what is the cumulative hierarchy? The constructible universe?
>
> No, the constructible universe is a proper subclass of the cumulative
> hierarchy.
Although, if one belives V=L, then one does believe that the
cumulative hierarchy is the constructible universe. I've heard that
Devlin believes V=L and, as I recally, a few other people.
MoeBlee
Stephen J. Herschkorn
>"Stephen J. Herschkorn" <sjher...@netscape.net> writes:
>
>
>>Let x be a set satisfying the axiom of
>>inifinity. Let y = {z in x: z != z}, which exists by separation.
>>y is the empty set.
>>
>>
> <>
> There's no need for x to be a set satisfying the axiom of infinity.
Here is the OP's list of axioms:
>Axiom of extensionality,
>foundation,
>separation ,
>pairing,
>union,
>collection,
>infinity,
>powerset
>
Which of these, besides the axiom of infinty, implies that any set exists?
MoeBlee
wrote:
> Aatu Koskensilta wrote:
> >"Stephen J. Herschkorn" <sjhersc...@netscape.net> writes:
>
> >>Let x be a set satisfying the axiom of
> >>inifinity. Let y = {z in x: z != z}, which exists by separation.
> >>y is the empty set.
>
> > <>
> > There's no need for x to be a set satisfying the axiom of infinity.
>
> Here is the OP's list of axioms:
>
> >Axiom of extensionality,
> >foundation,
> >separation ,
> >pairing,
> >union,
> >collection,
> >infinity,
> >powerset
>
> Which of these, besides the axiom of infinty, implies that any set exists?
First order logic alone proves that "there exists an object":
Let P(x) be any logically true formula.
ExP(x) by logic alone.
As to 'set' (in a context where urelements are excluded by the axiom
of extensionality), it would depend on the definition of 'set'. One
definition might be:
x is a set <-> Ez xez.
Then in Z (ZF, etc., but not in class theories such as NBG) every
object is a set by the pairing axiom (or pairing theorem if we derive
it from the schema of replacement), or by the power set axiom.
Anyway, moreover, we don't need the axiom of infinity to prove the
existence of an empty set, as I posted a proof earlier today (didn't
you and I already discuss this a few years ago?).
MoeBlee
Stephen J. Herschkorn
>Let P(x) be any logically true formula.
>
>ExP(x) by logic alone.
>
>
How so? How do we know the universe is not empty?
>Anyway, moreover, we don't need the axiom of infinity to prove the
>existence of an empty set, as I posted a proof earlier today
>
Here it is:
>AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
>schema of separation
>EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>
>
In this last step, and below (corrected per your later post), what is
x? It seems to be a free variable. Aren't you assuming the universe is
non-void?
>Ay(yeb <-> (yex & (yey & ~yey)) ... existential instantiation
>yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>yeb <-> (yey & ~yey) ... sentential logic
>yeb ... supposition toward a contradiction
>yey & ~yey ... sentential logic
>~yeb ... by contradiction
>Ay ~yeb ... universal generalization
>EbAy ~yeb .... existential generalization
>
Don't universal extantiation and existential generalization assume a
non-void universe? (I am really not familiar with logic at this level.)
>(didn't you and I already discuss this a few years ago?)
>
Not that I recall. (That doesn't mean it didn't happen! I am having
techincal difficulties searching my e-mail right now.)
MoeBlee
wrote:
> MoeBlee wrote:
> >Let P(x) be any logically true formula.
>
> >ExP(x) by logic alone.
>
> How so? How do we know the universe is not empty?
Semantically, the universe is not empty by stipulation, as built into
our rules for structures for a language. Syntactically, the existence
of at least one object is provable as it is built into our rules of
quantification.
It is a meta-theorem of first order logic that for any formula P,
if |- P then |- ExP.
For a particular instance (let's use, say, xex -> xex as 'P'):
Ex(xex -> xex)
> >Anyway, moreover, we don't need the axiom of infinity to prove the
> >existence of an empty set, as I posted a proof earlier today
>
> Here it is:
>
> >AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
> >schema of separation
> >EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>
> In this last step, and below (corrected per your later post), what is
> x? It seems to be a free variable. Aren't you assuming the universe is
> non-void?
I'm using only first order logic applied to an instance of the axiom
schema of specification. 'x' was bound in the axiom, then I applied
universal instantiation. This is 100% kosher first order logic.
> >Ay(yeb <-> (yex & (yey & ~yey)) ... existential instantiation
> >yeb <-> (yex & (yey & ~yey)) ... universal instantiation
> >yeb <-> (yey & ~yey) ... sentential logic
> >yeb ... supposition toward a contradiction
> >yey & ~yey ... sentential logic
> >~yeb ... by contradiction
> >Ay ~yeb ... universal generalization
> >EbAy ~yeb .... existential generalization
>
> Don't universal extantiation and existential generalization assume a
> non-void universe? (I am really not familiar with logic at this level.)
The rules are formulated so that they provide that the universe is non-
empty. I.e., the proof rules are formulated to match the semantic rule
that provides that any universe is non-empty.
For an alternative, in which the logic does not require that a
universe of discourse be non-empty, see the subject of "free logic".
However, ordinary formal set theory, such as ZFC, does not use free
logic.
MoeBlee
Jesse F. Hughes
Surely, you can see that your paper does not even begin to answer the
OP's question. By posting this link, you are merely wasting his time.
If you think otherwise, perhaps you could explain its relevance to his
question. You do understand his question, yes?
--
"Looking at their behavior I see them endangering not only their own
futures, but that of their families, and now, considering my latest
result, the future of people all over the world." -- James S. Harris,
on the shortsightedness of his mathematical critics
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > On 1 jul, 11:47, gratis mezn <zeroni...@googlemail.com> wrote:
> >> Hello all,
> >> I am sure this has been asked many times but couldn't find
> >> exactly what I am looking for.
>
> >> Basically, I am trying to understand what sets are (as opposed to
> >> classes). As I understand :
>
> >> 1) If you can define it by a formula its a class
>
> >> 2) its a set only if it is in the cumulative hierarchy.
>
> >> Now, how do we know that there exists atleast one set in the
> >> cumulative hierarchy. How do we establish the existence of the null
> >> set.
>
> >> For instance, {x : x \neq x} would be empty. How do we know its a set.
> >> Which axiom says it is a set?.
>
> >> We cant use separation here(???). I cant think how this becomes a set.
>
> > For what you will find it's worth:
>
> >http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf
>
> Surely, you can see that your paper does not even begin to answer the
> OP's question. By posting this link, you are merely wasting his time.
That's for the OP to decide, I think.
> If you think otherwise, perhaps you could explain its relevance to his
> question. You do understand his question, yes?
The relevance is "Something about Set Theory", huh ..
Han de Bruijn
David C. Ullrich
Erm, thanks. I was wondering what it said in _his_ version...
Anyway, the point is that once you know a set exists then
you _can_ use separation to show that the empty set exists
(by a much simpler argument than the one already posted,
of course). Any version of the axiom of infinity says
"There exists a set such that..." and hence shows that
a set exists.
Yes, of course the standard formulation of FOL stipulates
that something exists. But doing what we want to do here
explicitly from some axiom, for example the axiom
of infinity given his list of axioms, seems much better.
Why? Same as in that recent exchange between the two
of us. Your version is sort of "implementation-dependent";
it depends on the fact that we've formulated ZFC in
such a way that the "universe" consists of nothing but
sets. _If_ ZFC is formulated as a first-order theory
in which everything is a set then yes, the existence of
a set follows from nothing but standard logic. But
there's no reason that ZFC has to be formulated that
way! We can trivially give an equivalent formulation
in which we do not assume that everything is a set
and all the axioms have "Ax..." replaced by
"Ax Set(x) -> ...". That gives an identical set
theory, but in that version we can't just use the
fact that "something exists". Otoh the axiom
of infinity plus separation work in _any_
formulation.
In fact, if I recall correctly, you've recently seen
an expressed enthusiasm for a formalization of
set theory in which your "something exists,
therefore a set exists" is not true: In that pdf
that was pointed out to us by I-forget-the-name
over in sci.logic it's not assumed that everything
is a set.
>
>MoeBlee
David C. Ullrich
<sjher...@netscape.net> wrote:
>Aatu Koskensilta wrote:
>
>>"Stephen J. Herschkorn" <sjher...@netscape.net> writes:
>>
>>
>>>Let x be a set satisfying the axiom of
>>>inifinity. Let y = {z in x: z != z}, which exists by separation.
>>>y is the empty set.
>>>
>>>
>> <>
>> There's no need for x to be a set satisfying the axiom of infinity.
>
>
>Here is the OP's list of axioms:
>
>>Axiom of extensionality,
>>foundation,
>>separation ,
>>pairing,
>>union,
>>collection,
>>infinity,
>>powerset
>>
>
>Which of these, besides the axiom of infinty, implies that any set exists?
Moeblee's answer to this is technically correct _if_ we assume the
axioms are formalized in just the right way - they're not always
formulated that way. See my reply to him elsewhere in the thread.
I _think_ that to clarify what's puzzling you about Aatu's
comment you need to look at _both_ things he said:
>
>This is a baffling proof. If there is something that has the empty set
>as an element clearly the empty set exists.
The proof you gave _is_ a little baffling - if you assume
the axiom of infinity in exactly the form you stated it
then there's no reason to go through the argument you
gave...
>> Or here's another. Let x be a set satisfying the axiom of
>> inifinity. Let y = {z in x: z != z}, which exists by separation.
>> y is the empty set.
>
>There's no need for x to be a set satisfying the axiom of infinity.
Anyway, a point, which I suspect may be his point, is that
all you need is the existence of a set, period. If s is a
set then separation shows that {x in s : x <> x} is a set,
and hence the empty set exists.
"There's no need for x to be a set satisfying the axiom of infinity"
is not quite the same thing as "there's no need for the axiom
of infinity".
David C. Ullrich
The fact that the words "set theory" appear in the OP and also
in that pdf does not make the pdf relevant to the question.
It's not.
Unless you can explain to us exactly _what_ in that
pdf says something about the question "How do you
prove the existence of the empty set in ZFC?".
This may be interesting, since the paper is explicitly
_not_ about ZFC...
>Han de Bruijn
Jesse F. Hughes
> On 2 jul, 04:34, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Han de Bruijn <umum...@gmail.com> writes:
>>
>> > On 1 jul, 11:47, gratis mezn <zeroni...@googlemail.com> wrote:
>> >> Hello all,
>> >> I am sure this has been asked many times but couldn't find
>> >> exactly what I am looking for.
>>
>> >> Basically, I am trying to understand what sets are (as opposed to
>> >> classes). As I understand :
>>
>> >> 1) If you can define it by a formula its a class
>>
>> >> 2) its a set only if it is in the cumulative hierarchy.
>>
>> >> Now, how do we know that there exists atleast one set in the
>> >> cumulative hierarchy. How do we establish the existence of the null
>> >> set.
>>
>> >> For instance, {x : x \neq x} would be empty. How do we know its a set.
>> >> Which axiom says it is a set?.
>>
>> >> We cant use separation here(???). I cant think how this becomes a set.
>>
>> > For what you will find it's worth:
>>
>> >http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf
>>
>> Surely, you can see that your paper does not even begin to answer the
>> OP's question. By posting this link, you are merely wasting his time.
>
> That's for the OP to decide, I think.
Precisely what I mean when I say you're wasting his time.
You understand his question and you know the contents of that paper,
so surely you should suggest that paper only if you see that it is
relevant to his question.
>> If you think otherwise, perhaps you could explain its relevance to his
>> question. You do understand his question, yes?
>
> The relevance is "Something about Set Theory", huh ..
Oh, well, if that's the measure, then perhaps he should read thousands
of published articles on the topic as well.
Really, your suggestion that he read your paper simply wastes his
time.
--
Jesse F. Hughes
"Certainly he who can digest a second or third fluxion need
not, methinks, be squeamish about any point in divinity."
George Berkeley, 1734
MoeBlee
> On Wed, 1 Jul 2009 12:43:59 -0700 (PDT), MoeBlee
>
> Yes, of course the standard formulation of FOL stipulates
> that something exists. But doing what we want to do here
> explicitly from some axiom, for example the axiom
> of infinity given his list of axioms, seems much better.
>
> Why? Same as in that recent exchange between the two
> of us. Your version is sort of "implementation-dependent";
> it depends on the fact that we've formulated ZFC in
> such a way that the "universe" consists of nothing but
> sets. _If_ ZFC is formulated as a first-order theory
> in which everything is a set then yes, the existence of
> a set follows from nothing but standard logic. But
> there's no reason that ZFC has to be formulated that
> way!
I did touch on that in a way. My point is that we don't need any set
theoretic axioms to prove that there exists some OBJECT or to prove
that there is an object that has no members. Then, we use some
definition of 'set', along with either pairing or power set (which, of
course, are set theoretic axioms), to prove that there exists a SET.
Again, we don't need to assume infinity just to prove there exists a
set (we may use as little as pairing).
> We can trivially give an equivalent formulation
> in which we do not assume that everything is a set
> and all the axioms have "Ax..." replaced by
> "Ax Set(x) -> ...".
Of course, we can take 'Set' as an added primitive. That will allow us
to relativize the axiom of extensionality to sets so that it is not
ruled out that there exist urelements. (Or we can take '0' as
primitive and acheive the same thing with a different formulation; or
take 'is an urelement' as primitive.)
Is addding 'set' as an added primitive what you have in mind?
> That gives an identical set
> theory,
If 'set' is an added primitive, then that's not the same theory as Z
set theory without 'set' as primitive.
So, do you have in mind that 'set' is DEFINED, then all axioms are
relativized to 'set'?
> but in that version we can't just use the
> fact that "something exists". Otoh the axiom
> of infinity plus separation work in _any_
> formulation.
I only pointed out that logic gives us such formulas as Ex(xex ->
xex), but I didn't use such a formula in my derivation of the
existence of an empty object from the axiom schema of separation (then
that that object is a set follows from the definition of 'set' as
found in say, Bernays-style class theory (x is a set iff Ey xey) and
the pairing axiom (if we have pairing as an axiom) or from power set).
But, of course, if the axiom of infinity is formulated as "Ex(x is a
set & 0ex & Ayex yu{y}ex)" then the axiom of infinity proves the
existence of a set.
> In fact, if I recall correctly, you've recently seen
> an expressed enthusiasm for a formalization of
> set theory in which your "something exists,
> therefore a set exists" is not true:
Something exists therefore a set exists (given a suitable definition
of 'set' and, say, pairing or power set). Just to be clear, I don't
claim that the existence of a set follows from logic alone.
> In that pdf
> that was pointed out to us by I-forget-the-name
> over in sci.logic it's not assumed that everything
> is a set.
I don't recall that. Maybe IST? But that's a different situation with
an added primitive 'standard'.
But, of course, in Bernays class theory (in the same language as Z set
theory) it is a theorem that not every object is a set. I don't at all
dispute such things.
MoeBlee
MoeBlee
> Maybe IST? But that's a different situation with
> an added primitive 'standard'.
But even in IST, "Ax x is a set" is a theorem, given the definition "x
is a set iff Ey xey".
Though, this is aside the point anyway, since my remarks have never
been that "Ax x is a set" is a theorem of all set or class theories
(indeed it is not a theorem in theories that allow that there may be
urelements and in theories that have proper classes); but rather that
"Ex x is a set" (given such definitions as "x is a set iff Ey xey") is
derivable in such ordinary theories without having to resort to the
axiom of infinity.
MoeBlee
David C. Ullrich
<jazz...@hotmail.com> wrote:
>On Jul 2, 5:37�am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Wed, 1 Jul 2009 12:43:59 -0700 (PDT), MoeBlee
>>
>> <jazzm...@hotmail.com> wrote:
>
>> Yes, of course the standard formulation of FOL stipulates
>> that something exists. But doing what we want to do here
>> explicitly from some axiom, for example the axiom
>> of infinity given his list of axioms, seems much better.
>>
>> Why? Same as in that recent exchange between the two
>> of us. Your version is sort of "implementation-dependent";
>> it depends on the fact that we've formulated ZFC in
>> such a way that the "universe" consists of nothing but
>> sets. _If_ ZFC is formulated as a first-order theory
>> in which everything is a set then yes, the existence of
>> a set follows from nothing but standard logic. But
>> there's no reason that ZFC has to be formulated that
>> way!
>
>I did touch on that in a way. My point is that we don't need any set
>theoretic axioms to prove that there exists some OBJECT or to prove
>that there is an object that has no members.
Huh???????? How are we going to prove there is an object
"with no members" without any axioms regarding "membership"?
>Then, we use some
>definition of 'set', along with either pairing or power set (which, of
>course, are set theoretic axioms), to prove that there exists a SET.
????? What versions of those axioms do you have in mind?
The ones I know are these:
If x, y are sets then there exists a set with exactly x and y as
elements.
If x is a set then there exists a set P such that for every y,
y is an element of P if and only if every element of y
is an element of x.
Using those to prove that there exists a set is going to be
tricky.
Oh. I just saw the "formal" part at the bottom of one of
your posts, where you claim to be using separation to
show that there exists a set. In that derivation you are
_assuming_ that everything is a set - my whole point
is that one might not want to assume this.
And you say that you're not using the general
"there exists something" from logic explicitly,
but you _do_ use that when you apply
universal instantiation:
AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
schema of separation
This is separation _if_ we're assuming that everything is a set.
EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
You can't erase the initial "Ax" in the first line except
in a context where we're assuming that something exists.
So your formal proof _does_ use "there exists something, by
logic", and it _does_ use "everything is a set". Which makes
it a little silly - since you _are_ using those two facts
it's immediate that there exists a set. (And silly or not,
my point is that it seems preferable to use things that
we assume _about sets_, because that gives a proof
that's valid in any equivalent formalization.)
>Again, we don't need to assume infinity just to prove there exists a
>set (we may use as little as pairing).
In, for example,
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
every axiom except for infinity begins with a universal
quantifier.
>> We can trivially give an equivalent formulation
>> in which we do not assume that everything is a set
>> and all the axioms have "Ax..." replaced by
>> "Ax Set(x) -> ...".
>
>Of course, we can take 'Set' as an added primitive. That will allow us
>to relativize the axiom of extensionality to sets so that it is not
>ruled out that there exist urelements. (Or we can take '0' as
>primitive and acheive the same thing with a different formulation; or
>take 'is an urelement' as primitive.)
>
>Is addding 'set' as an added primitive what you have in mind?
My point was just that there are _many_ ways to formalize
ZFC in which your "standard conventions in logic imply
that there exists something, hence since we assume that
everything is a set, there exists a set" is simply invalid.
The specific example I was referring to was
http://www.mathematik.tu-darmstadt.de/~blumensath/st.pdf
, where everything is a class and some classes are sets.
>> That gives an identical set
>> theory,
>
>If 'set' is an added primitive, then that's not the same theory as Z
>set theory without 'set' as primitive.
Oh come on now. The appearance of the marks on the page is
different - you don't see that the two are isomorphic?
To be incredibly explicit: Say T is the formal set theory
defined on the wikipedia page cited above. Let T' be
the theory obtained from T by adding Set as a primitive
and replacing every Ax by "Ax Set(x) ->" and replacing
Ex by "Ex Set(x) and". Then the things that T' says
about sets are exactly the same as the things that T says
about sets.
Again, to be explicit: If T'' consists of T' plus the
axiom "Ax Set(x)" then every theorem of T is
also a theorem of T''.
>So, do you have in mind that 'set' is DEFINED, then all axioms are
>relativized to 'set'?
>
>> but in that version we can't just use the
>> fact that "something exists". Otoh the axiom
>> of infinity plus separation work in _any_
>> formulation.
>
>I only pointed out that logic gives us such formulas as Ex(xex ->
>xex), but I didn't use such a formula in my derivation of the
>existence of an empty object from the axiom schema of separation (then
>that that object is a set follows from the definition of 'set' as
>found in say, Bernays-style class theory (x is a set iff Ey xey) and
>the pairing axiom (if we have pairing as an axiom) or from power set).
>
>But, of course, if the axiom of infinity is formulated as "Ex(x is a
>set & 0ex & Ayex yu{y}ex)" then the axiom of infinity proves the
>existence of a set.
What's an example of an alternate formulation of the
axiom of infinity that does _not_ have the existence of
a set as a trivial consequence?
