Re: the return of the master : tommy1729

On Jul 6, 11:58 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jul 3, 2:05 pm, Transfer Principle <lwal...@xxxxxxxxx> wrote:
Let's go back to MoeBlee's alleged proof that TST is
inconsistent now. One of tommy1729's axioms is:
Ax x=[x]
MoeBlee's alleged proof involves instantiating to the
case x=0, so that we have 0={0}.
Is that what I did? Please refer to the specific post so that I can
have the context.
I don't recall using the empty set in my argument (maybe I did; but I
don't recall it).
Again, please link or at least name the thread and post numbers so
that we can evaluate what was actually posted and not your re-
interpretation of what was posted.

Post numbers? Maybe those newsreaders that I can't afford
have post numbers, but I have no access to them.

So let me instead give a date. The thread was titled at
least three different times (nothing anyone would want to
read (or: crank boxing (or: the death of the dance))),
and the post in which MoeBlee gave his proof was given on
December 19, 2008, at 2121 Greenwich Mean Time:

1 Ax x=[x] .... axiom
2 Ay(y in [x] <-> y=x) ... from tommy1729's own qualifying statement
3 Ax x in [x] ... from 2
4 Ex Ay y not-in x ... axiom
5 Ay y not-in x ... existential instantiation from 4
6 x = [x] .... from 1
7 Ay y not-in [x] ... from 5, 6
8 ~Ey y in [x] ... from 7
9 x in [x] ... from 2
10 Ey y in [x] ... from 9
But 8 and 10 is a contradiction.
You are welcome to say exactly what is not first order logic applied
to his axioms.
Or in simple English:
There is an x such that for all y, y not in x (_empty set axiom_). So
[emphasis mine]
let x be such that for all y, we have y not in x. But for all x, we
have x=[x] and x in [x], so x in x. So x itself is a y such that y in
x. Contradiction.

Already, we can see that MoeBlee's proof _does_ use the
empty "set" after all. Essentially, he claims that from
"x=[x]" we prove that every object is a "singleton," yet
the empty "set" is not a "singleton," hence what he
believes to be a contradiction.

Looking at MoeBlee's lines 1-10 again:

1 Ax x=[x] .... axiom
2 Ay(y in [x] <-> y=x) ... from tommy1729's own qualifying statement
3 Ax x in [x] ... from 2
4 Ex Ay y not-in x ... axiom
5 Ay y not-in x ... existential instantiation from 4
6 x = [x] .... from 1
7 Ay y not-in [x] ... from 5, 6
8 ~Ey y in [x] ... from 7
9 x in [x] ... from 2
10 Ey y in [x] ... from 9

the errors are in lines 2 and 4. MoeBlee believes that
these are accurate renderings of tommy1729's axioms from
English to symbolic language of the theory, but I don't
believe that these are accurate at all.

In December, MoeBlee pointed out that his line 2 is
derived from tommy1729's remark, as follows:

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

To MoeBlee, or anyone accustomed to working in ZFC, this
would appear to define [x] as a singleton, a set whose
lone element is x. But in mereology, we _can't_ conclude
from this remark that [x] is a "singleton" in the sense
of line 2, that any y in [x] must be x itself.

Similarly, line 4 is based on a ZFC definition of empty
set, but we can't conclude in mereology that [] is a set
such that for any y, y not-in [].

Indeed, the use of the word "in" to represent the lone
two-place predicate of the theory is a bit awkward,
since it doesn't work the same way as "e" in ZFC. This
is why galathaea suggested that tommy1729 use the symbol
"c" to represent the lone two-place predicate, so that
someone like MoeBlee wouldn't have made the mistakes that
he ends up making.

So what are the correct interpretations of tommy1729's
remarks in English that MoeBlee should have used instead
of his lines 2 and 4. Let's compare MoeBlee's incorrect
ZFC-based Empty Set Axiom with the correct mereological
Empty Set Axiom:

ZFC Empty Set:
4 Ex Ay y not-in x ... axiom

Mereological Empty Set:
Ex Ay xcy

So this object x is a part (not an "element," but a
_part_) of every set. At first, this doesn't necessarily
sound like an empty set. But we can prove that this set
must be the smallest possible set.

