Re: Which axiom prohibits this kind of construction?

On Oct 13, 6:41 pm, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
galathaea <galath...@xxxxxxxxx> writes:
On Oct 13, 5:16 pm, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
"Jesse F. Hughes" <je...@xxxxxxxxxxxxx> writes:

(1) It is possible to have a theory where x = [x].

This all depends. What does [x] mean?

If it means: the set z satisfying the below condition

(Ay)(y in z <-> y = x)

then, no, you cannot have an interesting (set-like) theory in which
this is true. It is inconsistent with the existence of an empty set.
If we have the axiom of pairing together with x = [x], then we can
prove (Ax)(Ay)( x = y ). And so on.

Note Tommy's own words on what *he* means:

[x] = is the set that contains x ( only )
however x may be a set itself.

From <27191727.1216068180786.JavaMail.jaka...@xxxxxxxxxxxxxxxxxxxxxx>

yep

now look over your response
with my interpretation in terms of a mereological theory on urelements

so tommy is defining a new theory of collection
where [x] is a collection that contains only x
which may also be a collection

x = [x] = [[x]] = ...

perfectly acceptable

all singleton urelements are unwrappable

No clue what this means.



now look at

[x, [y]] = [[[x]]], y = [[x], [[y]]] = ...

It is obvious to me that you mean something else when you write [x]
than I mean. I've *given* you my interpretation. I've defined it
explicitly. Here it is:

(Ay)( y in [x] <-> y = x ).

If you mean something else, say what you mean.

i'll change things around a bit for formality

here's a language:

constants: a0, a1, ...
variables: x0, x1, ...
names can be made through any alphabetic string (except the lone c)
quantifiers: forAll, thereExists
equational relation: =
logical connectives: /\, \/, ~
optional logical connectives(?): ->, <->
(optional?) logical collectors: (, )

now watch this upsetting move for a set theory

intersection: //\\
(note
i'm breaking with tommy i'm sure on that one)
connective symbol: ,
primitive subcollection relation: c
optional mereological collectors: [, ]

now that's the giveaway
because i've named it disposible

a collection is given in list form as
a0, a1, ..., an

using the symbol when name assigning

A = [a0, a1, ..., an]

{ where i've taken some additional notational conveniences }

helps break apart the pieces for conceptual clarity
and visually may assist in parsing complex pages of equations

in the end
it has the same right being there
as -> does in many logical calculi where it is eliminatable
(as a notational convenience to tie semantic tools to)

but it's not needed
as it is
for instance
in classical set theory

..

and then
to give an illustration of starting to formulate in this language

look at a definition of urelements

(x0 c urelement) <-> (forAll x1)((x1 c x0) <-> (x1 = x0))

that seems a nice possible form
but maybe there are better ones

maybe
urelement = [x0 | (forAll x1)((x1 c x0) <-> (x1 = x0))]
shows off another use of []?

the point of the urelements is
without the use of an element relation
the ability to discern "sizes"

that would be necessary for an axiom of infinity

an axiom of union is wanted
so something like
(forAll x0)(forAll x1)(thereExists x2)(x2 = x0, x1)

or i could have written that

(forAll x0)(forAll x1)(thereExists x2)(x2 = [x0, x1])

;)

or i'm sure some other ways of building the existential net
would be more agreeable to some around here versus others

but now
i'll just point that i've gotten this far in the explanation
and i still haven't clarified some aspects
that may be easily assumed wrongly

does [x0, x1] = [x1, x0]?

and how do we even count elements
to formulate an axiom of infinity?

i can explore those points deeper
and see if i could interpret tommy's desires for these
but that's not the point here

the point is that this isn't what is being discussed
even though it should be

the point is
there are certainly formal approaches
to ideas tommy has explored

and i don't hold a grudge against tommy
for not getting the formalism down the first few tries

nobody
when they first are exploring a theory
has a very clear idea if what they are doing makes sense

it may be easy to scoff
when he says he is working on something

but when there is no attempt to show him how to put it all together
who benefits?

do others reading the post benefit?

is it purely the sportsmanship?

again
a perfectly acceptable and consistent state of affairs

As long as we leave [...] completely unspecified, yeah, I guess so.
Any axioms? Any definitions? Anything to work with here? All I can
see is that you have a language with an unrestricted function symbol
(a funny kind, with no fixed arity, but that's obviously analogous to
{...}).

I was hoping for something substantial.

it seems clear to me you dismissed a legitimate approach to collection
through uncharitable interpretation

No. My interpretation has been explicit. I can show you post after
post where I explicitly said *this* is how I interpret [x] and these
are the consequences. Here are four messages in which I clearly
stated my interpretation:

<87tzsdkol2....@xxxxxxxxxxxxx>, <87skud19cw....@xxxxxxxxxxxxx>,
<87lk013jp5....@xxxxxxxxxxxxx>, <87d4l439vk....@xxxxxxxxxxxxx>

In every one of these four messages except the first listed, I
explicitly asked Tommy to correct me if my interpretation was wrong.

I'll ask you to do the same. It's still unclear to me what you mean
by [x]. I know what it means in set theory. I've never seen anything
like [x], {x} or (x) used in mereology (about which I know next to
nothing). So, give me a more explicit definition than you have so
far, since I can't work with "[x] is a collection that contains only x
which may also be a collection". I can't see how this statement
distinguishes [x] from the usual singleton, aside from the fact that
you have used the word "collection" instead of "set".

In a nutshell: at least I've offered an interpretation. You have
not. You've left [x] completely undefined.

i can expand on any of the above
provide ebnf syntax and everything
if you need it more formally

nothing is said of logic really
but that is one of the parts that tommy gets to grow

he likes 3-valued logic so i'd expect he attach that on

will someone say it is stoopid?

probably

can it be done consistently?

?

as i've said several times
i'm not defending tommy's erratic responses
or his own struggle for consistency

(i see him as still having learning to do like all of us)

i'm pointing out that those who requested he have more knowledge
themselves did not show an understanding
or pursuit of understanding
that could have given better light to the issues involved

Right. Carefully stepping through a proof that the evident
interpretation of his axiom yields deeply unsatisfactory results is
not helpful.

yes
i know all the good that was done along the way
you don't have to throw it up in defense

i know there were some explanations
on why immediate interpretations with many assumptions
were not viable

there was also a lot of haughtiness on both sides

it devolved from there

it's that end i dislike

they weren't there to guide him to better ideas
but instead
many were there simply to participate in uncharitable interpretation

I haven't seen what a charitable interpretation looks like and I'm not
clever enough to give one myself. Thus, I did the best thing I could:
I explicitly stated my interpretation and some of its consequences and
asked for correction.

if cleverness was the barrier
then tommy should be feeling pretty good right now

but i want to add
because i don't know if i am clear
that i'm not doing this rant against you

my ranting is about the dynamic that keeps getting promoted

participants play various roles in this dynamic
and i don't think you played
for instance
the bite-them-when-they're-down attack dog type
(there are others who play that role)

you were just the
unclever-enough-to-give-anything-but-uncharitable-views
role

it's more of a formal role

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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