>> In fact, if I recall correctly, you've recently seen
>> an expressed enthusiasm for a formalization of
>> set theory in which your "something exists,
>> therefore a set exists" is not true:
>
>Something exists therefore a set exists (given a suitable definition
>of 'set' and, say, pairing or power set). Just to be clear, I don't
>claim that the existence of a set follows from logic alone.
_How_ does the existence of something, perhaps not a set,
together with pairing or power set, imply that there
exists a set?
Pairing, for example, says that for any two _sets_ x and y, etc.
Note that in, for example, that pdf, proper classes are not
allowed to be elements of sets.
>> In that pdf
>> that was pointed out to us by I-forget-the-name
>> over in sci.logic it's not assumed that everything
>> is a set.
>
>I don't recall that. Maybe IST? But that's a different situation with
>an added primitive 'standard'.
>
>But, of course, in Bernays class theory (in the same language as Z set
>theory) it is a theorem that not every object is a set. I don't at all
>dispute such things.
How _do_ you get the existence of a set in Bernays class theory
just from separation and pairing?
gratis mezn
> On 2 jul, 04:34, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
>
>
> > Han de Bruijn <umum...@gmail.com> writes:
>
> > > On 1 jul, 11:47, gratis mezn <zeroni...@googlemail.com> wrote:
> > >> Hello all,
> > >> I am sure this has been asked many times but couldn't find
> > >> exactly what I am looking for.
>
> > >> Basically, I am trying to understand what sets are (as opposed to
> > >>classes). As I understand :
>
> > >> 1) If you can define it by a formula its a class
>
>
> > >> Now, how do we know that there exists atleast onesetin the
> > >> cumulative hierarchy. How do we establish the existence of the null
> > >>set.
>
> > >> For instance, {x : x \neq x} would be empty. How do we know its aset.
> > >> Which axiom says it is aset?.
>
> > >> We cant use separation here(???). I cant think how this becomes aset.
>
> > > For what you will find it's worth:
>
> > >http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf
>
> > Surely, you can see that your paper does not even begin to answer the
> > OP's question. By posting this link, you are merely wasting his time.
>
> That's for the OP to decide, I think.
>
> > If you think otherwise, perhaps you could explain its relevance to his
> > question. You do understand his question, yes?
>
>
> Han de Bruijn
I couldn't find anything relevant. I read the first para and then
glanced through it.
Stephen J. Herschkorn
>On Jul 1, 6:28 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
>wrote:
>
>
>>MoeBlee wrote:
>>
>>
>>>Let P(x) be any logically true formula.
>>>
>>>
>>>ExP(x) by logic alone.
>>>
>>>
>>How so? How do we know the universe is not empty?
>>
>>
>
>Semantically, the universe is not empty by stipulation, as built into
>our rules for structures for a language. Syntactically, the existence
>of at least one object is provable as it is built into our rules of
>quantification.
>
>It is a meta-theorem of first order logic that for any formula P,
>if |- P then |- ExP.
>
>
Others have commented basically the same thing. This is new to me
(though I don't doubt it). What does one make of the theory consisting
of the sole axiom (*), "For all x, x != x"? This theory does have a
model, e.g., the empty set.
Less obviously, one might posit a theory (all of whose axioms are
universally quantified) that has (*) as a theorem, though one might not
realize it at first. For example, consider a set theory which consists
only of Extensionality, Union, Replacement, Power, Foundation (=
Regularity), and the axiom, "For all x, x in x." This implies "For
all x, x in x and not(x in x)," which implies that the universe must
be void. What do we make of such a theory? Why do we limit ourselves
to the assumption of a non-void universe?
Note that in ZF, Infinity guarantees the existence of a set, so we avoid
all this. (In the preceding thread there was some objection to a proof
that used Infinity to apply separation to *some* set. The claim is that
Infinity is unnecessary.)
Herbert Newman
> Others have commented basically the same thing. This is new to me
> (though I don't doubt it). What does one make of the theory consisting
> of the sole [non logical] axiom (*), "For all x, x != x"?
Depends. If your framework (i.e. the framework of the theory) is FOPL _with
identity_ your theory would be inconsistent. Since
Ax(x = x)
is a theorem in /FOPL with identity/. And (hence) it does not have a model.
(Note that the domain of a model for a FO theory must contain at least one
object.)
So let's assume our framework is just FOPL (with "=" and/or "=/=" in its
_language_). Then your axiom just states concerning the "relation" =/= that
For all x, x =/= x.
So what? Since there is no other axiom related to "=/=" there's no
"conflict" whatsoever. It's like adopting the axiom
AxR(x,x) ,
for some 2-ary predicate letter R.
Of course this theory does have a model. (Interpret "R" as identity on the
domain, where the domain is arbitraty, but not empty).
On the other hand, if you adopt the axiom
Ax~(x = x) ,
there's again no problem. (Interpret "=" as empty set, i.e. empty relation,
where the domain is arbitraty, but not empty).
> This theory does have a model, e.g., the empty set.
Nope. There are no empty models in FO theories.
> Note that in ZF, Infinity guarantees the existence of a set ...
Sure.
Though...
> In the preceding thread there was some objection to a proof that used
> Infinity to apply separation to *some* set. The claim is that Infinity
> is unnecessary.
Right. Since the framework of ZF(C) usually (always?) is FOPL (usually WITH
identity), we do not need any "set existence" axiom - though sometimes
axiom systems for ZF(C) _do_ contain such an axiom! :-)
I guess, the idea is that ZF(C) should "hold" [at least "in principle"]
even in a "FO" framework which allows for empty domains.
I guess the (historical) REASON is that historically ZF was born _before_
FOPL! (With other words, tradition is stronger than "logical necessity" in
this case. Since such provisions are simply not necessary in the framework
of FOPL.)
Herb
Aatu Koskensilta
> Depends. If your framework (i.e. the framework of the theory) is FOPL
> _with identity_ your theory would be inconsistent. Since
>
> Ax(x = x)
>
> is a theorem in /FOPL with identity/. And (hence) it does not have a model.
> (Note that the domain of a model for a FO theory must contain at least one
> object.)
Right, but this is just a more or less arbitrary convention. It is
possible to formulate first-order logic so that Ex(x = x) is not a
logical truth, and the theory with (x)(x =/= x) as its sole axiom is
consistent on such a formulation.
David C. Ullrich
<sjher...@netscape.net> wrote:
>In sci.math, MoeBlee wrote:
>
>>On Jul 1, 6:28 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net>
>>wrote:
>>
>>
>>>MoeBlee wrote:
>>>
>>>
>>>>Let P(x) be any logically true formula.
>>>>
>>>>
>>>>ExP(x) by logic alone.
>>>>
>>>>
>>>How so? How do we know the universe is not empty?
>>>
>>>
>>
>>Semantically, the universe is not empty by stipulation, as built into
>>our rules for structures for a language. Syntactically, the existence
>>of at least one object is provable as it is built into our rules of
>>quantification.
>>
>>It is a meta-theorem of first order logic that for any formula P,
>>if |- P then |- ExP.
>>
>>
>
>Others have commented basically the same thing. This is new to me
>(though I don't doubt it). What does one make of the theory consisting
>of the sole axiom (*), "For all x, x != x"? This theory does have a
>model, e.g., the empty set.
With the standard conventions this theory is inconsistent; it
has no model because the empty set is not allowed. No
reason it has to be that way, that's the way it's done.
(I imagine because Ax P(x) seems like it "should"
imply Ex P(x).)
>Less obviously, one might posit a theory (all of whose axioms are
>universally quantified) that has (*) as a theorem, though one might not
>realize it at first. For example, consider a set theory which consists
>only of Extensionality, Union, Replacement, Power, Foundation (=
>Regularity), and the axiom, "For all x, x in x." This implies "For
>all x, x in x and not(x in x)," which implies that the universe must
>be void. What do we make of such a theory? Why do we limit ourselves
>to the assumption of a non-void universe?
Possibly because the theory of the empty set is not regarded
as all that interesting?
>Note that in ZF, Infinity guarantees the existence of a set, so we avoid
>all this. (In the preceding thread there was some objection to a proof
>that used Infinity to apply separation to *some* set. The claim is that
>Infinity is unnecessary.)
David C. Ullrich
Aatu Koskensilta
> Possibly because the theory of the empty set is not regarded as all
> that interesting?
Possibly. A more important reason is probably that the possibility that
variables don't refer introduces all sorts of complications. (Recall
that most deductive systems for first-order logic involve in an
essential way deductions with open formulas.)
MoeBlee
David, please take this in a constructive spirit: I wonder whether
you're taking me to be claiming more than I really am claiming. What
I've said has been, for the most part, about derivability of certain
formulas. Let's consider those particular claims and try to weed out
from those more general things that I have not actually said.
But I did make one misstatement in my last post, for which I don't
blame you for objecting to it:
> On Thu, 2 Jul 2009 10:42:07 -0700 (PDT), MoeBlee
> >we don't need any set
> >theoretic axioms to prove that there exists some OBJECT or to prove
> >that there is an object that has no members.
>
> Huh???????? How are we going to prove there is an object
> "with no members" without any axioms regarding "membership"?
I meant to say that we don't need set theoretic axioms to prove that
there exists an object (exactly, that there exists some object having
some property: "ExPx", where 'P' could be, say, 'x=x') and that we
don't need the axiom of infinity to prove that there is an object that
has no members (exactly, that there is an object such that nothing is
related to it (i.e., related from the "left") by the epsilon relation,
whatever that epsilon relation may be: "ExAy~yex"). Of course a set
theoretic axiom is needed for that: I used an instance of the axiom
schema of separation. I had meant to say that the axiom of infinity is
not needed.
> >Then, we use some
> >definition of 'set', along with either pairing or power set (which, of
> >course, are set theoretic axioms), to prove that there exists a SET.
>
> ????? What versions of those axioms do you have in mind?
> The ones I know are these:
>
> If x, y are sets then there exists a set with exactly x and y as
> elements.
The version I use, which is common in many a textbook on set theory,
doesn't use a relativization to 'set'. So the axiom is: For all x any
y, there exists an object with exactly x and y as members:
AxyEzAv(vez <-> (v=x or v=y))
Then, if we take some appropriate definition of set, such as:
x is a set <-> Ey xey
then, we can prove there exists a set:
Take any x (whether it is a set or not), then x in {x x} = {x}.
So I have not assumed there exists a set. Also, as the existence of
some object, my point is that I don't use some axiom such as:
ExPx.
I.e., I don't use some axiom that says there exists an object having
some property.
That such a statement is a THEOREM of logic alone is not disputed by
mean, indeed, in the discussion, I've been affirming at all along.
> Oh. I just saw the "formal" part at the bottom of one of
> your posts, where you claim to be using separation to
> show that there exists a set.
Not just separation, but a definition of 'is a set' and the axiom of
pairing (or power set would do too).
> In that derivation you are
> _assuming_ that everything is a set
No, I don't. Though, it turns out to be a theorem that everything is a
set.
Again, to be explicit:
Def: x is a set <-> Ez xez
Axiom: AxyEzAv(vez <-> (v=x or v=y))
Theorem: Ex x is a set
Proof:
EzAv(vez <-> (v=x or v=y))
Let Av(vez <-> (v=x or v=y))
xez
x is set
Ex x is a set
And that everything is just as easy to prove.
I'm not attaching any great philosphic or even mathematical importance
to that. I'm not claiming any special characterization of it. I'm not
claiming that this is the most general way to approach the subject.
I'm just saying that in Z set theory (with its axioms as often given
without relativization to 'set'), and given such a definition of 'is a
set', we do have the proof above.
> And you say that you're not using the general
> "there exists something" from logic explicitly,
> but you _do_ use that when you apply
> universal instantiation:
David, in all constructive spirit, please consider exactly what I've
said. I've AGREED that the logic provides for the existence of at
least one object. What I said is that I haven't used any axiom that
says VERBATIM: ExPx, for some 'P'. But, yes, such a statement is a
THEOREM (that comes right out of the mechanics of our rules/logical
axioms, such as universal instatiation). But I didn't even make use of
such a theorem; rather I let existence of an object come right out of
the logic, and then existence of a set come right out of (via, of
course, the logic) the definition I gave of 'is a set' and the pairing
axiom. And I've not claimed that this is profound or anything like
that. Just that it is the case. That's all I'm really saying. Please
consider that you might have thought I've claimed something very much
more than that.
> AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
> schema of separation
>
> This is separation _if_ we're assuming that everything is a set.
It is an exact instance of the axiom schema of separation. Purely, an
instance of an infinite set of formulas, none of which mention the
predicate 'is a set'. I've not used anything about 'set'.
> EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>
> You can't erase the initial "Ax" in the first line except
> in a context where we're assuming that something exists.
I'm just using universal instantiation. That the rule of universal
instantiation goes along semantically with the semantic stipulation
that the domain of discourse is nonempty is not contested by me,
indeed; I've affirmed it all along.
> So your formal proof _does_ use "there exists something, by
> logic",
I've not disputed that. All I've said is that I don't make use of any
verbatim axiom (or even theorem) of the form "ExPx".
Again, I've said all along, the logic itself provides theorems of the
form "ExPx". And I've agreed all along that the logic works hand in
glove with a semantics that stipulates a nonempty domain of discourse.
> and it _does_ use "everything is a set".
No, I did not assume everything is a set.
> >Again, we don't need to assume infinity just to prove there exists a
> >set (we may use as little as pairing).
>
> In, for example,
>
> http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>
> every axiom except for infinity begins with a universal
> quantifier.
I'll take your word for that. But it doesn't in itself dispute that we
don't need the axiom of infinity to prove that there exists a set.
> To be incredibly explicit: Say T is the formal set theory
> defined on the wikipedia page cited above. Let T' be
> the theory obtained from T by adding Set as a primitive
> and replacing every Ax by "Ax Set(x) ->" and replacing
> Ex by "Ex Set(x) and". Then the things that T' says
> about sets are exactly the same as the things that T says
> about sets.
Sure, and I'm not disputing that one bit.
> Again, to be explicit: If T'' consists of T' plus the
> axiom "Ax Set(x)" then every theorem of T is
> also a theorem of T''.
I think so, yes.
I agree with such things, and they don't contradict what I've said.
> >I only pointed out that logic gives us such formulas as Ex(xex ->
> >xex), but I didn't use such a formula in my derivation of the
> >existence of an empty object from the axiom schema of separation (then
> >that that object is a set follows from the definition of 'set' as
> >found in say, Bernays-style class theory (x is a set iff Ey xey) and
> >the pairing axiom (if we have pairing as an axiom) or from power set).
>
> >But, of course, if the axiom of infinity is formulated as "Ex(x is a
> >set & 0ex & Ayex yu{y}ex)" then the axiom of infinity proves the
> >existence of a set.
>
> What's an example of an alternate formulation of the
> axiom of infinity that does _not_ have the existence of
> a set as a trivial consequence?
I don't know of one offhand. My purpose was just to be clear that I
affirm your version does prove that there exists a set. (Actually,
better just to disregard that particular quoted comment by me; it's
not wrong, but it doesn't make any point really.)
> >But, of course, in Bernays class theory (in the same language as Z set
> >theory) it is a theorem that not every object is a set. I don't at all
> >dispute such things.
>
> How _do_ you get the existence of a set in Bernays class theory
> just from separation and pairing?
I didn't claim to be able to do so. My remarks were about Z set
theories, just as the poster himself specified his system to be ZF.
MoeBlee
MoeBlee
> And that everything is just as easy to prove.
That should be: that everything is a set is as easy to prove.
MoeBlee
David C. Ullrich
<jazz...@hotmail.com> wrote:
>On Jul 3, 4:36 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>
>David, please take this in a constructive spirit: I wonder whether
>you're taking me to be claiming more than I really am claiming. What
>I've said has been, for the most part, about derivability of certain
>formulas. Let's consider those particular claims and try to weed out
>from those more general things that I have not actually said.
Fine. But if you want to clarify things you need to clarify
a few things. Below I can't tell whether or not you're
using the "something exists" convention from logic,
and also whether you're considering a set theory
in which everything is a set.
There are books with this property which are _not_ talking
about set theories in which everything is a set?
That version of the axiom is _not_, for example, part
of (what I assume to be the standard version of)
Godel=Bernays set theory. See for example
http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
where it's made very clear that proper classes are not allowed
to be elements of anything. (Or that pdf cited below, where
proper classes are not allowed to be elements of anything.)
>Then, if we take some appropriate definition of set, such as:
>
>x is a set <-> Ey xey
>
>then, we can prove there exists a set:
>
>Take any x (whether it is a set or not), then x in {x x} = {x}.
>
>So I have not assumed there exists a set. Also, as the existence of
>some object, my point is that I don't use some axiom such as:
>
>ExPx.
>
>I.e., I don't use some axiom that says there exists an object having
>some property.
>
>That such a statement is a THEOREM of logic alone is not disputed by
>mean, indeed, in the discussion, I've been affirming at all along.
>
>> Oh. I just saw the "formal" part at the bottom of one of
>> your posts, where you claim to be using separation to
>> show that there exists a set.
>
>Not just separation, but a definition of 'is a set' and the axiom of
>pairing (or power set would do too).
>
>> In that derivation you are
>> _assuming_ that everything is a set
>
>No, I don't. Though, it turns out to be a theorem that everything is a
>set.
For heaven's sake. You're talking about a set theory where everything
is a set. I don't see how it matters whether that's an axiom or a
theorem - the argument does not apply to, for example, Godel-Bernays
set theory.
>Again, to be explicit:
>
>Def: x is a set <-> Ez xez
>Axiom: AxyEzAv(vez <-> (v=x or v=y))
>Theorem: Ex x is a set
>Proof:
>EzAv(vez <-> (v=x or v=y))
>Let Av(vez <-> (v=x or v=y))
>xez
>x is set
>Ex x is a set
>
>And that everything is just as easy to prove.
>
>I'm not attaching any great philosphic or even mathematical importance
>to that. I'm not claiming any special characterization of it. I'm not
>claiming that this is the most general way to approach the subject.
>I'm just saying that in Z set theory (with its axioms as often given
>without relativization to 'set'), and given such a definition of 'is a
>set', we do have the proof above.
>
>> And you say that you're not using the general
>> "there exists something" from logic explicitly,
>> but you _do_ use that when you apply
>> universal instantiation:
>
>David, in all constructive spirit, please consider exactly what I've
>said. I've AGREED that the logic provides for the existence of at
>least one object.
I don't understand the point to writing AGREED that way.
My whole point is that it would be better _not_ to use this
fact.
Because, for example, every statement _about sets_ that's
true in ZFC is also true in GB. But the fact that logic
provides for the existence of at least one object is
_not_ enough to show that a set exists in GB (or if
it is nobody's explained how).
>What I said is that I haven't used any axiom that
>says VERBATIM: ExPx, for some 'P'. But, yes, such a statement is a
>THEOREM (that comes right out of the mechanics of our rules/logical
>axioms, such as universal instatiation). But I didn't even make use of
>such a theorem; rather I let existence of an object come right out of
>the logic, and then existence of a set come right out of (via, of
>course, the logic) the definition I gave of 'is a set' and the pairing
>axiom. And I've not claimed that this is profound or anything like
>that. Just that it is the case. That's all I'm really saying. Please
>consider that you might have thought I've claimed something very much
>more than that.
>
>> AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
>> schema of separation
>>
>> This is separation _if_ we're assuming that everything is a set.
>
>It is an exact instance of the axiom schema of separation. Purely, an
>instance of an infinite set of formulas, none of which mention the
>predicate 'is a set'. I've not used anything about 'set'.
But you're using that axiom. And it's _not_ an axiom in staandard
set theories in which not everything is a set.
>> EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>>
>> You can't erase the initial "Ax" in the first line except
>> in a context where we're assuming that something exists.
>
>I'm just using universal instantiation. That the rule of universal
>instantiation goes along semantically with the semantic stipulation
>that the domain of discourse is nonempty is not contested by me,
I didn't say it was concocted by you.
>indeed; I've affirmed it all along.
>
>> So your formal proof _does_ use "there exists something, by
>> logic",
>
>I've not disputed that.
Fine. Then I don't see how any of this is relevant to what I
thought you were replying to.
>All I've said is that I don't make use of any
>verbatim axiom (or even theorem) of the form "ExPx".
>
>Again, I've said all along, the logic itself provides theorems of the
>form "ExPx". And I've agreed all along that the logic works hand in
>glove with a semantics that stipulates a nonempty domain of discourse.
