Re: Contradiction or paradox

On May 18, 3:32 am, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
(did you ever answer that poster who about a week ago gave you a point
blank critique?).

Say what?

A post on May 8, 2007 by TXLogic in the thread 'ZFC Is Consistent - No
Axiom Infers The Negation Of Any Axiom', which is a thread you
started, and (as I now answer my own question) you did not answer the
post (at least not in that thread).

[begin post by TXLogic:]

On May 8, 5:29 am, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:

On 7 May 2007 07:00:13 -0700, Charlie-Boo <shymath...@xxxxxxxxx>
wrote:
...
The assumption is that ~(x e x) is a wff

Wow, this again. The term "wff" has a perfectly standard
definition, and ~(x e x) _is_ a wff by that definition.
If you claim it's not then you're using the word wff to
mean something other than what everybody else means by it.
In which case you have a lot of explaining to do.
Starting with this:

Exactly what is your definition of "wff"?

C-B appears to be a bit confused on this point, and on numerous
others. In his much cited (by him) Arxiv paper (http://arxiv.org/
html/
cs/0003071), he provides what is apparently supposed to be a BNF for
his "Program Calculus":

Programs will be represented using a semi-formal imaginary
miniature programming language. A program consists of a series
of lines. Each line consists of commands, each followed by one or
more arguments separated by commas. There are four different
commands: for, set, if and write.

for set-definition : For each element of a set defined by
set-definition, the variables in set-definition assume that value
and
the rest of the line is executed.

set variable = expression : The variable is assigned the value of
expression and the rest of the line is executed.

if expression : If the expression is true then continue executing
the
current line. Otherwise, continue at the previous for command on
the
current line, or the next line if there is no previous for on this
line.

write (expression , ...) : Output the tuple consisting of the
values
of
(expression , ...).

But nowhere in the "BNF are we told what a "set-definition", or even
an "expression", is; the BNF is lacking some crucial terminals here.

He also seems to think a casual discussion of the intended semantics
of his language suffices to support his many strong claims about what
his system CBL can do:

I. Program Transformations

Let P be an arbitrary predicate, such as a given number is prime or
there exists an employee who earns more than his manager. Let PP
be any computer program (program) that solves P by determining and
reporting to us whether P is true or false. (We decide upon a
single
programming language for all of our discussions.)

Assertion: There is a procedure (program) that will transform (map)
any such PP into another program PP' that solves ~P, the formal
negation of P.

Demonstration: A number of possible procedures come to mind. For
one, we could change every point where program PP is about to
report to us that P is true and change it to false, and vice-
versa.
Alternately, we could call program PP as a subroutine, and then
report the opposite of what PP does.

We call this assertion the NOT rule and signify it by P->P, where
in
general A->B means "any programs that solve A can be transformed
into a program that solves B".

We will consider two distinct types of programs: those that solve a
predicate, by reporting true or false, and those that solve a set,
by
listing its elements. We indicate what a program solves with a wff
(well-formed formula) of the Predicate Calculus.

While the standard syntax of the Predicate Calculus is used (~ not,
^ and, v or, $ there exists, @ for all, A B C ... variables, "..."
literals), we extend the semantics of wffs by giving special
meaning
to certain variables:

Unquantified input variables I, J, K, ... represent values that
must
be
supplied to the program as input. Unquantified output variables x,
y,
z, ... represent values that are output by the program.

We use single letter names P, Q, R ... to represent arbitrary
relations, and multiple letter names to represent specific
relations that we define. Thus, to solve P(I) a program would
have to take in a value I and output a value of true or false.
To solve Q(I,x) a program would take in a value I and output
every value for x. Note that in general I and x represent a
tuple of any number of individual values. Furthermore,
additional input variables, not explicitly represented, may be
present without altering the general principles and manipulations
being discussed.

A subscripted H is added to a wff containing output variables,
as in Q(I,x)_H, to indicate that the program must always halt.
(The set of values output must necessarily always be finite.)

For any wff W, the expression -W means that no program can
exist that solves W.

Notably, we are never provided with a rigorous definition of what a
program is, nor does the semantics provide a rigorous definition of
what it means for a program to halt; that is, there is no actual
mathematical definition of the notation Q(I,x)_H.

Of course, without an actual syntax for his theory and only an
informal, quasi-mathematical semantics, it is pretty much impossible
to verify all of the alleged theorems that follow, let alone C-B's
oft
repeated claims about how CBL proves the undecidability of the
halting
problem, Gödel's theorem, Rosser's theorem, etc. There is in
particular, AFAICS, no definition of the notion of a Turing machine;
in Section VI, the term "Turing Machine" suddenly appears without
definition in the assertion "We synthesize a program to list all
Turing Machines that halt no on themselves, proving that this set is
recursively enumerable." Numerous assertions follow that appear to
be
quantifying over TMs. But a definition of "Turing Machine" is not to
be found.

There is also no notion of natural number, so it is hard to see how
anything about incompleteness gets any purchase. Rather, an
"Incompleteness Axiom" just appears on the scene: -~YES(x,x). But
what the values of x are supposed to be here is unstated. YES itself
simply appears in the preceding sentence:

Thus, P(x) is solvable when there is a literal "?" such that
DEF:P(a) , YES("...",a).

Perhaps I just lack imagination, but I simply cannot see what is
going
on here. The system appears to be at best a desultory series of
assertions using a lot of traditional terminology but without any
definitions and with no actual foundation in any genuine mathematics.

[end post by TXLogic]

And that thread is yet another example of your ineptitude, as I and
other posters showed how very silly is your basic argument there.

Your mindless and irresponsible claims about Norm
Megill's system is another example.

I pointed out specifics. You have not.

Your ineptitude (or dishonesty?) is again glaring there, as you failed
to take account of the fact that Megill's theorem has certain
hypotheses, which was pointed out to you by a few other posters but
for which you have no coherent response.

It's not too pleasant digging up
such examples, but still not hard to do.

You have proven nothing except that you are content to deal in
personal attacks rather than technical issues.

I have comments about you personally, but I talk about technical
issues with a number of posters quite often. And, for just one example
in which I did engage you technically, look again at the 'ZFC Is
Consistent - No Axiom Infers The Negation Of Any Axiom' thread in
which I and a number of people showed you TECHNICALLY how silly your
argument is there. Moreover, our VERY FIRST CONVERSATION was as to the
technical matter of the definition of 'wff of the language of ZFC' in
which I answered your question but you never (even after I asked you a
few times) responded to my question as to what your point is in asking
such a question and having people provide you with such information.

MoeBlee

.

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