Proof:
1 Ex Ay xcy ... Mereological Empty Set
2 Ay xcy ... Existential Instantiation
3 Ax Ay (xcy & ycx) -> x=y ... Antisymmetry Axiom
(mentioned in my last post)
4 Ay (xcy & ycx) -> x=y ... Universal Instantiation
5 Ay xcy -> (ycx -> x=y) ... equivalent to step 4
(since "(P&Q)->R" and "P->(Q->R)" are equivalent --
checking the truth tables, both evaluate to false when
P=Q=true, R=false and to true otherwise)
6 ycx -> x=y ... Modus Ponens (steps 5,2)

So we see that x (which has been instantiated to be an
empty set) has the property that if y is a part of x,
then y must be x itself. This object has only one part,
namely itself. And since x must be a part of every set,
x must be the smalllest possible set -- no smaller set
is possible since no set can avoid having x as a part.

Now we'd like to _define_ [], the empty set, to be
this particular set x. Of course, I already know that
MoeBlee doesn't accept definitions without a proof of
existence and uniqueness. We've already proved that an
empty set exists, so now we must prove it's unique. To
do so, we let y be an empty set as well, by rewriting
the Empty Set Axiom with a change of dummy variables so
that we can instantiate to another set y, in order to
prove that if x and y are both empty sets, then x=y.

7 Ey Ax ycx ... Mereological Empty Set, change variables
8 Ax ycx ... Existential Instantiation
9 ycx ... Universal Instantiation
10 x=y ... Modus Ponens (steps 6,9)

The proof is very similar to the proof that the identity
of a group must be unique. We've proved that any set y
that's a part of x must be x itself, but y, being an
empty set, must be a part of every set. And so y is a
part of x, and since the only part of x is x itself, y
must equal x. QED

Therefore the empty set is unique, and so we can
justifiably write the definition:

[] =def the unique x s.t. Ay xcy

What about a definition of [x]? We are reminded of the
definition of the notation {} in ZFC as given by Suppes,
who gives a separate definition for {x}, {x,y}, {x,y,z},
and {x,y,z,w}. Formally, we actually have four separate
function symbols, a one-place symbol {}, a two-place
symbol {,}, a three-place symbol {,,}, and finally, a
four-place symbol {,,,}.

So we can define a one-place function symbol for []. I
have been trying to figure out a good definition for
[x] that satisfies MoeBlee's eliminability requirement
for acceptable definitions. And the best definition so
far appears to be simply:

[x] =def x

Can't get more "eliminable" than that!

The first non-trivial definition for [], therefore,
would be a two-place function symbol [,]. But first, we
would need a definition for atomic, which is to be a
one-place predicate:

x atomic <->def Ey ~xcy & Ay (ycx <-> y=x v Az ycz)

or, using the previously defined symbol []:

x atomic <->def ~x=[] & Ay (ycx <-> y=x v y=[])

The "~x=[]" (or "Ey ~xcy") part is to prevent [] itself
from being an atom. We want to avoid [] being an atom
for the same reason that we avoid 1 being a prime.

Then we can define the two-place function symbol [x,y]:

[x,y] =def the z s.t. Aw (w atomic & wcz -> wcx v wcy)

Existence and uniqueness? Most mereological theories
have an axiom guaranteeing existence, so there's no
reason that TST can't have one. Such an axiom would be
the analog of ZFC's Pairing Axiom. Uniqueness is
guaranteed by the analog of ZFC's Extensionality Axiom,
which I mentioned in a previous post.

It also might be more elegant if we could avoid
mentioning "atomic" and write something like:

[x,y] =def the z s.t. Av zcv <-> xcv & ycv

Now this might require something like a Separation
Schema to prove existence and uniqueness. The former
might be more intuitive, while the latter would be a
little bit shorter. I'm not sure which one tommy1729
(or galathaea, the poster who had been helping him
with his theory) would prefer.

Once [x,y] has been defined, we actually could write a
different definition of [x]:

[x] =def [x,x]

(from analogy with the definition {x} =def {x,x} given
by Suppes for ZFC) and prove as a theorem, tommy1729's
infamous remark:

Ax x=[x].

But we can just take this as a definition until enough
axioms to prove it as a theorem have been agreed upon.

In any rate, these are the correct definitions for []
and [x] that are required for the proof to work. The
correct definitions don't satisfy MoeBlee's lines 2 or
4 of his proof, and so MoeBlee's proof does not show
that TST is inconsistent.
.

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