>
>> and it _does_ use "everything is a set".
>
>No, I did not assume everything is a set.
Aargh. I didn't say you "assumed" that. You're
using the fact that you _are_ in a set theory where
everything is a set.
>> >Again, we don't need to assume infinity just to prove there exists a
>> >set (we may use as little as pairing).
>>
>> In, for example,
>>
>> http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>
>> every axiom except for infinity begins with a universal
>> quantifier.
>
>I'll take your word for that. But it doesn't in itself dispute that we
>don't need the axiom of infinity to prove that there exists a set.
It shows that we _do_ need the axiom of infinity _if_ we're
going to avoid using the "logic says that something
exists".
Or: It shows that we _do_ need the axiom of infinity
to prove that there exists a set in that theory where
all the axioms of ZFC have had "Ax" replaced by
"Ax Set(x) ->". Those theories are equivalent _wrt_
what they say about sets, but any structure with no
sets is a model of the second theory minus infinity.
Here's the point: The distinction between the theories
T and T' below is of interest only to logicians;
it's a purely _formal_ distinction. An actual mathematician
doing actual mathematics doesn't care about the difference
between "everything is a set" and "sets are the only
things we're talking about today". A proof that's valid
in one context but not the other is "implementation
dependent", like what we were talking about the
other day, a proof of something about the reals
that's valid if the reals were defined as Dedekind
cuts but not if the reals were defined as equivalence
classes of Cauchy sequences of rationals.
Which is to say that the proof using the axiom of
infinity shows that the existence of a set "really
does" follow from the axioms of ZFC; it's
sufficiently robust that it works under various
equivalent formalizations of the axioms,
while a proof using "something exists, by logic",
and/or "everything is a set" does not have this
robustness, it's "implementation dependent".
It seems to me that it seems to you that I'm
quibbling over the same sort of details as you're
quibbling over. I don't _care_ whether you're
using "logic says something exists" or using
"Ex x=x"; using one is _equivalent_ to using
the other! Imo it would be much better not to
use this (or if you insist, not to use "either of
these").
Similarly for what seems to me the even more
tenuous distinction between using "everything
is a set" and "we're talking about a set theory
in which everything is a set"...
>> To be incredibly explicit: Say T is the formal set theory
>> defined on the wikipedia page cited above. Let T' be
>> the theory obtained from T by adding Set as a primitive
>> and replacing every Ax by "Ax Set(x) ->" and replacing
>> Ex by "Ex Set(x) and". Then the things that T' says
>> about sets are exactly the same as the things that T says
>> about sets.
>
>Sure, and I'm not disputing that one bit.
>
>> Again, to be explicit: If T'' consists of T' plus the
>> axiom "Ax Set(x)" then every theorem of T is
>> also a theorem of T''.
>
>I think so, yes.
>
>I agree with such things, and they don't contradict what I've said.
You seem to be missing my point. Given that the two theories
"say" the same things "about" sets, doesn't it seem like a proof
of the existence of the empty set that works in both theories
would be preferable to a proof that works in one but not
the other?
>> >I only pointed out that logic gives us such formulas as Ex(xex ->
>> >xex), but I didn't use such a formula in my derivation of the
>> >existence of an empty object from the axiom schema of separation (then
>> >that that object is a set follows from the definition of 'set' as
>> >found in say, Bernays-style class theory (x is a set iff Ey xey) and
>> >the pairing axiom (if we have pairing as an axiom) or from power set).
>>
>> >But, of course, if the axiom of infinity is formulated as "Ex(x is a
>> >set & 0ex & Ayex yu{y}ex)" then the axiom of infinity proves the
>> >existence of a set.
>>
>> What's an example of an alternate formulation of the
>> axiom of infinity that does _not_ have the existence of
>> a set as a trivial consequence?
>
>I don't know of one offhand. My purpose was just to be clear that I
>affirm your version does prove that there exists a set. (Actually,
>better just to disregard that particular quoted comment by me; it's
>not wrong, but it doesn't make any point really.)
>
>> >But, of course, in Bernays class theory (in the same language as Z set
>> >theory) it is a theorem that not every object is a set. I don't at all
>> >dispute such things.
>>
>> How _do_ you get the existence of a set in Bernays class theory
>> just from separation and pairing?
>
>I didn't claim to be able to do so. My remarks were about Z set
>theories, just as the poster himself specified his system to be ZF.
>
>MoeBlee
David C. Ullrich
Aatu Koskensilta
> It shows that we _do_ need the axiom of infinity _if_ we're going to
> avoid using the "logic says that something exists".
Sure. In fact, the axiom, redundant in presence of infinity,
There exists an empty set.
is often included among the axioms of ZFC when presented in ordinary
mathematical English. If this is not done we're led to such rather odd
conclusions as that the axiom of infinity is needed to prove {{},{{}}}
exists and so on.
This whole discussion is a perfect illustration of something I like to
babble about interminably, that we need to exercise our best judgment
and use our good sense to figure out whether this or that formal
technicality or detail is related in any informative or significant way
to anything in our actual mathematical reasoning, or whether it's just
that, a technicality of no intrinsic interest.
David C. Ullrich
<aatu.kos...@uta.fi> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>
>> It shows that we _do_ need the axiom of infinity _if_ we're going to
>> avoid using the "logic says that something exists".
>
>Sure. In fact, the axiom, redundant in presence of infinity,
>
> There exists an empty set.
>
>is often included among the axioms of ZFC when presented in ordinary
>mathematical English. If this is not done we're led to such rather odd
>conclusions as that the axiom of infinity is needed to prove {{},{{}}}
>exists and so on.
>
>This whole discussion is a perfect illustration of something I like to
>babble about interminably, that we need to exercise our best judgment
>and use our good sense to figure out whether this or that formal
>technicality or detail is related in any informative or significant way
>to anything in our actual mathematical reasoning, or whether it's just
>that, a technicality of no intrinsic interest.
Precisely. Although, curiously, I'm not entirely certain which
side you're wagging your finger at here. My interpretation
of the relevance of that paragraph is that you're agreeing
with me that getting the existence of the empty set from
the convention that something is a technicality of no
intrinsic interest, but it seems _possible_ that you're
instead referring to my insistence that the "right" way
to do it is to use the axiom of infinity, because that
shows it "really does follow from the axioms"...
Dave Seaman
> David C. Ullrich <dull...@sprynet.com> writes:
>> It shows that we _do_ need the axiom of infinity _if_ we're going to
>> avoid using the "logic says that something exists".
> Sure. In fact, the axiom, redundant in presence of infinity,
> There exists an empty set.
> is often included among the axioms of ZFC when presented in ordinary
> mathematical English. If this is not done we're led to such rather odd
> conclusions as that the axiom of infinity is needed to prove {{},{{}}}
> exists and so on.
The usual statement of the axiom of infinity is Ex(0 in x and ...), thus
presuming that the empty set exists before the AoI can even be stated.
Therefore, I don't see how we can claim that the existence of the empty
set "follows from" the AoI in its usual form.
I think the AoI can be formulated as "there exists an inductive set",
meaning "there exists a nonempty set that is closed under the successor
operation". This establishes that at least one set exists without
mentioning the empty set.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
MoeBlee
I'll start from the middle:
> Here's the point: The distinction between the theories
> T and T' below is of interest only to logicians;
> it's a purely _formal_ distinction. In actual mathematician
> doing actual mathematics doesn't care about the difference
> between "everything is a set" and "sets are the only
> things we're talking about today". A proof that's valid
> in one context but not the other is "implementation
> dependent", like what we were talking about the
> other day, a proof of something about the reals
> that's valid if the reals were defined as Dedekind
> cuts but not if the reals were defined as equivalence
> classes of Cauchy sequences of rationals.
I quite understand that. And I never disputed it. But I don't have to
make my remarks concern only mathematicians who are not interested in
certain logical matters. Nor am I going to in all cases make my
remarks so that they can be generalized without reliance on some
"implementation dependency" or another. My remarks were correct: In ZF
(even just in Z) we don't need the axiom of infinity to prove there is
a set (given a certain definition of 'is a set'). I made no claim that
that is not dependent on the specifics of three things: first order
logic, the specific axioms, and a certain definition.
Generally, if you wish to continue pointing out that certain of my
remarks in various threads are implementation dependent, then okay, I
can already say in advance "Yes, I've understood that, thank you,
noted".
However, about the axiom of infinity, while using it to prove that
there exists a set is, in the particular sense you state, more
general, it also has the DISadvantage of making the existence of a set
depend on the OVERKILL assumption that there exists a successor
inductive (hence, infinite) set. One may wish not to stake one's view
that there exists at least one set on there being an INFINITE set. So
my remarks are pertinent in at least that way. Moreover, my remarks
are instructive as to the point of logic, especially where the poster
was asking about proof in the particular system ZF.
Back to an example of the more general matter you've raised, if I talk
about ordered pairs, and I use the Kuratowski definition (either by
explicitly mentioning it or by taking it for granted as it is
virtually unanimously used in modern set theory textbooks) to prove
some particular point, such that {1} is a function (since {1} = {<0
0>}), then you might point out that my remarks are implementation
dependent. And I'll say, "Yes, noted, thank you", and we should then
be able to move on, right?
Now, some of the rest of the squabble:
> Below I can't tell whether or not you're
> using the "something exists" convention from logic,
> and also whether you're considering a set theory
> in which everything is a set.
I'm using classical first order logic with identity. And I'm using the
set theoretic axioms of ordinary Z set theory. And I've said that my
remarks pertain where we have some suitable definition of 'is a
set' (such as "x is a set <-> Ey xey"). David, nothing less, nothing
more. So, of course the logic provides that something exists (both
semantically and in such theorems as ExP). I've said this already.
> >The version I use, which is common in many a textbook on set theory,
> >doesn't use a relativization to 'set'. So the axiom is: For all x any
> >y, there exists an object with exactly x and y as members:
>
> >AxyEzAv(vez <-> (v=x or v=y))
>
> There are books with this property which are _not_ talking
> about set theories in which everything is a set?
Suppes allows that there may exist urelements, and, as I recall gives
the above formulation of pairing I just gave. But that's a different
situation anyway, since he takes an additional primitive (it's either
'0' or 'is a set', I don't recall at the moment).
But this is aside the point. I've made no claim about theories in
which there are, or may be, proper classes or urelements. I stated
that I'm talking about ordinary Z set theories.
> That version of the axiom is _not_, for example, part
> of (what I assume to be the standard version of)
> Godel=Bernays set theory.
And I SAID I'm talking about Z set theories, not about NBG. Just as
the POSTER said specifically he was asking about ZF.
> For heaven's sake. You're talking about a set theory where everything
> is a set. I don't see how it matters whether that's an axiom or a
> theorem - the argument does not apply to, for example, Godel-Bernays
> set theory.
No, YOU "for heaven's sake". I SAID I was talking about Z set
theories, just as the POSTER had said his context is ZF. And I agree
that for this purpose it doesn't matter whether "everything is a set
is axiomatic or a theorem". That is MY point.
I'm sorry, but you seem to have got some idea in your head that I've
been claiming something about all kinds of various theories, even
though I have not been, and even though I have made clear that I have
not been, as I've said already plenty already that I'm talking about
ordinary Z set theories, and just as I responded in that CONTEXT in
which the poster said he's talking about ZF.
> >David, in all constructive spirit, please consider exactly what I've
> >said. I've AGREED that the logic provides for the existence of at
> >least one object.
>
> I don't understand the point to writing AGREED that way.
> My whole point is that it would be better _not_ to use this
> fact.
I said I agree that the logic provides for the existence of at least
one object. As to what is "better" to do, I've not opined. The logic
has a rule of universal instantiation, and I used it. And I used it in
regard to a correct claim I made, which is that we don't require the
axiom of infinity to prove there is an object or a set.
Sure, if the point is convince people who don't much about logic that
there exists an object, then it may be better to convince them of that
by pointing to an axiom that explicitly states existence. I've not
said otherwise. Rather, my point is the plain, correct, raw claim that
we can derive the existence of a set without the axiom of infinity.
One can even see in certain set theory books that the author mentions
both that existence comes from logic, but let's also make our own self-
standing assumption that there exists an object (or set, whatever the
case); and it seems to me that the author is doing that for the very
purpose of not baffling people who don't know the specifics of first
order semantics and the way the rule of universal instantiation
supports that.
But again, my remarks were not intended to be pedagogically exhaustive
in that way. I merely made a correct claim about what is provable from
what and that is the only intent and only "message" I wished to be
conveyed by my remarks.
> Because, for example, every statement _about sets_ that's
> true in ZFC is also true in GB. But the fact that logic
> provides for the existence of at least one object is
> _not_ enough to show that a set exists in GB (or if
> it is nobody's explained how).
And I haven't been talking about NBG. The poster was talking about ZF,
and I responded in that context. And I didn't claim that using logic
to provide the existence of an object is the best explanation in a
context in which we may wish to include NBG, etc. I merely made a
plain, raw, correct claim about what can be proven from what in Z set
theories with a given definition of 'is a set'.
> But you're using that axiom [separation]. And it's _not_ an axiom in staandard
> set theories in which not everything is a set.
I didn't say it is! It's an axiom schema of the system I said my
remarks pertain to! (Or, actually, depending on the particular
axiomatization of ZF, it's a theorem schema).
> >I'm just using universal instantiation. That the rule of universal
> >instantiation goes along semantically with the semantic stipulation
> >that the domain of discourse is nonempty is not contested by me,
>
> I didn't say it was concocted by you.
I said "contested", not "concocted".
> >> So your formal proof _does_ use "there exists something, by
> >> logic",
>
> >I've not disputed that.
>
> Fine. Then I don't see how any of this is relevant to what I
> thought you were replying to.
It seems you've somehow read WAY PAST whatever I wrote to think that
I've been saying things I have not been saying. That's where the lack
of relevance stems from.
> >All I've said is that I don't make use of any
> >verbatim axiom (or even theorem) of the form "ExPx".
>
> >Again, I've said all along, the logic itself provides theorems of the
> >form "ExPx". And I've agreed all along that the logic works hand in
> >glove with a semantics that stipulates a nonempty domain of discourse.
>
> >> and it _does_ use "everything is a set".
>
> >No, I did not assume everything is a set.
>
> Aargh. I didn't say you "assumed" that. You're
> using the fact that you _are_ in a set theory where
> everything is a set.
"assumed" or the word you used, which is "using", whatever. I didn't
claim not be using the fact that I'm in set theory in which everything
is a set!!!
What I said is that I've used classical first order logic, ordinary
axioms of Z set theory, and a certain definition of 'is a set'.
> It shows that we _do_ need the axiom of infinity _if_ we're
> going to avoid using the "logic says that something
> exists".
Fine! I never said otherwise.
> Or: It shows that we _do_ need the axiom of infinity
> to prove that there exists a set in that theory where
> all the axioms of ZFC have had "Ax" replaced by
> "Ax Set(x) ->". Those theories are equivalent _wrt_
> what they say about sets, but any structure with no
> sets is a model of the second theory minus infinity.
And I didn't say otherwise.
MoeBlee
MoeBlee
> You seem to be missing my point. Given that the two theories
> "say" the same things "about" sets, doesn't it seem like a proof
> of the existence of the empty set that works in both theories
> would be preferable to a proof that works in one but not
> the other?
No, I'm not missing your point, because I didn't claim anything about
PREFERABILITY and I very well understand why certain people might
prefer one approach to the other. My remarks were not about what is
preferable but merely as to the plain fact that the axiom of infinity
is indeed not REQUIRED to do this.
Anyway, in my previous post, I mentioned a certain sense in which
using the axiom of infinity is not preferable. As to whehter that
makes one more preferable in the balance, I've not opined.
MoeBlee
MoeBlee
> On Tue, 07 Jul 2009 18:14:19 +0300, Aatu Koskensilta
>
> >David C. Ullrich <dullr...@sprynet.com> writes:
>
> >> It shows that we _do_ need the axiom of infinity _if_ we're going to
> >> avoid using the "logic says that something exists".
>
> >Sure. In fact, the axiom, redundant in presence of infinity,
>
> > There exists an empty set.
>
> >is often included among the axioms of ZFC when presented in ordinary
> >mathematical English. If this is not done we're led to such rather odd
> >conclusions as that the axiom of infinity is needed to prove {{},{{}}}
> >exists and so on.
>
> >This whole discussion is a perfect illustration of something I like to
> >babble about interminably, that we need to exercise our best judgment
> >and use our good sense to figure out whether this or that formal
> >technicality or detail is related in any informative or significant way
> >to anything in our actual mathematical reasoning, or whether it's just
> >that, a technicality of no intrinsic interest.
>
> Precisely. Although, curiously, I'm not entirely certain which
> side you're wagging your finger at here. My interpretation
> of the relevance of that paragraph is that you're agreeing
> with me that getting the existence of the empty set from
> the convention that something is a technicality of no
> intrinsic interest, but it seems _possible_ that you're
> instead referring to my insistence that the "right" way
> to do it is to use the axiom of infinity, because that
> shows it "really does follow from the axioms"...
But I am on no "side" that claims that using separation and universal
instantiation is "the right way" or a "better way" or "preferable" to
using infinity. Prior to this morning, I merely pointed out that the
axiom of infinity is not REQUIRED. Sheesh! And then this morning I've
also pointed out a drawback to using the axiom of infinity for this,
though, still, I've not made any claim about what is "the right way"
or a "better way" or "preferable".
MoeBlee
MoeBlee
> The usual statement of the axiom of infinity is Ex(0 in x and ...), thus
> presuming that the empty set exists before the AoI can even be stated.
VERY technically (some people may charge that it is purely pedantic),
and using the Fregean method, '0' is just a 0-place function symbol.
So even IF we had not proven the existence/uniqueness theorem E!xAy
~yex, the axiom of infinity would still be STATABLE. The axiom of
infinity would still state that there exists some set that has 0
(WHATEVER 0 might be) in it and that is closed under successor.
MoeBlee
MoeBlee
>{1} is a function (since {1} = {<0 0>})
Oops, of course I meant (with Von Neumann nats and Kuratowski ordered
pairs) that {1} is an ordered pair and {{1}} is a function.
MoeBlee
Stephen J. Herschkorn
> The usual statement of the axiom of infinity is Ex(0 in x and ...),
> thus presuming that the empty set exists before the AoI can even be
> stated. Therefore, I don't see how we can claim that the existence of
> the emptyset "follows from" the AoI in its usual form.
>
>I think the AoI can be formulated as "there exists an inductive set",
>meaning "there exists a nonempty set that is closed under the successor
>operation". This establishes that at least one set exists without
>mentioning the empty set.
>
"0 in x" is just short for "Ey in x [Az ~(z in y)]." It follows by
basic logic that Ey [Az ~(z in y)], which is precisely the statement
of the existence of an empty set. (Extensionality implies that the
empty set is unique.)
Transfer Principle
> Han de Bruijn <umum...@gmail.com> writes:
> > That's for the OP to decide, I think.
> so surely you should suggest that paper only if you see that it is
> relevant to his question.
I disagree with Hughes. Not only is HdB's paper relevant to
the discussion at hand, but it is _more_ relevant than any
other link given in this thread!
Why? The central discussion in this thread is whether the
Axiom of Infinity is required to prove that the empty set
exists in ZF. If Hughes actually clicked on the link, then
he'd know that the paper concerns a finitist set theory
which is essentially ZF-Infinity. And who would be more
likely to be concerned with whether Infinity is required
to prove the existence of 0 than a _finitist_?
Think about it. The OP states that he's an adherent of ZF,
a theory with an Axiom of Infinity. And so it doesn't even
_matter_ whether Infinity is needed to prove that 0 exists
at all -- we know that ZF proves the existence of 0. Of
this there's no debate. ZF proves that 0 exists regardless
of whether Infinity is required in the proof or not.
In ZFC we can prove the existence of a wellorder of R, a
nonmeasurable set, and a nonprincipal ultrafilter. So why
does it matter to set theorists whether AC is needed in
any of those proofs? It's because some set theorists --
the constructivists -- don't accept AC. Therefore, set
theorists want to know which proofs will hold up without
AC and which proofs require AC. To whom does it matter the
most which proofs require AC? The constructivists. Indeed,
the OP states that he's working in ZF, not ZFC, so it'd be
unreasonable to expect the OP to accept or prove the
existence of a wellorder of R, a nonmeasureable set, or a
nonprincipal ultrafilter. His chosen theory doesn't prove
the existence of such sets.
So, just as whether AC is required to in an existence
proof matters most to constructivists, whether Infinity is
required in an existence proof matters to finitists. For
surely, even a finitist like HdB wants the _empty set_ to
exist, and so he must worry about whether Infinity is
needed to prove that 0 exists. Otherwise, it could be that
ZF-Infinity doesn't prove the existence of 0, and HdB
would be hard-pressed to convince anyone to use a theory
in which one can't prove the existence of any object, not
even the empty set.
Can HdB prove that 0 exists in his theory. Click on the
link, and we see that he can indeed? How? He explicitly
includes an Empty Set Axiom. Therefore, 0 exists.
How is this relevant to the discussion at hand? HdB's
inclusion of an Empty Set Axiom seems to suggest that
without it, HdB believes that he cannot prove that 0
exists in the theory. So a theory without Infinity might
not be able to prove the existence of 0. This is what
Ullrich believes as well. Infinity is needed to prove
that 0 exists.
> Really, your suggestion that he read your paper simply wastes his
> time.
Hughes, like MoeBlee, often calls discussions about
theories other than ZF(C) "wastes of time," especially
if they are (ultra)finitist theories. Of course,
someone's likely to call me a "troll" if I state that
MoeBlee and Hughes call these discussions "wastes of
time" _because_ they are about theories other than
ZF(C), so let me simply let their repeated use of the
phrase "waste of time" speak for itself.
Peter Webb
not be able to prove the existence of 0. This is what
Ullrich believes as well. Infinity is needed to prove
that 0 exists.
****************************
No. There are many different but equivalent formulations of the Axioms of ZF
and ZFC. Quite commonly, the Axiom of Infinity is formulated in such a way
as to directly prove the existence of the empty set, so its not needed as an
Axiom. Different Axiom sets may be able to prove the existence of the empty
set in different ways. If they don't, then its easy to just add it in as an
axiom.
You don't "need" infinity to prove the existence of the empty set. It just
happens that under the most common formulation of ZF and ZFC, its easy to
prove using the Axiom of Infinity. If you want to take out the Axiom of
Infinity, either just make sure that the existence of the empty set can be
proved from the remaining axioms that you choose to keep, or modify one of
those axioms, or add it in the existence of the empty set as an Axiom.
There is no deep connection between the existence of the empty set and the
existence of an infinite set; it is just an artifact of how the axioms are
sometimes worded that the axiom of infinity allows a direct proof that the
empty set exists - you certainly don't need to allow infinite sets in order
to have empty sets.
Jesse F. Hughes
> On Jul 2, 7:13 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Han de Bruijn <umum...@gmail.com> writes:
>> > That's for the OP to decide, I think.
>> You understand his question and you know the contents of that paper,
>> so surely you should suggest that paper only if you see that it is
>> relevant to his question.
>
> I disagree with Hughes. Not only is HdB's paper relevant to
> the discussion at hand, but it is _more_ relevant than any
> other link given in this thread!
>
> Why? The central discussion in this thread is whether the
> Axiom of Infinity is required to prove that the empty set
> exists in ZF. If Hughes actually clicked on the link, then
> he'd know that the paper concerns a finitist set theory
> which is essentially ZF-Infinity. And who would be more
> likely to be concerned with whether Infinity is required
> to prove the existence of 0 than a _finitist_?
In fact, the paper is nothing more than a sketch of a model of
ZFC-Infinity. Such a sketch of a single, particular model does not
settle the question of whether ZFC - Infinity |- (Ex)(Ay)(~ y in x).
Han's paper is utterly irrelevant to the OP's question.
--
"Sale or rental of this disc is ILLEGAL. If you have rented or
purchased this disc, please call the MPAA at 1-800-NO-COPYS."
-- The MPAA begins a new anti-piracy program,
found on a DVD purchased in China
Jesse F. Hughes
>> Really, your suggestion that he read your paper simply wastes his
>> time.
>
> Hughes, like MoeBlee, often calls discussions about
> theories other than ZF(C) "wastes of time," especially
> if they are (ultra)finitist theories.
Why, you are either stupid or a liar.
--
"[I]n mathematics there are two types of integers: primes and
composites. [...] It's like how in the world there are mostly two
kinds of people: male and female [...] and lots of reasons for
interest in the differences." -- JSH on math/biology
Transfer Principle
> On Jul 7, 8:55 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> > Precisely. Although, curiously, I'm not entirely certain which
> > side you're wagging your finger at here. My interpretation
> > of the relevance of that paragraph is that you're agreeing
> > with me that getting the existence of the empty set from
> > the convention that something is a technicality of no
> > intrinsic interest, but it seems _possible_ that you're
> > instead referring to my insistence that the "right" way
> > to do it is to use the axiom of infinity, because that
> > shows it "really does follow from the axioms"...
> But I am on no "side" that claims that using separation and universal
> instantiation is "the right way" or a "better way" or "preferable" to
> using infinity.
Speaking of "side," I never thought the day would come when
I'd actually be on David Ullrich's side of a debate. And
yet this is precisely where I find myself.
This debate concerns whether the Axiom of Infinity is
required to prove that the empty set exists. There appears
to be three schools of thought here:
1. An explicit Empty Set Axiom is required to prove that
the set 0 exists in ZFC.
2. The existence of 0 is provable from the axioms of
Infinity and Separation Schema.
3. The existence of 0 is provable from the axioms of
FOL= and Separation Schema.
Ullrich obviously adheres to 2. Another set theorist who
adheres to 2 is Randall Holmes, who also uses an Axiom of
Infinity to prove that the empty set exists in his theory
PST, Pocket Set Theory. Notice that PST proves the
existence of proper classes, and the cornerstone of
Ullrich's argument is that FOL= and Separation Schema are
not sufficient to prove in a _proper class theory_ that 0
exists and is a _set_.
MoeBlee obviously adheres to 3. Suppes, the textbook
which MoeBlee often cites, does something completely
different -- Suppes lets 0 be a _primitive_ and defines a
"set" to be either 0 or an object with elements. Then
Suppes uses Separation Schema to prove that 0 is actually
an _empty_ set. Although this isn't how MoeBlee presents
his argument, Suppes does support his side since Infinity
is used nowhere in the proof.
Notice that Ullrich mentions a theory T' which consists
of the axioms of ZFC relatived to a new primitive called
"set," and a theory T'' which adds to the theory T' an
axiom guaranteeing that every object is a set. I actually
mentioned these exact theories to MoeBlee in a previous
thread, where I was talking about the so-called "crank"
Srinivasan and the ex-"crank" zuhair where discussing
particular models of what turned out to be Ullrich's
theory T'. And now the same theory appears in this thread
as well.
And so we see that once we leave MoeBlee's comfort zone
of ZF(C), we enter a world in which an axiom such as the
Axiom of Infinity is required to prove that 0 exists and
is a set. The theories in which Infinity is required
include class theories such as NBG and PST, Ullrich's
theory T', as well as the theories created or suggested
by Srinivasan and zuhair. And this is why finitists such
as HdB include an explicit Empty Set Axiom, for without
it they can't prove that 0 exists.
And so we see that the bulk of the evidence is on
Ullrich's side of the debate. Only Suppes supports
MoeBlee's argument, and even Suppes does it differently
by letting 0 be a primitive and defining "set" such that
0 is by definition a set.
> Prior to this morning, I merely pointed out that the
> axiom of infinity is not REQUIRED. Sheesh! And then this morning I've
> also pointed out a drawback to using the axiom of infinity for this,
> though, still, I've not made any claim about what is "the right way"
> or a "better way" or "preferable".
Bull! MoeBlee's made this exact same denial when I told
MoeBlee how he finds it "preferable" for theories other
than ZFC to provide for an axiomatization for an
application to the sciences. Sure, MoeBlee might avoid
using the exact words "preferable" or "better way," but
as often as he mentions calculus for the sciences, it's
obvious to me that MoeBlee really does "prefer" that
proposed theories are so applicable, just as it was
obvious to Ullrich that MoeBlee's "preference" is for
the existence of 0 to be derivable without Infinity. So
applicability to the sciences and the derivability of
the empty set's existence without Infinity represent
MoeBlee's desiderata in the same way that the repeated
preferences of the "cranks" represent their desiderata.
To conclude this post, let's look at MoeBlee's alleged
proof that 0 exists without Infinity:
> AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
> schema of separation
> EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
and Ullrich's response:
"You can't erase the initial "Ax" in the first line except
in a context where we're assuming that something exists."
So according to Ullrich, we can't use the universal
instantiation rule unless we know that something exists.
But doesn't MoeBlee's use of UI here look familiar. Let's
go back to the Nam Nguyen debate. MoeBlee writes:
> 1 AxAy x=y ... axiom
> 2 Ay x=0 ... universal instantiation
But the only axiom of Marshall's theory is "AxAy x=y",
which begins with a universal quantifier. So we don't know
that something exists in Marshall's theory -- which means
that MoeBlee's use of UI in line 2 is invalid!
Therefore, MoeBlee's alleged proof of "Ax x+y=0" in the
theory with language {"+", "0"}, and axiom "AxAy x=y" is
actually invalid! And so yet another of MoeBlee's proofs
falls apart!
David Bernier
> Dave Seaman wrote:
>
>> The usual statement of the axiom of infinity is Ex(0 in x and ...),
>> thus presuming that the empty set exists before the AoI can even be
>> stated. Therefore, I don't see how we can claim that the existence of
>> the emptyset "follows from" the AoI in its usual form.
>>
>> I think the AoI can be formulated as "there exists an inductive set",
>> meaning "there exists a nonempty set that is closed under the successor
>> operation". This establishes that at least one set exists without
>> mentioning the empty set.
>>
>
> "0 in x" is just short for "Ey in x [Az ~(z in y)]." It follows by
> basic logic that Ey [Az ~(z in y)], which is precisely the statement
> of the existence of an empty set. (Extensionality implies that the
> empty set is unique.)
>
I'd be pleased for any enlightenment about what appears below.
Without AoI, I'm not sure that the axiom of pairing, or say the axiom of
separation, will prove that a set exists.
So say without AoI, how might it be possible to show:
(1) Ex: Ay: y e x <==> ( y =/= y) , expanded:
(2) Ex: Ay: y e x <===> ( Az: (z e y) <===> not(z e y) ).
If the universe of discourse U (or V ... ) has nothing,
IOW if U is a "void", I'd say by the common rules of
logic (1) is false, because in such a case it is
not possible to find a y with: ( y =/= y) .
So I'd say (1) is False if the Universe of discourse
is vacuous; obviously, we want a non-void universe U.
So I'd like to get opinions/ideas about the question:
Is (1) provable in ZF - AoI ?
At the moment, what I think is that the Axiom of Separation
looks promising for proving (1), but how to apply Separation
if we don't already "know" that a set exists?
So another way of looking is to try to show:
(1b) Ex: x=x .
in ZF - AoI.
Maybe it can be done, but I'm not convinced.
David Bernier
David C. Ullrich
<dse...@no.such.host> wrote:
>On Tue, 07 Jul 2009 18:14:19 +0300, Aatu Koskensilta wrote:
>> David C. Ullrich <dull...@sprynet.com> writes:
>
>>> It shows that we _do_ need the axiom of infinity _if_ we're going to
>>> avoid using the "logic says that something exists".
>
>> Sure. In fact, the axiom, redundant in presence of infinity,
>
>> There exists an empty set.
>
>> is often included among the axioms of ZFC when presented in ordinary
>> mathematical English. If this is not done we're led to such rather odd
>> conclusions as that the axiom of infinity is needed to prove {{},{{}}}
>> exists and so on.
>
>The usual statement of the axiom of infinity is Ex(0 in x and ...), thus
>presuming that the empty set exists before the AoI can even be stated.
>Therefore, I don't see how we can claim that the existence of the empty
>set "follows from" the AoI in its usual form.
_If_ the axiom is formulated exactly that way then one needs to
show the empty set exists first, yes. (That doesn't contraidict
the fact that the existence of the empty set follows from the axiom,
but never mind that.)
But there are many different ways of formulating all these things.
In _every_ formulation (every one I've seen, and every one I
can imagine) the axiom of infinity begins "Ex...". Hence the
axiom of infinity does imply that there exists a set. None of
the other axioms have the property that in every formulation
they imply the existence of at least one set.
>I think the AoI can be formulated as "there exists an inductive set",
>meaning "there exists a nonempty set that is closed under the successor
>operation". This establishes that at least one set exists without
>mentioning the empty set.
David C. Ullrich
David C. Ullrich
Several times today you say you haven't said anything about
what's preferable to what. So _say_ something about that!
In particular, answer the following question - I asked
yesterday but I don't see a reply (perhaps you thought
it was just a rhetorical question):
We have two theories, T and T'. The theories both
"say" exactly the same things "about" sets. (See below
for details on that). We have two proofs of the
existence of the empty set. One works in T but not
in T'. The other works in both. Which proof is
preferable?
(Details: Of course if, say, T had the existence of the
empty set as an axiom while T' did not it would be
a silly question. But that's not the case here; in
a sense I have a hard time defining both precisely
and abstractly, the axioms for each theory also
"say" the same things about sets as the axioms
for the other theory. In particular, T' consists
just of T with every "Ax" replaced by
"Ax Set(x) ->"; the same applies to the
aciomatizations.)
>MoeBlee
David C. Ullrich
There are other possible reasons. According to
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
, in Kunen, "Set Theory: An Introduction to Independence Proofs"
the author explicitly includes the existence of a set as an axiom.
You really think that in _that_ book he's assuming the reader
doesn't know much about logic?
David C. Ullrich
Stephen J. Herschkorn
>http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>
>, in Kunen, "Set Theory: An Introduction to Independence Proofs"
>the author explicitly includes the existence of a set as an axiom.
>You really think that in _that_ book he's assuming the reader
>doesn't know much about logic?
>
>
Careful about citing secondary sources. The statement of the axiom
(which Kunen labels Axiom 0) is followed by:
This axiom says that our universe is non-void. Under most
developments of formal logic, this is derviable from the logical
axioms alone and thus redundant to state here, but we do so for
emphasis.
I wonder why he doesn't mention that Infinity renders it redundant as well.
MoeBlee
> Several times today you say you haven't said anything about
> what's preferable to what. So _say_ something about that!
I'm really not sufficiently motivated at this time. I don't mean that
as flip, but rather that sincerely, of all the things to talk about,
that subject is not one that I'm very motivated to post about at this
time. And not that I'm saying that it is not a worthwhile subject or
that I am not interested in it (actually, I am).
> In particular, answer the following question - I asked
> yesterday but I don't see a reply (perhaps you thought
> it was just a rhetorical question):
>
> We have two theories, T and T'. The theories both
> "say" exactly the same things "about" sets. (See below
> for details on that). We have two proofs of the
> existence of the empty set. One works in T but not
> in T'. The other works in both. Which proof is
> preferable?
That is covered by my remarks overall. In a certain sense, of course,
the proof that generalizes more is preferable in that sense But then
there may be other considerations too. I mentioned all that
previously.
And just, hopefully finally, to be clear: When I mentioned that Z set
theory proves the existence of the empty set without having to use the
axiom of infinity, my point was not to say what proof is preferable in
any given context. Rather, I'm just reporting a plain fact about what
is derivable from what, especially since the very question of
derivablity of the empty set from the axiom schema of separation had
been asked about.
MoeBlee
MoeBlee
> , in Kunen, "Set Theory: An Introduction to Independence Proofs"
> the author explicitly includes the existence of a set as an axiom.
> You really think that in _that_ book he's assuming the reader
> doesn't know much about logic?
No, I didn't claim (or at least I did not mean to convey) that only
reason one would include the empty set axiom is to communicate better
with people who don't know that the rule of universal instantiation
may be used to toward an existence proof ("ExPx" for certain formulas
P).
Different authors have different axiomatizations and some of them
provide explanations for their approach on this very matter. I am
happy just to read the various approaches.
MoeBlee
MoeBlee
wrote:
> "0 in x" is just short for "Ey in x [Az ~(z in y)]."
It is equivalent to what you wrote when we are equipped with the
ordinary definition of '0'.
> It follows by
> basic logic that Ey [Az ~(z in y)],
I think some people may view it that way. However, where '0' is
defined, not primitive, on a certain very rigorous approach to defined
symbols (such as '0'), the details shake out differently from your
approach. I gave the details in a post in our discussion from a few
years ago (the one in which we both discovered that the axiom schema
of replacement has different formulations that are not all as strong
as one another).
> which is precisely the statement
> of the existence of an empty set. (Extensionality implies that the
> empty set is unique.)
But WHATEVER the properties of 0, it is a unique object just by the
fact that '0' is a term of the language ("E!x c=x" is a theorem of
logic alone, no matter what 'c' is, such as 'c' could be '0'). The
axiom of extensionality is not to prove that 0 is unique but rather
that there is a unique x such that Ay~yex.
Note to David Ullrich: (I'm not being sarcastic.) Please understand
that my remarks here are not meant to pertain to the most general
context possible but rather only to the specific contexts that would
apply and only within the particular techinical specifications I've
alluded to.
MoeBlee
MoeBlee
> On Jul 2, 7:13 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > Han de Bruijn <umum...@gmail.com> writes:
> > > That's for the OP to decide, I think.
> > You understand his question and you know the contents of that paper,
> > so surely you should suggest that paper only if you see that it is
> > relevant to his question.
>
> I disagree with Hughes. Not only is HdB's paper relevant to
> the discussion at hand, but it is _more_ relevant than any
> other link given in this thread!
>
> Why? The central discussion in this thread is whether the
> Axiom of Infinity is required to prove that the empty set
> exists in ZF.
That came up soon, but the actual question put by the poster was more
general.
> If Hughes actually clicked on the link, then
> he'd know that the paper concerns a finitist set theory
> which is essentially ZF-Infinity. And who would be more
> likely to be concerned with whether Infinity is required
> to prove the existence of 0 than a _finitist_?
Whatever HdB is "concerned about", his posturings on such matters are
based in his ignorance of the subject.
> Think about it. The OP states that he's an adherent of ZF,
He says he "works in ZF". I guess that's what you mean by an
"adherent".
So that makes me an "adherent" of ZFC, Z-regularity+countable choice,
NBG, IST, PA, PRA, Robinson arithmetic, first order group theory, the
theory of partial orderings, the pure predicate calculus, identity
thoery, second order logic, intuitionistic first order logic, Boolean
algebra, etc., etc.
> a theory with an Axiom of Infinity. And so it doesn't even
> _matter_ whether Infinity is needed to prove that 0 exists
> at all -- we know that ZF proves the existence of 0.
If '0' is not taken as a primitive, then ZF proves the existence of a
unique set that has no members, then we define '0' as that set.
> In ZFC we can prove the existence of a wellorder of R, a
> nonmeasurable set, and a nonprincipal ultrafilter. So why
> does it matter to set theorists whether AC is needed in
> any of those proofs? It's because some set theorists --
> the constructivists -- don't accept AC.
That is one reason. It doesn't preclude that there may be other
reasons.
> Therefore, set
> theorists want to know which proofs will hold up without
> AC and which proofs require AC. To whom does it matter the
> most which proofs require AC? The constructivists.
Provide actual evidence that constructivists care more about what is a
theorem in ZFC and not in ZF than do non-constructivists.
> Indeed,
> the OP states that he's working in ZF, not ZFC, so it'd be
> unreasonable to expect the OP to accept or prove the
> existence of a wellorder of R, a nonmeasureable set, or a
> nonprincipal ultrafilter. His chosen theory doesn't prove
> the existence of such sets.
How do you know that he doesn't ALSO work in ZFC?
> So, just as whether AC is required to in an existence
> proof matters most to constructivists, whether Infinity is
> required in an existence proof matters to finitists. For
> surely, even a finitist like HdB wants the _empty set_ to
> exist, and so he must worry about whether Infinity is
> needed to prove that 0 exists.
Interesting that you think HdB has even thought about it that far.
> Otherwise, it could be that
> ZF-Infinity doesn't prove the existence of 0, and HdB
> would be hard-pressed to convince anyone to use a theory
> in which one can't prove the existence of any object, not
> even the empty set.
Interesting that you think HdB even understands how to prove things in
ZF-axiom_infinity.
By the way, HdB is UTTERLY confused even about his own theory, as he
is UTTERLY confused about the difference between a theory and model of
that theory.
> Can HdB prove that 0 exists in his theory. Click on the
> link, and we see that he can indeed? How? He explicitly
> includes an Empty Set Axiom. Therefore, 0 exists.
Hahaha! Exactly. Irrelevant to the discussion. The poster surely knows
that adopting an empty set axiom proves the existence of an empty set!
> How is this relevant to the discussion at hand? HdB's
> inclusion of an Empty Set Axiom seems to suggest that
> without it, HdB believes that he cannot prove that 0
> exists in the theory.
It does? Maybe it suggests that HdB doesn't mind redundant axioms
(redundant axioms are found occasionally in mathematics; some authors
even point out that they are adopting redundant axioms). Maybe it
suggest HdB doesn't even THINK about such things.
> So a theory without Infinity might
> not be able to prove the existence of 0. This is what
> Ullrich believes as well.
Ullrich's point is NOT that the axiom of infinity is REQUIRED to prove
the existence of an empty set.
> Infinity is needed to prove
> that 0 exists.
It is? Who says?
> > Really, your suggestion that he read your paper simply wastes his
> > time.
>
> Hughes, like MoeBlee, often calls discussions about
> theories other than ZF(C) "wastes of time," especially
> if they are (ultra)finitist theories.
STOP PUTTING WORDS IN MY MOUTH! I've been asking you to stop putting
words in my mouth for at least about a year now.
I don't call a discussion a waste of time merely for being a
discussion about theories alternative to ZFC or even for being about
finitist or ultrafinitis theories. I am INTERESTED in such subjects.
Rather, what I've called a waste of time are certain particular
ramblings by certain posters (and I didn't even say that all postings
by such people are a waste of time).
> Of course,
> someone's likely to call me a "troll" if I state that
> MoeBlee and Hughes call these discussions "wastes of
> time" _because_ they are about theories other than
> ZF(C), so let me simply let their repeated use of the
> phrase "waste of time" speak for itself.
Whether troll or not, you are an INCORRIGIBLE *LIAR* about such
things.
MoeBlee
David C. Ullrich
<jazz...@hotmail.com> wrote:
>On Jul 8, 4:03�am, David C. Ullrich <dullr...@sprynet.com> wrote:
>
>> Several times today you say you haven't said anything about
>> what's preferable to what. So _say_ something about that!
>
>I'm really not sufficiently motivated at this time. I don't mean that
>as flip, but rather that sincerely, of all the things to talk about,
>that subject is not one that I'm very motivated to post about at this
>time. And not that I'm saying that it is not a worthwhile subject or
>that I am not interested in it (actually, I am).
>
>> In particular, answer the following question - I asked
>> yesterday but I don't see a reply (perhaps you thought
>> it was just a rhetorical question):
>>
>> We have two theories, T and T'. The theories both
>> "say" exactly the same things "about" sets. (See below
>> for details on that). We have two proofs of the
>> existence of the empty set. One works in T but not
>> in T'. The other works in both. Which proof is
>> preferable?
>
>That is covered by my remarks overall.
Really? You say uncountably many times that you have not said anything
about what's better than what, and that covers a question about
what's better than what?
>In a certain sense, of course,
>the proof that generalizes more is preferable in that sense
We're not talking about a proof that "gneralizes". T and T' are
_equivalent_ regarding sets - the "generalization" is from a proof
that the empty set exists to a proof that the empty set exists.
MoeBlee
<webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> So a theory without Infinity might
> not be able to prove the existence of 0. This is what
> Ullrich believes as well. Infinity is needed to prove
> that 0 exists.
Wait, be careful. Ullrich is saying (in my own paraphrase now) that
the axiom of infinity is the only one that - no matter how we choose
from among the certain various not too uncommon axiomatizations -
proves that there exists an object that has no members and that is a
SET.
(For that matter, in certain axiomatizations of NB, that 0 is a SET is
an AXIOM).
Ullrich does NOT dispute that given formal first order Z set theory,
with 'e' as the only non-logical primitive, the axiom schema of
separation proves ExAy ~yex, and then with extensionality E!xAy ~yex
and then that with 'set' defined as 'x is set <-> Ey xey' we have E!x
(x is a set & Ay ~yex) from the previous along with, say, pairing or
power set (also, infinity could be used at this juncture, but it is
not REQUIRED).
Indeed, you will find in certain widely referenced textbooks a proof
of E!xAy ~yex from the axiom schema of separation and axiom of
extensionality (of course, using the rule of universal instantiation
and other ruels from first ordedr logic).
> No. There are many different but equivalent formulations of the Axioms of ZF
> and ZFC. Quite commonly, the Axiom of Infinity is formulated in such a way
> as to directly prove the existence of the empty set,
Ex(0ex & Azex zu{z}ex)
proves that there exists a SET (with the definition of 'set' as 'x is
a set <-> Ey yex') since, by the axiom, 0 is in some object x.
However (and PLEASE note some disclaimers I make below) the axiom
itself does not entail that 0 is an empty set nor that there exists an
empty set. In Z set theory (as opposed to, say NB or NBG), the
existence of an empty object comes from either the axiom schema of
separation (or an empty set axiom if we don't mind a redundant axiom),
and that that object is a SET (given the definition of 'x is set' as
'Ey xey') comes from pairing, power set, or infinity.
That is, just because the axiom of infinity mentions something NAMED
'0' does not entail that the object referred to as '0' has the
property of not having any members or any property at all other than
logical properties had by all objects. To establish that there IS an
object that has no members requires proof.
Now, just to avoid an unnecessary squabble, I'll say that I grant that
some people may take using the 0-place function symbol '0' in itself
is an assertion that there exists a set having no members. So what I
am saying is applicable only if one is interested in a certain very
rigorous and literal application of certain procedures in mathematical
logic having to do with DEFINED symbols such as '0'; and, of course,
in context here where '0' is defined, not primitive.
MoeBlee
MoeBlee
> On Jul 7, 11:05 am, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jul 7, 8:55 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> > > Precisely. Although, curiously, I'm not entirely certain which
> > > side you're wagging your finger at here. My interpretation
> > > of the relevance of that paragraph is that you're agreeing
> > > with me that getting the existence of the empty set from
> > > the convention that something is a technicality of no
> > > intrinsic interest, but it seems _possible_ that you're
> > > instead referring to my insistence that the "right" way
> > > to do it is to use the axiom of infinity, because that
> > > shows it "really does follow from the axioms"...
> > But I am on no "side" that claims that using separation and universal
> > instantiation is "the right way" or a "better way" or "preferable" to
> > using infinity.
>
> Speaking of "side," I never thought the day would come when
> I'd actually be on David Ullrich's side of a debate. And
> yet this is precisely where I find myself.
>
> This debate concerns whether the Axiom of Infinity is
> required to prove that the empty set exists.
NO! That is NOT the "debate" I've had with Ullrich. You're not even
READING The posts!
> There appears
> to be three schools of thought here:
>
> 1. An explicit Empty Set Axiom is required to prove that
> the set 0 exists in ZFC.
> 2. The existence of 0 is provable from the axioms of
> Infinity and Separation Schema.
> 3. The existence of 0 is provable from the axioms of
> FOL= and Separation Schema.
These aren't even mutually exclusive.
> Ullrich obviously adheres to 2.
So do I!!!
(1) The "existence of 0" is from logic alone ANYWAY. Ex x=0, no matter
WHAT 0 is.
(2) The existence of an object that has no members is provable from
the separation schema.
(3) That there is a unique such object is then provable from
extensionality.
(4) That object is given the name '0'.
(5) That that object is a SET (upon the definition 'x is a set <-> Ey
xey') then follows from pairing or from power set or from infinity.
> Another set theorist who
> adheres to 2 is Randall Holmes, who also uses an Axiom of
> Infinity to prove that the empty set exists in his theory
> PST, Pocket Set Theory.
So what! The the question of PROVABILITY was not about pocket theory,
but about ZF.
Then Ullrich said explicitly that he doesn't dispute that in a given
system we may prove the existence of an empty set without the axiom of
infinity. Rather, his point is that the axiom of infinity is BETTER
(NOT the ONLY one) to use for the purpose of a derivation that is
GENERAL across various systems. And I didn't even dispute THAT point
with Ullrich. I even AGREE that the axiom of infinity provides greater
generality in that sense. The only thing left is that I declined to
state a view as to whether that generality makes one approach better
than other approaches.
> Notice that PST proves the
> existence of proper classes, and the cornerstone of
> Ullrich's argument is that FOL= and Separation Schema are
> not sufficient to prove in a _proper class theory_ that 0
> exists and is a _set_.
And I AGREE with Ullrich about that! Sheesh! THAT was not the
"debate".
> MoeBlee obviously adheres to 3.
There's hardly any "adhering" that needs to be done. It is a plain
finite fact that in a certain common formulation of Z set theory
(perforce in ZF, which was the STATED context) and with the mentioned
definition of 'is a set', one can derive the existence of a unique
empty set without having to use the axiom of infinity. Period. That's
not even a matter of dispute with Ullrich or with anyone who has
simply read the proof.
> Suppes, the textbook
> which MoeBlee often cites, does something completely
> different -- Suppes lets 0 be a _primitive_ and defines a
> "set" to be either 0 or an object with elements.
Yes, and I mentioned that when I very first recommended the book to
you.
> Then
> Suppes uses Separation Schema to prove that 0 is actually
> an _empty_ set. Although this isn't how MoeBlee presents
> his argument, Suppes does support his side since Infinity
> is used nowhere in the proof.
It doesn't even MATTER! There's no rational other "side" as to the
question that one may derive the existence of an empty set without
using the axiom of infinity.
> Notice that Ullrich mentions a theory T' which consists
> of the axioms of ZFC relatived to a new primitive called
> "set," and a theory T'' which adds to the theory T' an
> axiom guaranteeing that every object is a set. I actually
> mentioned these exact theories to MoeBlee in a previous
> thread, where I was talking about the so-called "crank"
> Srinivasan and the ex-"crank" zuhair where discussing
> particular models of what turned out to be Ullrich's
> theory T'. And now the same theory appears in this thread
> as well.
So what?! Such things are obvious anyway.
> And so we see that once we leave MoeBlee's comfort zone
> of ZF(C), we enter a world in which an axiom such as the
> Axiom of Infinity is required to prove that 0 exists and
> is a set.
You jerk, this is not outside my "comfort zone". If you had asked bout
this YEARS ago I could have explained it to YOU. All of this is not a
matter of debate. You didn't even bother to READ the exchanges you're
commenting on! (Or your reading comprehension and retention is
seriously impaired.)
> The theories in which Infinity is required
> include class theories such as NBG and PST,
I don't recall about NBG, but NB in one of its system forms does NOT
require the axiom of infinity to prove there is a unique empty set,
since the class version of separation plus extensionality prove that
there exists a unique class having no members, then "O is a set" comes
from EITHER the axiom of infinty OR from a redundant AXIOM '0 is a
set'.
> Ullrich's
> theory T', as well as the theories created or suggested
> by Srinivasan and zuhair. And this is why finitists such
> as HdB include an explicit Empty Set Axiom, for without
> it they can't prove that 0 exists.
No, IF HdB's theory really is Z-axiom-infinity or stronger, then the
existence of a unique empty set (given the mentioned definition of
'set') is provable without an empty set axiom.
You are COMPLETELY confused about all of this, while you're acting a
jerk telling me what is in my "comfort zone".
> And so we see that the bulk of the evidence is on
> Ullrich's side of the debate. Only Suppes supports
> MoeBlee's argument,
(1) As I've said, what you THINK was at difference between Ullrich and
me is not even what turns out to be at difference, indeed even if it
turns out there WAS a substantive difference, i.e., other than our
talking past one another in an unfortunate way.
(2) You are an UTTER clown when you sway "only Suppes". There are
OTHER textbooks in set theory that derive "ExAy ~yex" from separation.
It is a WELL KNOWN fact about set theory that the existence of an
empty set is derivable from separation (that it is a SET, upon the
mentioned definition, then would come from pairing, power or
infinity).
> and even Suppes does it differently
> by letting 0 be a primitive and defining "set" such that
> 0 is by definition a set.
So what?! This is not even the kind of thing that is at issue.
> > Prior to this morning, I merely pointed out that the
> > axiom of infinity is not REQUIRED. Sheesh! And then this morning I've
> > also pointed out a drawback to using the axiom of infinity for this,
> > though, still, I've not made any claim about what is "the right way"
> > or a "better way" or "preferable".
>
> Bull! MoeBlee's made this exact same denial when I told
> MoeBlee how he finds it "preferable" for theories other
> than ZFC to provide for an axiomatization for an
> application to the sciences.
YOU LIAR! Cut it out!
(1) I didn't say one approach is preferable to another here. I
recognized Ullrich's sense and I mentioned another sense that
disagrees, and I said I decline to state an overall preference.
(2) As to axiomatization of mathematics for the sciences, go back to
what I ACTUALLY said about such things, not to YOUR
MIScharacterizations.
>Sure, MoeBlee might avoid
> using the exact words "preferable" or "better way," but
> as often as he mentions calculus for the sciences, it's
> obvious to me that MoeBlee really does "prefer" that
> proposed theories are so applicable,
You LYING fool. Your claim is IDIOTIC. First order group theory does
not provide an axiomatization for the mathematics for the science. PRA
does not provide an axiomatization for mathematics for the sciences.
PA does not provide an axiomatization for mathematics for the
sciences. Etc. Etc. But I never claimed that I "prefer" that they do!
I never said that a proposed theory must axiomatize mathematics for
the sciences or that I prefer that it do. There are all kinds of
different theories with different roles to play. The matter of
axiomatization of mathematics for the sciences concerns a proposed
theory in CONTEXT OF that theory being proposed as an alternative to
ZFC. Read my EXACT remarks on the subject to see what I ACTUALLY said.
> just as it was
> obvious to Ullrich that MoeBlee's "preference" is for
> the existence of 0 to be derivable without Infinity.
How could it be obvious? Please point out EXACTLY where I said such a
thing. Please ENOUGH with your continual reading INTO my remarks what
is NOT there.
> So
> applicability to the sciences and the derivability of
> the empty set's existence without Infinity represent
> MoeBlee's desiderata in the same way that the repeated
> preferences of the "cranks" represent their desiderata.
I mentioned a certain sense in which resort to infinity for this is
not preferable. I made it EXPLICIT that I don't necessarily take that
as DETERMINATIVE for what I prefer.
> To conclude this post, let's look at MoeBlee's alleged
"alleged". Oh please, grow up.
> proof that 0 exists without Infinity:
>
> > AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
> > schema of separation
> > EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation
>
> and Ullrich's response:
>
> "You can't erase the initial "Ax" in the first line except
> in a context where we're assuming that something exists."
That is NOT an issue as to whether the proof is correct. Rather, it
concerns how one would characterize the underlying assumptions in the
method of such a proof.
And when the smoke clears on that, I don't find a disagreement between
Ullrich and me on that point. Yes, it is fully granted that the first
order logic comes with a standard semantics in which every domain of
discourse has at least one member, and the rule of universal
instantiation supports that and our quantifier rules must conform to
that semantics if we wish to have soundness of our logistic system
with that semantics.
> So according to Ullrich, we can't use the universal
> instantiation rule unless we know that something exists.
And according to MOEBLEE the rule of universal instantiation would not
be valid in a semantics in which there are empty domains of discourse.
And according to MOEBLEE the rule of universal instantiation would be
inconsistent if for every formula P we had ~Ex P.
> But doesn't MoeBlee's use of UI here look familiar. Let's
> go back to the Nam Nguyen debate. MoeBlee writes:
>
> > 1 AxAy x=y ... axiom
> > 2 Ay x=0 ... universal instantiation
>
> But the only axiom of Marshall's theory is "AxAy x=y",
> which begins with a universal quantifier. So we don't know
> that something exists in Marshall's theory -- which means
> that MoeBlee's use of UI in line 2 is invalid!
No, it's valid in plain, ordinary, first order logic, you ignoramus!
Read the damn formulation of the rule, then read the clauses in the
damn proof of the soundness theorem in which the rule is proven
VALID.
> Therefore, MoeBlee's alleged proof of "Ax x+y=0" in the
> theory with language {"+", "0"}, and axiom "AxAy x=y" is
> actually invalid! And so yet another of MoeBlee's proofs
> falls apart!
You are SUCH a fool.
ASK ANY LOGICIAN whether plain first order logic permits the rule of
universal instantiation to be applied as I did. And whether the
soundness theorem for first order logic includes a proof that the rule
of universal instantiation is valid. Better yet, just get a book on
the subject and read for YOURSELF. Or better yet, just start with
plain common sense:
If a property is true of ALL objects, then it is true of any
particular object. Sheesh! Even Aristotle could tell you that!
MoeBlee
MoeBlee
> Stephen J. Herschkorn wrote:
> > Dave Seaman wrote:
>
> >> The usual statement of the axiom of infinity is Ex(0 in x and ...),
> >> thus presuming that the empty set exists before the AoI can even be
> >> stated. Therefore, I don't see how we can claim that the existence of
> >> the emptyset "follows from" the AoI in its usual form.
>
> >> I think the AoI can be formulated as "there exists an inductive set",
> >> meaning "there exists a nonempty set that is closed under the successor
> >> operation". This establishes that at least one set exists without
> >> mentioning the empty set.
>
> > "0 in x" is just short for "Ey in x [Az ~(z in y)]." It follows by
> > basic logic that Ey [Az ~(z in y)], which is precisely the statement
> > of the existence of an empty set. (Extensionality implies that the
> > empty set is unique.)
>
> I'd be pleased for any enlightenment about what appears below.
>
> Without AoI, I'm not sure that the axiom of pairing, or say the axiom of
> separation, will prove that a set exists.
In a formal Z set theory in which the only non-logical symbol is 'e',
and with a definition of 'x is a set' as 'Ey xey', there is a proof
that there exists a unique empty set, using only the axiom schema of
separation, the axiom of extensionality, and the pairing axiom. This
is not a matter of debate. It's a plain finite fact about sequences of
formulas.
> So say without AoI, how might it be possible to show:
>
> (1) Ex: Ay: y e x <==> ( y =/= y) , expanded:
>
> (2) Ex: Ay: y e x <===> ( Az: (z e y) <===> not(z e y) ).
I already gave a variation of that. I'll give it for your specific
formulation:
First, I take it you don't need me to prove
~Az(zey <-> ~zey).
That is, we'll take that as already proven.
AzExAy(yex <-> (yez & ~yez)) ... instance of separation
ExAy(yex <-> (yez & ~yez)) ... universal instantiation
Let Ay(yex <-> (yez & ~yez)) ... letting 'x' be the x mentioned above
(since 'x' not previously free)
yex <-> (yez & ~yez) ... universal instantiation
~yex ... sentential logic
yex <-> Az(zey <-> ~zey) ... sentential logic and our already proven
theorem ~Az(zey <-> ~zey).
Ay(yex <-> Az(zey <-> ~zey)) ... universal instantiation (since y was
not free previous to instantiation to it)
ExAy(yex <-> Az(zey <-> ~zey) ... existential generalization (since
the conditions on the variables are satisfied)
Note: The use of 'z' in the formula P ("~zey") is not disallowed, only
'x' in P is disallowed. But if one is squeemish about 'z' used that
way, then just use any other contradiction or negation of a theorem
you like adjusted suitably, blah blah blah...
> If the universe of discourse U (or V ... ) has nothing,
Nope. The official logic for formal Z set theory is plain first order
logic. Any universe of discourse for a structure for the language is
non-empty.
Of course, if we used a logic that were built otherwise, we might not
have this guarantee of a nonvoid universe of discourse.
> So I'd like to get opinions/ideas about the question:
> Is (1) provable in ZF - AoI ?
Since, by definition, ZF is a theory in first order logic, the answer
is yes.
But, of course, if you prefer to ponder the axioms of ZF with some
other logic, then your results will be different.
> At the moment, what I think is that the Axiom of Separation
> looks promising for proving (1), but how to apply Separation
> if we don't already "know" that a set exists?
We don't need to know a set exists. Rather we work in a logic that
ensures that some object exists. Then upon a suitable definition of
'is a set' we also get that the unique empty object is a set. Or, many
(probably most) authors speak more informally in such contexts so that
they speak of everything as a set, and of 'classes' in general
(including proper classes) as a certain kind of figure of speech; or,
they present not Z set theory but rather NB or NBG or MK or even ZF
(with possible urelements) where it is not the case that everything is
a set.
MoeBlee
MoeBlee
> >> We have two theories, T and T'. The theories both
> >> "say" exactly the same things "about" sets. (See below
> >> for details on that). We have two proofs of the
> >> existence of the empty set. One works in T but not
> >> in T'. The other works in both. Which proof is
> >> preferable?
>
> >That is covered by my remarks overall.
>
> Really?
Yes, in one of my posts I mentioned whether or not we relativize
certain axioms to 'set'. So I didn't "cover" the subject with all the
specifics you've mentioned, but the subject is addressed in my own
remarks too. Now that the dialectic between us has progressed, I
really don't see a substantive difference between us.
> You say uncountably many times that you have not said anything
> about what's better than what, and that covers a question about
> what's better than what?
Some number of times now I've said that I haven't opined as to which
is the better approach, yes. I think certain approaches better in
certain senses and others in other senses; I'm not interested in
committing to an overall
In my remark about "covering" I meant that I've addressed the subject
of relativizing to 'set'. I didn't mean to say that that covers the
issue of what is what approach I might prefer overall. My lack of
exactness then for not making clear that when I mentiond 'cover' I was
speaking about the first part of your paragraph and not the specific
question at the end. (Though, I did address the question eventually,
even if, as should be reasonably allowed, it is to say that I am not
motivated to opine at this time about overall preferability.)
> >In a certain sense, of course,
> >the proof that generalizes more is preferable in that sense
>
> We're not talking about a proof that "gneralizes". T and T' are
> _equivalent_ regarding sets - the "generalization" is from a proof
> that the empty set exists to a proof that the empty set exists.
That last comment by you is what I meant by 'generalizes' in that
context. Perhaps a reasonably charitable reading of my remarks would
reveal that.
MoeBlee
David Bernier
This intrigues me: the possibility that things in FOL may be different
from what I thought they were. So if I have the time, I'll have a look
at books on logic, set theory, classes, types, NBG, FOL and so on.
David Bernier
>> So I'd like to get opinions/ideas about the question:
>> Is (1) provable in ZF - AoI ?
>
> Since, by definition, ZF is a theory in first order logic, the answer
> is yes.
>
> But, of course, if you prefer to ponder the axioms of ZF with some
> other logic, then your results will be different.
[...]
MoeBlee
> logic, set theory, classes, types, NBG, FOL and so on.
And if you are interested in logic in which we don't stipulate that
every domain of discourse is non-empty, then look up the subject "free
logic".
MoeBlee
>
David C. Ullrich
I understood that that's what you meant by generalizes.
But I really don't think that that's the right word:
Say you have two proofs of something, one of which
generalzes to prove something else and one of which
does not. The non-generalizing proof might still be
better.
But that's not the situation her at all! The point is that
the theorm in T and the corresonding theorem in T'
are going to look like _exactly the same theorem_ to
anyone who's not for some reason worrying about
the details of a _formalization_ of the theorem in
first-order logic. Here the proof does not just
"not generalize", it is insufficient to prove the theorem
itself, if the theorem is just restated in a way that
would not look to most people like a restatement
at all.
The proof works _if_ we're assuming that everything
is a set, and assume certain axioms about sets.
It does _not_ work if we assume exactly the same
axioms about sets, without assuming everything
is a set.
That's just not the right proof.
David C. Ullrich
<sjher...@netscape.net> wrote:
>David C. Ullrich wrote:
>
>>http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>
>>, in Kunen, "Set Theory: An Introduction to Independence Proofs"
>>the author explicitly includes the existence of a set as an axiom.
>>You really think that in _that_ book he's assuming the reader
>>doesn't know much about logic?
>>
>>
>
>Careful about citing secondary sources. The statement of the axiom
>(which Kunen labels Axiom 0) is followed by:
>
> This axiom says that our universe is non-void. Under most
> developments of formal logic, this is derviable from the logical
> axioms alone and thus redundant to state here, but we do so for
> emphasis.
I really don't see what your point is. That quote supports my point:
Note he says "in most developments of formal logic". He doesn't
say it's a _true fact_ about logic, only that it's a very common
convention. A proof deriving something from the axioms and
actual _valid logical reasoning_ is much better than a proof that
depends on a _formal convention_.
>I wonder why he doesn't mention that Infinity renders it redundant as well.
No doubt because that's silly - simply assuming that there exists a
set makes much more sense.
(Then why have _I_ been deriving it from the axiom of infinity?
Because the original question was how to show that a set exists
in ZF; presumably in that context the axiomatization does not
include the assumption that a set exists, or the question would
be incredibly silly.)
Stephen J. Herschkorn
>On Wed, 08 Jul 2009 11:12:01 -0400, "Stephen J. Herschkorn"
><sjher...@netscape.net> wrote:
>
>
>
>>David C. Ullrich wrote:
>>
>>
>>
>>>http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>>
>>>, in Kunen, "Set Theory: An Introduction to Independence Proofs"
>>>the author explicitly includes the existence of a set as an axiom.
>>>You really think that in _that_ book he's assuming the reader
>>>doesn't know much about logic?
>>>
>>>
>>>
>>>
>>Careful about citing secondary sources. The statement of the axiom
>>(which Kunen labels Axiom 0) is followed by:
>>
>> This axiom says that our universe is non-void. Under most
>> developments of formal logic, this is derviable from the logical
>> axioms alone and thus redundant to state here, but we do so for
>> emphasis.
>>
>>
>
>
>
>
>>I wonder why he doesn't mention that Infinity renders it redundant as well.
>>
>>
>
>No doubt because that's silly - simply assuming that there exists a
>set makes much more sense.
>
>
I disagree that this is silly or makes more sense. How does it make
sense to introduce an "axiom" that is a direct consequence of another
axiom? Standard ZF does not have an Axiom of Existence (which is why
Kunen labels it "Axiom 0"), probably because it is unnecessary in the
presence of the Axiom of Infinity, which *is* standard. I think that
Kunen introduces it only so that he can obtain some results before
discussing Infinity without having to preface every theorem with "If a
set exists, then..." If he is going to take the time to mention that
the axiom is unnecessary in the context of general logic, why not
mention that is unnecessary in face of other axioms? I think you agree
that the latter remark is more satisfying. In fact, isn't that what you
have been saying?
BTW, when you characterize something as "silly," your comment comes off
as rather dismissive, perhaps even a dig at the OP.
>(Then why have _I_ been deriving it from the axiom of infinity?
>Because the original question was how to show that a set exists
>in ZF; presumably in that context the axiomatization does not
>include the assumption that a set exists, or the question would
>be incredibly silly.)
>
--
MoeBlee
> I understood that that's what you meant by generalizes.
> But I really don't think that that's the right word:
That's fine. I was speaking pretty informally; not trying to pin down
the exact wording, since I figured we would understand each other in
context.
> Say you have two proofs of something, one of which
> generalzes to prove something else and one of which
> does not. The non-generalizing proof might still be
> better.
>
> But that's not the situation her at all! The point is that
> the theorm in T and the corresonding theorem in T'
> are going to look like _exactly the same theorem_ to
> anyone who's not for some reason worrying about
> the details of a _formalization_ of the theorem in
> first-order logic. Here the proof does not just
> "not generalize", it is insufficient to prove the theorem
> itself,
Of course, my sense of 'generalize' was not meant in a technical
sense. I meant just what you are saying - I meant "cutting across both
theories" in the way you are mentioning. Etc.
> if the theorem is just restated in a way that
> would not look to most people like a restatement
> at all.
>
> The proof works _if_ we're assuming that everything
> is a set,
Which proof? The proof I gave? I didn't assume everything is a set.
> and assume certain axioms about sets.
Of course.
> It does _not_ work if we assume exactly the same
> axioms about sets, without assuming everything
> is a set.
> That's just not the right proof.
You're welcome to your determination of "the right proof". Meanwhile,
the proof I gave is a correct proof in the context that I stated it,
and for the main purpose of showing that the axiom schema of
separation could be used for the purpose, just as the poster ASKED
that specific question; also showing that the axiom of infinity is not
required to make the proof (of course, again, in the context I
mentioned). As to what proof is preferable to give or in some sense
"the right proof" to give, I'm not motivated to opine. But in the
context of the poster ASKING whether separation could be used, and in
context of it having been said that the axiom of infinity is needed, I
gave a proof (which, of course, is only as good as the formal context
in which is given) that answers those questions.
Maybe we can better get to the heart of this exchange: What specific
assertion (given its stated or reasonably surmised context) have I
made, if any, that you think is incorrect?
MoeBlee
MoeBlee
> I didn't assume everything is a set.
Perhaps though, you are using 'assume' in a sense that includes the
consequences of assumptions. So, though I have not used an axiom "Ax x
is a set", of course, the axioms I used do entail "Ax x is a
set" (with the definition 'x is a set <-> Ey xey').
Please tell me what you disagree with, if anything, in the following:
(1) Using only first order logic, from the axiom schema of separation
(not in a form that relativizes the quantifiers to 'is a set'), we can
prove:
ExAy ~yex
(2) The above proof uses just first order logic that does not require
an axiom of the form "ExP" and the proof uses no logical nor non-
logical axiom of the form "ExP". But ordinary (and throughout, I mean
plain ordinary) first order semantics stipulates nonempty domains of
discourse and the quantification rules of the logical calculus support
that requirement and would not be valid without that requirement.
(3) Then using the axiom of extensionality (not in a form that
relativizes the quantifiers 'is a set') we can go on to prove:
E!xAy ~yex
(4) Using the pairing axiom (not in a form that relativizes the
quantifiers to 'is a set') or using the power set axiom (not in a form
that relativizes the quantifiers to 'is a set'), and with a definition
"x is a set <-> Ey xey", we can go on to prove:
(a) E!x(x is a set & Ay ~yex)
and
(b) Ax x is a set
(5) [Among your own points, and on which particular point I, of
course, agree:] In certain common and notable systems that relativize
the quantifiers of certain axioms to 'is a set', the above cumulative
proof does not work; therefore the above cumulative proof lacks the
"across the board" (or whatever you, David, want to call it) feature
that a proof using the axiom of infinity (in certain ordinary forms)
offers.
MoeBlee
Transfer Principle
> not be able to prove the existence of 0. This is what
> Ullrich believes as well. Infinity is needed to prove
> that 0 exists.
> ****************************
> No. There are many different but equivalent formulations of the Axioms of ZF
> and ZFC. Quite commonly, the Axiom of Infinity is formulated in such a way
> as to directly prove the existence of the empty set, so its not needed as an
> Axiom. Different Axiom sets may be able to prove the existence of the empty
> set in different ways. If they don't, then its easy to just add it in as an
> axiom.
OK, I see what you mean here. Of course one literally
doesn't need Infinity to prove the existence of 0, since
in HdB's theory there's no Axiom of Infinity, yet the
empty set clearly exists -- since there's an explicit
Empty Set Axiom.
So it would be more precise to say that a proof in ZF(C)
that 0 exists uses the Axiom of Infinity, and so if we
remove Infinity from the list of axioms, we can no
longer prove that any set exists at all. At least, this
is what the debate is all about.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > Hughes, like MoeBlee, often calls discussions about
> > theories other than ZF(C) "wastes of time," especially
> > if they are (ultra)finitist theories.
> Why, you are either stupid or a liar.
The latter, apparently, based on MoeBlee's comments in
this thread.
To me, it appears that there's some relationship between
whether a discussion is about ZF(C) or an alternate theory
and whether the phrase "waste of time" appears somewhere
in the discussion. But trying to point out that such a
relationship exists makes me a "liar." And so, once again
I find myself having to correct these "lies."
MoeBlee suggests that I:
"Read my EXACT remarks on the subject to see what I ACTUALLY said."
So I search for these exact remarks on the subject of
whether theories other than ZFC are a waste of time, but
the only post I found was in the thread titled "Cantor's
argument is erroneous," and the post is dated April 14,
2009, at 1621 Greenwich Mean Time. MoeBlee asked:
"But let's appropriately up the ante. Is Katz's theory sufficient as
foundational theory from which to derive mathematics for the
sciences?"
and later in the post, wrote:
"I asked because it is a basis of comparison among theories. A theory
that does NOT provide for an axiomatization for the mathematics for
the sciences is in that way not comparable to a theory that does.
Whatever features a theory may have (such has having infinite sets
but
not uncountables sets) are put in perspective of what ELSE that
theory
is or is not capable of, such as providing an axiomatization for the
mathematics for the sciences. This in no way entails that I think
theories that don't provide an axiomatization for the mathematics for
the sciences are not worth consideration or are "crank". As I already
mentioned, first order PA does not provide an axiomatization for the
mathematics for the sciences but I don't regard first order PA as
unworthy of study."
So we see that the key words are "foundation" and "comparision,"
so that in this April 14th post, MoeBlee is suggesting that a
theory that doesn't axiomatize for math of the sciences isn't
_comparable_ to ZFC as a _foundational_ theory.
And so let me attempt to correct the "lies" that I've been
accused of making in this thread, based on the post from
April 14th. It's possible that my corrections are still
"lies" and require further correction.
"Lie": Hughes/MoeBlee prefer working in ZFC.
Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
"Lie": Hughes/MoeBlee don't consider theories other than ZFC to
be worth considering
Correction: Hughes/MoeBlee don't consider theories other than
ZFC to be worth considering as a _foundational theory_, unless
the proposed theory is as powerful as ZFC in axiomatizing for
math of the sciences.
"Lie": To Hughes/MoeBlee, discussions of theories other than
ZFC are wastes of time.
Correction: To Hughes/MoeBlee, discussions of theories other
than ZFC that don't axiomatize for math of the sciences are
wastes of time when looking for a _foundational theory_.
"Lie": ZFC is MoeBlee's "comfort zone."
Correction: ZFC is MoeBlee's _foundational theory_.
"Lie": Those who reject ZFC or provide alternate theories are
called "cranks."
Correction: Those who claim that ZFC doesn't axiomatize the
math for the sciences, or who can't give a rigorous proof
that their proposed theory axiomatizes of the same, so that
the theory serves as a foundational theory, are called "cranks."
Also, I can provide new definitions for some of my terms:
An "adherent of a theory T" is someone whose foundational
theory is T.
A "standard theorist" is "an adherent of ZFC," or of a
theory that differs little from ZFC (such as ZF) as far as
serving as a foundational theory is concerned.
Among the so-called "cranks," only AP has suggested an
alternative to ZFC as a foundational theory -- his
desiderata include _geometry_ as a foundation. But it's
difficult to tell whether it axiomatizes math for the
sciences -- in particular, whether the sciences whose math
is axiomatized is AP's controversial Atom Totality theory.
Of course, any "crank" who rejects ZFC rejects it
completely, including as a foundational theory.
And so my latest "lies" entail overgeneralizing the
desiderata of Hughes/MoeBlee for _foundational theories_
by "lying" that they won't consider any theory in _any_
circumstance unless they satisfy these desiderata.
I'll wait to see whether my "lies" need any further
corrections or not.
Jesse F. Hughes
> "Lie": Hughes/MoeBlee prefer working in ZFC.
> Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
You seem to be quoting Moe in order to determine what I believe.
To be honest, ZFC is not my favorite "foundational theory". I much
prefer category theory. Just a matter of taste.
But in any case, I'm not a practicing mathematician and so I "use" no
foundational theory at all.
>
> "Lie": Hughes/MoeBlee don't consider theories other than ZFC to
> be worth considering
> Correction: Hughes/MoeBlee don't consider theories other than
> ZFC to be worth considering as a _foundational theory_, unless
> the proposed theory is as powerful as ZFC in axiomatizing for
> math of the sciences.
Well, that seems almost reasonable. I'd say "powerful enough" rather
than "as powerful as ZFC", and it's really a vague notion. Category
theory isn't powerful in quite the same sense as ZFC: we don't prove
that a category of natural numbers exists in the same way that we
prove that a set of natural numbers exists. Nonetheless, CT is a
useful foundation, seems to em.
>
> "Lie": To Hughes/MoeBlee, discussions of theories other than
> ZFC are wastes of time.
> Correction: To Hughes/MoeBlee, discussions of theories other
> than ZFC that don't axiomatize for math of the sciences are
> wastes of time when looking for a _foundational theory_.
I suppose so.
> "Lie": ZFC is MoeBlee's "comfort zone."
> Correction: ZFC is MoeBlee's _foundational theory_.
Not about me, so I've no opinion.
> "Lie": Those who reject ZFC or provide alternate theories are
> called "cranks."
> Correction: Those who claim that ZFC doesn't axiomatize the
> math for the sciences, or who can't give a rigorous proof
> that their proposed theory axiomatizes of the same, so that
> the theory serves as a foundational theory, are called "cranks."
> Also, I can provide new definitions for some of my terms:
> An "adherent of a theory T" is someone whose foundational
> theory is T.
Most mathematicians have no foundational theory at all, right?
> A "standard theorist" is "an adherent of ZFC," or of a
> theory that differs little from ZFC (such as ZF) as far as
> serving as a foundational theory is concerned.
Some use a number of different theories that could be foundational
theories, right?
I've done work in category theory, ZFC and ZFA (Aczel's
non-well-founded set theory). Am I "standard" or not?
--
"Sale or rental of this disc is ILLEGAL. If you have rented or
purchased this disc, please call the MPAA at 1-800-NO-COPYS."
-- The MPAA begins a new anti-piracy program,
found on a DVD purchased in China
David C. Ullrich
Why do you keep saying that????
No, the words "everything is a set" have not appeared
in any of your presentations of the proof. But the proof
is valid only in formalizations where the universe of
discourse includes nothing but sets.
>> and assume certain axioms about sets.
>
>Of course.
>
>> It does _not_ work if we assume exactly the same
>> axioms about sets, without assuming everything
>> is a set.
>
>> That's just not the right proof.
>
>You're welcome to your determination of "the right proof". Meanwhile,
>the proof I gave is a correct proof in the context that I stated it,
>and for the main purpose of showing that the axiom schema of
>separation could be used for the purpose, just as the poster ASKED
>that specific question; also showing that the axiom of infinity is not
>required to make the proof (of course, again, in the context I
>mentioned). As to what proof is preferable to give or in some sense
>"the right proof" to give, I'm not motivated to opine. But in the
>context of the poster ASKING whether separation could be used, and in
>context of it having been said that the axiom of infinity is needed, I
>gave a proof (which, of course, is only as good as the formal context
>in which is given) that answers those questions.
>
>Maybe we can better get to the heart of this exchange: What specific
>assertion (given its stated or reasonably surmised context) have I
>made, if any, that you think is incorrect?
Did I say the proof was wrong?
David C. Ullrich
<sjher...@netscape.net> wrote:
>David C. Ullrich wrote:
>
>>On Wed, 08 Jul 2009 11:12:01 -0400, "Stephen J. Herschkorn"
>><sjher...@netscape.net> wrote:
>>
>>
>>
>>>David C. Ullrich wrote:
>>>
>>>
>>>
>>>>http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>>>
>>>>, in Kunen, "Set Theory: An Introduction to Independence Proofs"
>>>>the author explicitly includes the existence of a set as an axiom.
>>>>You really think that in _that_ book he's assuming the reader
>>>>doesn't know much about logic?
>>>>
>>>>
>>>>
>>>>
>>>Careful about citing secondary sources. The statement of the axiom
>>>(which Kunen labels Axiom 0) is followed by:
>>>
>>> This axiom says that our universe is non-void. Under most
>>> developments of formal logic, this is derviable from the logical
>>> axioms alone and thus redundant to state here, but we do so for
>>> emphasis.
>>>
>>>
>>
>>
>>
>>
>>>I wonder why he doesn't mention that Infinity renders it redundant as well.
>>>
>>>
>>
>>No doubt because that's silly - simply assuming that there exists a
>>set makes much more sense.
>>
>>
>
>I disagree that this is silly or makes more sense.
I was just guessing at Kunen's motivation. You'd have
to ask him for the real reason he included a redundant
axiom.
>How does it make
>sense to introduce an "axiom" that is a direct consequence of another
>axiom? Standard ZF does not have an Axiom of Existence (which is why
>Kunen labels it "Axiom 0"), probably because it is unnecessary in the
>presence of the Axiom of Infinity, which *is* standard. I think that
>Kunen introduces it only so that he can obtain some results before
>discussing Infinity without having to preface every theorem with "If a
>set exists, then..." If he is going to take the time to mention that
>the axiom is unnecessary in the context of general logic, why not
>mention that is unnecessary in face of other axioms? I think you agree
>that the latter remark is more satisfying. In fact, isn't that what you
>have been saying?
>
>BTW, when you characterize something as "silly," your comment comes off
>as rather dismissive, perhaps even a dig at the OP.
Huh??? How in the world is my guess as to why Kunen
included a redumdant axiom a dig at the OP?
>>(Then why have _I_ been deriving it from the axiom of infinity?
>>Because the original question was how to show that a set exists
>>in ZF; presumably in that context the axiomatization does not
>>include the assumption that a set exists, or the question would
>>be incredibly silly.)
>>
David C. Ullrich
MoeBlee
> > I didn't assume everything is a set.
>
> Why do you keep saying that????
>
> No, the words "everything is a set" have not appeared
> in any of your presentations of the proof. But the proof
> is valid only in formalizations where the universe of
> discourse includes nothing but sets.
The proof is valid from certain ordinary axioms and a certain
definition of 'is a set'. And in this context, yes, of course, the
theory does prove 'Ax x is set'.
I think (I hope, at least) that what I wrote in my followup post puts
us square on the matter:
"Perhaps though, you are using 'assume' in a sense that includes the
consequences of assumptions. So, though I have not used an axiom "Ax x
is a set", of course, the axioms I used do entail "Ax x is a
set" (with the definition 'x is a set <-> Ey xey')." - MoeBlee
MoeBlee
MoeBlee
> On Jul 7, 10:47 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > Transfer Principle <lwal...@lausd.net> writes:
> > > Hughes, like MoeBlee, often calls discussions about
> > > theories other than ZF(C) "wastes of time," especially
> > > if they are (ultra)finitist theories.
> > Why, you are either stupid or a liar.
>
> The latter, apparently, based on MoeBlee's comments in
> this thread.
>
> To me, it appears that there's some relationship between
> whether a discussion is about ZF(C) or an alternate theory
> and whether the phrase "waste of time" appears somewhere
> in the discussion. But trying to point out that such a
> relationship exists makes me a "liar."
Now you're even lying about THAT! You are incredible! I never said
that pointing out mere relationships between discussions of certain
theories and the appearance of the phrase "waste of time" makes one a
liar. Go back to the SPECIFIC things I said you were lying about, not
some OTHER woozy generalizations that you conjure now.
> And so, once again
> I find myself having to correct these "lies."
>
> MoeBlee suggests that I:
>
> "Read my EXACT remarks on the subject to see what I ACTUALLY said."
>
> So I search for these exact remarks on the subject of
> whether theories other than ZFC are a waste of time, but
> the only post I found was in the thread titled "Cantor's
> argument is erroneous," and the post is dated April 14,
> 2009, at 1621 Greenwich Mean Time. MoeBlee asked:
>
> "But let's appropriately up the ante. Is Katz's theory sufficient as
> foundational theory from which to derive mathematics for the
> sciences?"
SO WHAT? Where did I say that Katz's theory is a waste of time if it
doesn't derive mathematics for the sciences? I didn't say that. I
wouldn't say that, and I wouldn't say that because I don't believe it.
You are reading into my posts what is not actually there.
> and later in the post, wrote:
>
> "I asked because it is a basis of comparison among theories. A theory
> that does NOT provide for an axiomatization for the mathematics for
> the sciences is in that way not comparable to a theory that does.
> Whatever features a theory may have (such has having infinite sets
> but
> not uncountables sets) are put in perspective of what ELSE that
> theory
> is or is not capable of, such as providing an axiomatization for the
> mathematics for the sciences. This in no way entails that I think
> theories that don't provide an axiomatization for the mathematics for
> the sciences are not worth consideration or are "crank". As I already
> mentioned, first order PA does not provide an axiomatization for the
> mathematics for the sciences but I don't regard first order PA as
> unworthy of study."
>
> So we see that the key words are "foundation" and "comparision,"
> so that in this April 14th post, MoeBlee is suggesting that a
> theory that doesn't axiomatize for math of the sciences isn't
> _comparable_ to ZFC as a _foundational_ theory.
It is isn't comparable IN THAT SENSE, yes, of course. But that doesn't
entail that the theory is a waste of time, and I never said it does.
Look, first order Boolean algebra is not comparable as a foundation
with ZFC; first order group theory is not comparable as a foundation
with ZFC. But I don't claim that first order Boolean algebra or first
order group theory are a waste of time!
> And so let me attempt to correct the "lies" that I've been
> accused of making in this thread, based on the post from
> April 14th. It's possible that my corrections are still
> "lies" and require further correction.
>
> "Lie": Hughes/MoeBlee prefer working in ZFC.
> Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
For CERTAIN purposes. But I never said that I prefer ZFC as a
foundational theory for the mathematics for the SCIENCES. For that
PARTICULAR project I'm more inclined to Z-regularity+"countable
choice". (And see later in this post for other theories, etc.) And
it's not even as if I have such things set in stone, as if they are
hardbound political allegiances or something like that. Your entire
picture about how I think about such things is based on your own
PROJECTION, not what I've ACTUALLY WRITTEN.
> "Lie": Hughes/MoeBlee don't consider theories other than ZFC to
> be worth considering
> Correction: Hughes/MoeBlee don't consider theories other than
> ZFC to be worth considering as a _foundational theory_, unless
> the proposed theory is as powerful as ZFC in axiomatizing for
> math of the sciences.
NO!!! I NEVER SAID THAT!!! DAMN!!!
Your "correction" is ANOTHER LIE.
I never said a competing theory has to be just as powerful as ZFC!!! I
just said that we may include such comparisons in our evaluations
(and, of course, those evaluations may include OTHER desiderata or
criteria).
In fact, quite personally, I actually may prefer a WEAKER foundational
theory than ZFC, in context of mathematics for the sciences. Z-
regularity+"countable choice" may be suitable. And I am interested in
studying other theories, not even comparable in strength, including
constructive mathematics, reverse mathematics, etc. Much of that is
still on my "to do" list, some of it only scratched at the surface by
me, but at least I aspire to know about it. I CERTAINLY do not RULE
OUT such things as possibly viable foundational alternatives to ZFC,
especially, since for mathematics for the sciences, ZFC is not even my
preference! I do like to study ZFC, but I never claimed it to be my
preferred foundational theory for the mathematics for the SCIENCES.
You PURELY fabricated that I have ever said such a thing.
> "Lie": To Hughes/MoeBlee, discussions of theories other than
> ZFC are wastes of time.
> Correction: To Hughes/MoeBlee, discussions of theories other
> than ZFC that don't axiomatize for math of the sciences are
> wastes of time when looking for a _foundational theory_.
NO, I NEVER SAID THAT. PURE FABRICATION BY YOU.
Your "correction" is ANOTHER LIE.
A certain proposal for a foundation may fall short of axiomatizing the
mathematics for the sciences, but the theory may be worth studying for
other reasons, or even for the purpose of seeing how to strengthen the
theory so that it does provide an axiomatization for the mathematics
for the sciences.
Why don't you just quote exactly from where I said certain of your
postings (or at least parts of them) are a waste of time? THAT, and
NOT what YOU later characterize, is what I said is a waste of time.
Your extrapolations to whole classes of theories is not MINE.
> "Lie": ZFC is MoeBlee's "comfort zone."
> Correction: ZFC is MoeBlee's _foundational theory_.
WHERE did you ever read me declare that ZFC is "my foundational
theory"?
Your "correction" is ANOTHER LIE.
> "Lie": Those who reject ZFC or provide alternate theories are
> called "cranks."
> Correction: Those who claim that ZFC doesn't axiomatize the
> math for the sciences, or who can't give a rigorous proof
> that their proposed theory axiomatizes of the same, so that
> the theory serves as a foundational theory, are called "cranks."
Your "correction" is ANOTHER LIE.
I've never said anything REMOTELY LIKE what you just said.
Wow, you are just incredible.
I've said MANY TIMES what I mean by "crank". Refer to what I ACTUALLY
WROTE, rather than PURELY FABRICATE please!
You REALLY need to take a long, hard look at your reading and thinking
skills. I don't think your unintelligent, but you do have a serious
problem of often INJECTING into what people write, so that you ascribe
to them notions that they actually have not asserted or believe. And
then you even carry that over to form a characterization of people as
having some kind of intellectual intolerance that they don't actually
have, which then, of course, has become your gravamen. Its a gravamen
based on an almost systematic set of ill-premises you've rationalized
by your process of injecting into things what is not actually there.
MoeBlee
MoeBlee
> So it would be more precise to say that a proof in ZF(C)
> that 0 exists uses the Axiom of Infinity, and so if we
> remove Infinity from the list of axioms, we can no
> longer prove that any set exists at all. At least, this
> is what the debate is all about.
No, it's not. You're completely confused.
MoeBlee
Transfer Principle
> Your "correction" is ANOTHER LIE.
Obviously, my corrections aren't sufficient, and will need
further correction. But the further corrections I'll give
in this most might still contain some "lies." The hope is
that after finitely many corrections, all the "lies" will
be eliminated -- without having to resort to verbatim
quotes every time I want to refer to another's opinion.
> > "But let's appropriately up the ante. Is Katz's theory sufficient as
> > foundational theory from which to derive mathematics for the
> > sciences?"
> SO WHAT? Where did I say that Katz's theory is a waste of time if it
> doesn't derive mathematics for the sciences? I didn't say that. I
> wouldn't say that, and I wouldn't say that because I don't believe it.
> You are reading into my posts what is not actually there.
I wonder how MoeBlee would have responded in that old
thread if I had simply given the three-word answer "I
don't know" to his question as to whether Katz's theory
is sufficient as a foundation, instead of tried to guess
why he had asked the question and have my guess declared
a "lie." I doubt that MoeBlee would've called _that_
response a "lie," since one can't get much more honest
(in a math situation) than saying "I don't know."
(And of course "I don't know" whether Katz's theory is
sufficient, because Katz never mentions axiomatizing
for the sciences in the paper!)
> > "Lie": Hughes/MoeBlee prefer working in ZFC.
> > Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
> For CERTAIN purposes. But I never said that I prefer ZFC as a
> foundational theory for the mathematics for the SCIENCES. For that
> PARTICULAR project I'm more inclined to Z-regularity+"countable
> choice".
Isn't Regularity (i.e., Foundation) one of the axioms
that's not included in Zermelo's original theory (i.e.,
like Replacement Schema, Regularity is an axiom of ZF
but not of Z)?
If so, then Z-regularity+"countable choice" is
redundant -- so the correction to my "lie" is that
MoeBlee's foundational theory is Z+"countable choice",
which I might abbreviate as Z+CC. For completeness,
the correction to the Hughes part of the "lie" is that
Hughes's foundational theory is category theory, as
given by Hughes himself in this thread.
> Why don't you just quote exactly from where I said certain of your
> postings (or at least parts of them) are a waste of time? THAT, and
> NOT what YOU later characterize, is what I said is a waste of time.
> Your extrapolations to whole classes of theories is not MINE.
The reason I extrapolate is that, once I post something
considered to be a "waste of time," I want to know how
exactly what I posted is a "waste of time," so that I
can extrapolate to what I might say in future posts and
know how to avoid "wasting time" in the future. My guess
as to what makes my posts "wastes of time" turned out to
be wrong and declared a "lie," and I end up continuing
to make posts that are "wastes of time."
Upon MoeBlee's recommendation, I search for a relevant
quote, but the only one I can find is in the thread
"IST - different questions". MoeBlee posted this on
June 9, 2009, at 2000 Greenwich Mean Time:
"SHEESH!! I blame MYSELF for wasting my own time reading through your
contortions only to find that they're based on patently false
premises. Of course, no hope that you'd ever read a BOOK that
carefully develops these concepts in mathematical logic so that such
postings as this one of yours don't serve so regularly as invitations
for me to waste my time."
The context of this was that the so-called "crank"
Srinivasan was trying to prove that ZFC is inconsistent
based on models of ZFC and IST whose universes are
intimately related. The inconsistency proof turned out
to be invalid because it was based on misconceptions
about how models and universes work. I had tried -- and
failed -- to fix the proof attempt by eliminating the
misconceptions about models and universes. MoeBlee
called my post a "waste of time" because it contained
the same misconceptions that Srinivasan's original post
contained -- time could have been saved by not even
reading my post at all, since my post failed to advance
the discussion from what Srinivasan had written.
Based on this post, and what others have written, I can
try to correct the "lies" made about "wasting time,"
but once again, I cannot guarantee that the following
correction still doesn't contain "lies."
Correction: A post about an alternate theory, or that
seeks to prove an established theory inconsistent, is a
"waste of time" if it is based on misconceptions on how
models and universes work, or how logic works (the key
concern in most Cantor diagonal threads) -- or the post
lacks rigor due to such misconceptions. Time is wasted
especially if such misconceptions are repeated over and
over again. The poster repeating such misconceptions is
often called a "crank."
Correction: My posts are a "waste of time" if I fail to
eliminate these misconceptions in the posts I defend,
or repeat the same misconceptions, or introduce even
more misconceptions.
Case in point here is the exchange between Hughes/HdB:
"Lie": HdB is a "crank" because he's a finitist.
Correction: HdB is a "crank" because his belief that
ZFC is inconsistent is based on the misconception that
the theory ZF-Infinity has a unique model, a model in
which ~Infinity is true.
It would be interesting if there could be a finitist
posting on sci.math who's a finitist simply because
he is philosophically (or Platonistically) opposed to
the existence of infinite sets, and not because of
misconceptions about models of ZF-Infinity. Indeed, a
hypothetical finitist might even post, or include in a
paper, something like:
"Axiom of the Empty Set:
ExAy ~yex
The empty set axiom is not included in most known
axiomatizations of ZFC because the existence of 0 is
derivable from the Axiom of Infinity. But since we are
considering a finite set theory, we must explicitly
include an Empty Set Axiom."
Note that I don't claim that the above is _true_ --
only that it would be _relevant_ (and in particular,
something that a hypothetical finitist that's more
relevant than what HdB posted).
> [Y]ou even carry that over to form a characterization of people as
> having some kind of intellectual intolerance that they don't actually
> have, which then, of course, has become your gravamen.
"Lie": People have some sort of intellectual intolerance.
Correction: ???
I don't know how to correct this "lie." I tend to
believe that the very use of the word "crank" entails a
sort of intellectual intolerance, but it appears that
this belief needs a correction. But I don't know what
the use of the word "crank" really is, if it's not
intellectual intolerance.
I await to see whether my corrections are acceptable, or
whether they still contain "lies."
Transfer Principle
Since I'm still trying to correct the "lies" in this
thread, let me try it in this post.
"Lie": Ullrich believes that a proof in ZF that 0
exists requires the use of Infinity, while MoeBlee
believes that the existence of 0 is provable from
FOL= and the Separation Schema alone.
Correction:
MoeBlee believes that the existence of 0 is provable
only from FOL= and the Separation Schema alone. Of
this there is no debate. My "lies" involve trying to
determine _Ullrich's_ -- and later on Webb's --
beliefs regarding the provability of the existence of
0 and where exactly they disagree with MoeBlee.
I know that Ullrich's argument involves some class
theories in which not every object is a set, so that
one can't prove that 0 is even a set without the use
of Infinity or certain other axioms that depend on
the particular class theory. But still, I don't know
exactly where Ullrich and MoeBlee disagree, leading
to the debate that caused this thread to be so long
(even after ignoring my posts and counting only
MoeBlee's and Ullrich's posts). I'm more likely to
make another "lie" than pinpoint the actual
disagreement between the two mathematicians.
"Lie": Ullrich believes that UI can't be validly
used to derive P from Ax P unless one already knows
that at least one object exists, while MoeBlee
believes that deriving P from Ax P is always valid
without having to worry about whether at least one
object exists or whether terms actually refer
(indeed at least one object always exists, and terms
always refer).
Correction:
Once again, there's no debate as to what MoeBlee
believes, but rather what Ullrich believes. Once
again, I can't pinpoint exactly where Ullrich
disagrees with MoeBlee's use of UI, or why Ullrich
criticize MoeBlee right after the latter used UI.
Jesse F. Hughes
> For completeness, the correction to the Hughes part of the "lie" is
> that Hughes's foundational theory is category theory, as given by
> Hughes himself in this thread.
For completeness, you might realize that I never claimed to have a
foundational theory. I'm not sure what it means to say that this or
that theory is my foundational theory.
I've done a smidgen of work in ZFC and a dollop of work in category
theory. I prefer the latter. There's really nothing more to it than
that. I don't have any feeling of being an adherent or proponent of
this theory or that.
Sorry.
--
Jesse F. Hughes
"Dead men can't talk. Especially when they've been cremated."
--- From the 1944 radio program "Adventures By Morse"
MoeBlee
> On Jul 10, 12:24 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > Your "correction" is ANOTHER LIE.
>
> Obviously, my corrections aren't sufficient, and will need
> further correction. But the further corrections I'll give
> in this most might still contain some "lies."
What I called 'lies' are lies. I don't know whether scare quotes will
be needed around 'lies' for what else you're about to come up with.
> The hope is
> that after finitely many corrections, all the "lies" will
> be eliminated -- without having to resort to verbatim
> quotes every time I want to refer to another's opinion.
>
> > > "But let's appropriately up the ante. Is Katz's theory sufficient as
> > > foundational theory from which to derive mathematics for the
> > > sciences?"
> > SO WHAT? Where did I say that Katz's theory is a waste of time if it
> > doesn't derive mathematics for the sciences? I didn't say that. I
> > wouldn't say that, and I wouldn't say that because I don't believe it.
> > You are reading into my posts what is not actually there.
>
> I wonder how MoeBlee would have responded in that old
> thread if I had simply given the three-word answer "I
> don't know" to his question as to whether Katz's theory
> is sufficient as a foundation, instead of tried to guess
> why he had asked the question and have my guess declared
> a "lie." I doubt that MoeBlee would've called _that_
> response a "lie," since one can't get much more honest
> (in a math situation) than saying "I don't know."
Whatever your point about that, at least you know that I never claimed
a theory has to provide an axiomatization for the mathematics of the
sciences just for the theory to be worthy of study.
> > > "Lie": Hughes/MoeBlee prefer working in ZFC.
> > > Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
> > For CERTAIN purposes. But I never said that I prefer ZFC as a
> > foundational theory for the mathematics for the SCIENCES. For that
> > PARTICULAR project I'm more inclined to Z-regularity+"countable
> > choice".
>
> Isn't Regularity (i.e., Foundation) one of the axioms
> that's not included in Zermelo's original theory (i.e.,
> like Replacement Schema, Regularity is an axiom of ZF
> but not of Z)?
Z set theory is not necessarily taken to be the exact set of axioms
Zermelo himself gave in his original paper. Indeed, Zermelo did not
present a formal theory in his original paper. One may debate whether
regularity is to be counted among the axioms of formal Z set theory,
but at least some sources do list regularity among the axioms of
formal Z set theory.
> If so, then Z-regularity+"countable choice" is
> redundant
(1) '+"countable choice"' is not reduant. (2) '-regularity' is not
redundant given that it is common to include regularity as an axiom of
Z set theory; thus, no matter whether one takes regularity as part of
Z set theory or not, it provides definiteness to say whether one is or
is not including regularity in some particular axiom set.
>-- so the correction to my "lie" is that
> MoeBlee's foundational theory is Z+"countable choice",
No, you're leaving out all the rest I said about foundational theories
that contradict your characterizations of my thinking.
> > Why don't you just quote exactly from where I said certain of your
> > postings (or at least parts of them) are a waste of time? THAT, and
> > NOT what YOU later characterize, is what I said is a waste of time.
> > Your extrapolations to whole classes of theories is not MINE.
>
> The reason I extrapolate is that, once I post something
> considered to be a "waste of time," I want to know how
> exactly what I posted is a "waste of time," so that I
> can extrapolate to what I might say in future posts and
> know how to avoid "wasting time" in the future.
Oh, how very selfless of you. As if it doesn't have anything to do
with your program to depict as intellectually intolerant.
> Correction: A post about an alternate theory, or that
> seeks to prove an established theory inconsistent, is a
> "waste of time" if it is based on misconceptions on how
> models and universes work, or how logic works (the key
> concern in most Cantor diagonal threads) -- or the post
> lacks rigor due to such misconceptions. Time is wasted
> especially if such misconceptions are repeated over and
> over again. The poster repeating such misconceptions is
> often called a "crank."
That's at least closer to the general idea. However, I don't even say
that all such postings are a waste of time for all purposes
whatsoever; nor, of course, just to be clear, do I hold that 'waste of
time' is not a subjective evaluation.
> Correction: My posts are a "waste of time" if I fail to
> eliminate these misconceptions in the posts I defend,
> or repeat the same misconceptions, or introduce even
> more misconceptions.
That might be one component that might make a particular post (not
having much else to redeem it) a waste of time. Another kind of time
wasting post (not having much else to redeem it) is the kind in which
you go through a whole bunch of deductions just to arrive at the
conclusion you've already been told, or to arrive at the conclusion,
which you've already been told, that a certain crank's desiderata or
whatever does not come automatically from the crank's assumptions or
arguments. But again, there is no general rule I can state about this
any more than I can state a general rule whether, say, a certain movie
is or is not a waste of my time.
> Case in point here is the exchange between Hughes/HdB:
>
> "Lie": HdB is a "crank" because he's a finitist.
> Correction: HdB is a "crank" because his belief that
> ZFC is inconsistent is based on the misconception that
> the theory ZF-Infinity has a unique model, a model in
> which ~Infinity is true.
That is ONE of a whole set of claims and behaviors of HdB that led me
to say that he's a crank.
> It would be interesting if there could be a finitist
> posting on sci.math who's a finitist simply because
> he is philosophically (or Platonistically) opposed to
> the existence of infinite sets, and not because of
> misconceptions about models of ZF-Infinity. Indeed, a
> hypothetical finitist might even post, or include in a
> paper, something like:
>
> "Axiom of the Empty Set:
> ExAy ~yex
>
> The empty set axiom is not included in most known
> axiomatizations of ZFC because the existence of 0 is
> derivable from the Axiom of Infinity.
One may choose to put it that way; but I do not.
> But since we are
> considering a finite set theory, we must explicitly
> include an Empty Set Axiom."
It's not true that we MUST adopt an empty set axiom to prove
ExAy ~yex
from the axioms of Z set theory without the axiom of infinity.
> Note that I don't claim that the above is _true_ --
> only that it would be _relevant_ (and in particular,
> something that a hypothetical finitist that's more
> relevant than what HdB posted).
Okay, it would be relevant to something or other to mention such
things. Meanwhile, frankly, I'm lost as to what is supposed to be the
point of this hypothetical exercise.
> > [Y]ou even carry that over to form a characterization of people as
> > having some kind of intellectual intolerance that they don't actually
> > have, which then, of course, has become your gravamen.
>
> "Lie": People have some sort of intellectual intolerance.
> Correction: ???
The correction is just not to cut my thought off at one sentence.
Please refer to whatever fuller statement I made about that. Did I
even list this as a lie of yours? (I'd like to see exactly what I
posted in that regard.)
> I don't know how to correct this "lie." I tend to
> believe that the very use of the word "crank" entails a
> sort of intellectual intolerance, but it appears that
> this belief needs a correction. But I don't know what
> the use of the word "crank" really is, if it's not
> intellectual intolerance.
First, of course, different people are intellectually intolerant in
certain ways regarding certain subjects. So, just to be clear, I've
not claimed that in general all the mathematicians and logicians who
use the word 'crank' aren't possibly intellectually intolerant in some
respect and in some degree or another about certain subjects. But I do
disagree with the general (and in certain cases, particular)
characterizations you make about people you call 'standard theorists'.
Second, it is not intellectual intolerance merely to say that a
certain person is a crank or that cranks have a strong presence at
sci.math and sci.logic.
Ongoing now, would you please quote me instead of putting words in my
mouth, or, if you have questions about what I mean, then just ask me
rather than put words in my mouth?
MoeBlee
MoeBlee
> On Jul 10, 12:26 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jul 9, 6:31 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > > So it would be more precise to say that a proof in ZF(C)
> > > that 0 exists uses the Axiom of Infinity, and so if we
> > > remove Infinity from the list of axioms, we can no
> > > longer prove that any set exists at all. At least, this
> > > is what the debate is all about.
> > No, it's not. You're completely confused.
>
> Since I'm still trying to correct the "lies" in this
> thread, let me try it in this post.
>
> "Lie": Ullrich believes that a proof in ZF that 0
> exists requires the use of Infinity, while MoeBlee
> believes that the existence of 0 is provable from
> FOL= and the Separation Schema alone.
> Correction:
> MoeBlee believes that the existence of 0
No, I didn't refer to "the existence of 0" (at least not in the latest
talks). Even specifically explained as to the matter of talking about
"the existence of 0". You simply skipped right past what I said as,
yet again, you mischaracterize my view.
> is provable
> only from FOL= and the Separation Schema alone. Of
> this there is no debate.
No debate that that is what I said? Or no debate that "ExAy ~yex" is
derivable through FOL= with the axiom schema of separation alone?
As to the former, correct, there is no debate. That is, there is no
rational argument basis to claim that I couched the matter as "the
existence of 0" (at least lately). As to the latter, yes, there is no
rational argument for a claim that it is not the case that "ExAy~yex"
is derivable through FOL= with the axiom schema of separation alone.
> My "lies" involve trying to
> determine _Ullrich's_ -- and later on Webb's --
> beliefs regarding the provability of the existence of
> 0 and where exactly they disagree with MoeBlee.
It is in my posts that you'll find SPECIFICALLY what I've said are
lies.
> I know that Ullrich's argument involves some class
> theories in which not every object is a set, so that
> one can't prove that 0 is even a set without the use
> of Infinity or certain other axioms that depend on
> the particular class theory. But still, I don't know
> exactly where Ullrich and MoeBlee disagree,
Ullrich's view was that the proof I gave (using separation, then
pairing for sethood) is not as preferable as one using the axiom of
infinity. He didn't say my proof is incorrect. As to preferability, I
didn't even make a claim.
> leading
> to the debate that caused this thread to be so long
> (even after ignoring my posts and counting only
> MoeBlee's and Ullrich's posts). I'm more likely to
> make another "lie" than pinpoint the actual
> disagreement between the two mathematicians.
First, Ullrich is a mathematician; I am not a mathematician. Second,
the dispute we had, seems to me, more a matter of talking past each
other. The only remaining point of difference that I can see is that
Ullrich seems (I stress *seems*) to me to take it as important to
declare a preference among the particular proofs in this matter, while
I am not interested at this time in declaring what proof is preferable
other than to point out that my proof did address questions and claims
made in the thread.
> "Lie": Ullrich believes that UI can't be validly
> used to derive P from Ax P unless one already knows
> that at least one object exists,
What? I never said that is a lie.
> while MoeBlee
> believes that deriving P from Ax P is always valid
> without having to worry about whether at least one
> object exists or whether terms actually refer
> (indeed at least one object always exists, and terms
> always refer).
Yes, that is a lie. I never claimed that universal instantiation works
"properly" even if we allow empty domains. For what I said EXACTLY
about the matter, please just refer to what I actually posted. As to
terms referring, again, you'd be better to look up what I wrote than
rely on your hazy summary.
MoeBlee
MoeBlee
Meanwhile, Mr. Transfer Principle, I take it you still have not
bothered to correct your misconception that not all instances of
AxP
___
P
are correct applications of universal instantiation, as formulated in
a number of textbooks in mathematical logic. (Also, your misconception
that my different application of universal instantiation was incorrect
in the thread with Nam.)
Your clucking that I am "wrong" about the matter just announces that
you're quite the ignorant fool.
MoeBlee
Transfer Principle
> Meanwhile, Mr. Transfer Principle, I take it you still have not
> bothered to correct your misconception that not all instances of
> AxP
> ___
> P
> are correct applications of universal instantiation, as formulated in
> a number of textbooks in mathematical logic. (Also, your misconception
> that my different application of universal instantiation was incorrect
> in the thread with Nam.)
OK, OK then.
"Lie": Sometimes one can't validly derive P from Ax P.
Correction: In classical FOL=, one can validly derive P
from Ax P, without any further conditions. There might
be types of logic from which one can't derive P from
Ax P (such as "free logic," or Nam's alternate logic
that he was inventing in that thread), but none of these
have anything to do with classical FOL=, in which P from
Ax P is always derivable.
While I'm at it, here's another "lie" that I would like
to address right now:
"Lie": Some posters are opposed to the Usenet access via
free sources such as Google and MathForum, and would
prefer that everyone would use full-priced newsreaders.
Correction: ???
It was mentioned numerous times that I have oversimplified
the situation concerning free and full-priced newsreaders,
and so this has no easy correction.
The concept of the Eternal September is relevant here,
since the argument given by the killfilers is that back
in the 1980's and early 1990's, there were very few
so-called "cranks" on Usenet because most people who are
not university affilated didn't have access to the Usenet
or Internet in general. Once the Internet and Usenet were
easy to access in the mid-1990's, the average layperson
would post on Usenet, resulting in a decline in the
quality of most posts. So by killfiling anyone who used a
post-1990's method of accessing Usenet, such as a free or
web-based newsreader, one would reverse the effect of the
Eternal September and only keep the higher quality
messages that are posted by those who use 1980's methods
of accessing Usenet.
The reason for this discussion is that I've found yet
another poster criticizing Eternal September posts. In
this case, the victim is cartman18 (who has also posted
in the tommy1729 thread):
"Christ, another mathforum.org imbecile with a hotmail address."
We've already discussed mathforum as a source of Eternal
September posts. But this is a new one -- criticizing
someone for using a Hotmail address. And for me, this is
a "lie" waiting to happen:
"Lie" (that I never made, yet I was very tempted to make):
Some posters are opposed to free email accounts such as
Hotmail, and prefer that everyone would use full-priced
email accounts.
So now what? This poster has associated Hotmail with the
Eternal September of low quality posts. So I wonder what
sort of email account that the poster would consider to
be more prestigious and associate with high quality posts
rather than the Eternal September of low quality posts.
Also, I want to know what a new Usenet user should do,
someone who wasn't online in the 1980's or early 1990's,
in order to be taken seriously by someone who associates
Google, Mathforum, and other free web-based news access,
and Hotmail and other easily obtained email, with the
Eternal September of low quality posts. Non-web-based
newsreaders and more difficult-to-obtain email accounts
sound archaic to those who first went online in the
mid 1990's or later, but to those who've been around
longer they are uncanny at separating the high-quality
from the low-quality posters.
Musatov
Francisco Antônio Dória - Wikipedia, the free encyclopedia 24 Apr
2009 ...
If exotic P = NP together with axiomatic set theory is omega-
consistent, then axiomatic set theory + P = NP is consistent.
http://en.wikipedia.org/wiki/Francisco_Ant%25C3%25B4nio_D%25C3%25B3ria
Musatov
Transfer Principle wrote:
Francisco Antônio Dória - Wikipedia, the free encyclopedia 24 Apr
Aatu Koskensilta
> But in any case, I'm not a practicing mathematician and so I "use" no
> foundational theory at all.
Depending on how one understands the term I may or may not be a
"practicing mathematician", but I certainly use or have no foundational
theory. In fact, I would submit that "foundational theory" in the sense
lwalke seems to be using the term is pointless jargon of not much
substance. In fairness, I think the same of many "standard" terms such
as "meta-language" and what have you.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> No, the words "everything is a set" have not appeared in any of your
> presentations of the proof. But the proof is valid only in
> formalizations where the universe of discourse includes nothing but
> sets.
I'll freely admit I haven't been following this exciting exchange in any
detail, but doesn't Moe's proof rather rely on a curious formulation of
separation, which allows us to form "subsets" of not only sets but of
any object at all, the formalisation as a scheme of
For any property P and an object A, there exists a set B such that
x in B iff x is in A and P holds of x.
?
This principle is of course rather peculiar, and certainly not something
we'd expect to find in any treatment of set theory conducted in
mathematical English -- it simply makes no ordinary mathematical sense
to ask whether e.g. pi is a member of the Diophantine equation 3x^2 + x
= 0 and so on.
T.H. Ray has shrewdly surmised I don't do any mathematics, so my
opinions probably count for very little. Be that as it may, I do find it
odd that people seem to feel very strongly that in discussions about set
theory and elementary proofs in set theory it is necessary to drag in
all sorts of formal grudgery of no apparent immediate relevance. In
contrast, a post asking how to prove that x^n * x^m = x^(n+m) would
probably not occasion an extended discussion about the restriction on
free variables in instances of the induction schema in first-order
arithmetic, the use of G�del's beta-function to express exponentiation
in the language of arithmetic, and so on.
Aatu Koskensilta
> On Jul 7, 10:05�pm, Transfer Principle <lwal...@lausd.net> wrote:
>
>> In ZFC we can prove the existence of a wellorder of R, a
>> nonmeasurable set, and a nonprincipal ultrafilter. So why does it
>> matter to set theorists whether AC is needed in any of those proofs?
>> It's because some set theorists -- the constructivists -- don't
>> accept AC.
>
> That is one reason.
No it isn't. It's just another of lwalke's peculiar fantasies.
> It doesn't preclude that there may be other reasons.
There are indeed perfectly sensible reasons for caring whether choice is
invoked in this or that proof.
> STOP PUTTING WORDS IN MY MOUTH! I've been asking you to stop putting
> words in my mouth for at least about a year now.
How's that been working for you so far?
David C. Ullrich
<aatu.kos...@uta.fi> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>
>> No, the words "everything is a set" have not appeared in any of your
>> presentations of the proof. But the proof is valid only in
>> formalizations where the universe of discourse includes nothing but
>> sets.
>
>I'll freely admit I haven't been following this exciting exchange in any
>detail, but doesn't Moe's proof rather rely on a curious formulation of
>separation, which allows us to form "subsets" of not only sets but of
>any object at all, the formalisation as a scheme of
>
> For any property P and an object A, there exists a set B such that
> x in B iff x is in A and P holds of x.
>
>?
Yes and no. Yes, but it's in a context where the only
objects are sets, so it's just ordinary separation.
>This principle is of course rather peculiar, and certainly not something
>we'd expect to find in any treatment of set theory conducted in
>mathematical English -- it simply makes no ordinary mathematical sense
>to ask whether e.g. pi is a member of the Diophantine equation 3x^2 + x
>= 0 and so on.
This is a way of stating my whole point. The argument does
rely on everything being a set, so that in particular it _does_
make sense (in a strict logical sense) to ask whether some
element of the Riemann zeta function is a model of ZFC.
Which is formally correct, but simply "wrong" - it "shouldn't"
matter what something _is_ in a proof in that sense.
>T.H. Ray has shrewdly surmised I don't do any mathematics, so my
>opinions probably count for very little. Be that as it may, I do find it
>odd that people seem to feel very strongly that in discussions about set
>theory and elementary proofs in set theory it is necessary to drag in
>all sorts of formal grudgery of no apparent immediate relevance.
Yes.
>In
>contrast, a post asking how to prove that x^n * x^m = x^(n+m) would
>probably not occasion an extended discussion about the restriction on
>free variables in instances of the induction schema in first-order
>arithmetic, the use of G�del's beta-function to express exponentiation
>in the language of arithmetic, and so on.
David C. Ullrich
Aatu Koskensilta
> Precisely. Although, curiously, I'm not entirely certain which
> side you're wagging your finger at here.
I was wagging my finger in a general sort of way, at an unspecific
direction. This whole thread is, from my perspective, depressingly
trivial and pointless. In my morose mood I take recourse in Torkel
Franz�n's wise words
Every day hundreds of participants fall away, exhausted or disgusted by
the umpteenth appearance of the same old arguments and
counterarguments, and every day hundreds of fresh and eager new
contributors step in to take their place.
which again remind me that what is tired and stale to me has not yet
lost its novelty to those encountering this stuff for the first time,
allowing me to regain my composure.
> My interpretation of the relevance of that paragraph is that you're
> agreeing with me that getting the existence of the empty set from the
> convention that something is a technicality of no intrinsic interest,
> but it seems _possible_ that you're instead referring to my insistence
> that the "right" way to do it is to use the axiom of infinity, because
> that shows it "really does follow from the axioms"...
There is no right or wrong way. How the existence of the empty set is
proved is just a matter of convention and organisation of the
material. That said, I certainly agree with you that deriving the
existence of the empty set from the convention that the universe of
discourse is empty, or from a formal principle that does not formalise
anything sensible in ordinary mathematical English, is, well, of no
apparent interest to someone who's interested in set theory as a
mathematical subject and not, say, the erudite details of these and
those formal theories, deductive systems, etc.
MoeBlee
> On Jul 13, 1:36 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > Meanwhile, Mr. Transfer Principle, I take it you still have not
> > bothered to correct your misconception that not all instances of
> > AxP
> > ___
> > P
> > are correct applications of universal instantiation, as formulated in
> > a number of textbooks in mathematical logic. (Also, your misconception
> > that my different application of universal instantiation was incorrect
> > in the thread with Nam.)
>
> OK, OK then.
> "Lie": Sometimes one can't validly derive P from Ax P.
> Correction: In classical FOL=, one can validly derive P
> from Ax P, without any further conditions. There might
> be types of logic from which one can't derive P from
> Ax P (such as "free logic," or Nam's alternate logic
> that he was inventing in that thread),
What "alternative logic"? Nam was saying that his notions are aligned
with Shoenfield's book.
> but none of these
> have anything to do with classical FOL=, in which P from
> Ax P is always derivable.
So what in the world was your point in claiming my use of universal
instantiation was "wrong"? (Maybe you just wanted to pretend you had
shown me to be wrong on the matter even if it meant you'd have to use
as your gambit a quite incorrect claim about a subject you've not
studied?)
> While I'm at it, here's another "lie" that I would like
> to address right now:
>
> "Lie": Some posters are opposed to the Usenet access via
> free sources such as Google and MathForum, and would
> prefer that everyone would use full-priced newsreaders.
> Correction: ???
Just to be clear, for the record, the above and Mr. Transfer
Principle's ensuing discussion about it have nothing to do with
anything I've ever said.
MoeBlee
MoeBlee
> David C. Ullrich <dullr...@sprynet.com> writes:
>
> > No, the words "everything is a set" have not appeared in any of your
> > presentations of the proof. But the proof is valid only in
> > formalizations where the universe of discourse includes nothing but
> > sets.
>
> I'll freely admit I haven't been following this exciting exchange in any
> detail, but doesn't Moe's proof rather rely on a curious formulation of
> separation, which allows us to form "subsets" of not only sets but of
> any object at all, the formalisation as a scheme of
>
> For any property P and an object A, there exists a set B such that
> x in B iff x is in A and P holds of x.
Not "property" verbatim, but rather "formula". (I used "property" only
in another discussion in which I gave an explanation of an INFORMAL
sense of the axiom schema of separation. If the other poster in that
discussion were to followup, of course, I would explain that it is not
claimed that the FORMAL axiom schema of separation provides a set for
each given set and property.)
For any formula P in which b does not occur free, all closures of the
following are axioms:
EbAy(yeb <-> (yex & P))
That is equivalent to many an ordinary statement of the formal axiom
schema of separation.
> This principle is of course rather peculiar, and certainly not something
> we'd expect to find in any treatment of set theory conducted in
> mathematical English
It's just a statement of a schema, equivalent to those found in many a
treatment of the subject, whether that is "curious" or "peculiar" or
not. Merely a stipulation of a certain set of formulas. And I've not
opined as to mathematical English.
-- it simply makes no ordinary mathematical sense
> to ask whether e.g. pi is a member of the Diophantine equation 3x^2 + x
> = 0 and so on.
Who aksed that?
On the other hand, if syntax is done in certain set theories, then
symbols and formulas themselves are sets. Sure, that's what one might
call "artifact" and not essential to our broader ordinary
understanding of symbols and formulas, just as one may say that it is
"artifact" that numbers are sets, etc. I've not claimed otherwise. But
I didn't even mention symbols and formulas in this sense.
I've not claimed much more than this: Using first order logic, from a
particular instance of a set of formulas (all of which are defined by
a particular schema) we may prove ExAy ~yex. Then, with a certain
common definition of 'is a set', and using the pairing axiom (in exact
form AbcExAy(yex <-> (y=b or y=c))) we prove Ex(x is a set & Ay~yex).
That is, in formal Z set theory (with a certain common definition of
'is a set') we may prove "Ex(x is a set & Ay ~yex)" without using the
axiom of infinity.
Personally, I don't find this curious or peculiar, though I might have
found it curious or peculiar had I not been familiar with first order
logic and its application to axioms that are first order formulas.
MoeBlee