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fom

Posts: 1,033
Registered: 12/4/12

Re: distinguishability - in context, according to definitions
Posted: Feb 14, 2013 5:42 PM

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On 2/14/2013 9:32 AM, Shmuel (Seymour J.) Metz wrote:
> In <qImdnYCz5tRmvITMnZ2dnUVZ_oWdnZ2d@giganews.com>, on 02/11/2013
> at 10:53 AM, fom <fomJUNK@nyms.net> said:
>
>> Should I assume this question is contemptuous?
>
> More precisely, that you were slinging around symbols and words
> without knowing what they mean, and inventing a local language without
> providing definitions. That, by the way, is what some of the
> participants in the tree thread were doing.

That last sentence speaks to the motivation for looking
into "distinguishability" from a cited source in the first
place.

>
> You might start by thinking about what you meant by "9.999...", then
> asking whether "pattern matching" has any relevance.
>
> Similarly, what were you trying to say when you wrote 'Now, in what
> follows, the particular problem will be considering the nature of
> eventually constant sequences taken to be ontologically "the same."'?

Here are descriptions of the received paradigm
for use of the sign of equality -- not necessarily
one with which I agree, but one that I did follow in that
post.

This is from Morris explaining the received
paradigm with which he disagrees,

"On this account, an identity statement is a
statement about the objects referred to by referring
expressions with which it is made. It is not
in any sense a statement about those expressions
themselves. The do not figure in its truth
conditions."

The received paradigm was heavily influenced
by Russell, Wittgenstein, Carnap, and Quine.
In large part, that means logical atomism.
When representing the nature of that position,
Cochhiarella writes:

"That is, an object's self-identity or nonidentity
with any other object is invariant through all
possible worlds of a logical space containing
that object. We must distinguish this ontological
invariance from the varying semantical relation
of denotation (Bedeutung) between an object and a
(non-Tractarian) name or definite description.
The former must be accounted for within the formal
ontology, the latter only within its applications."

So, in answer to your question, given two names
purporting reference for one object, I took the
time to clarify that the analysis was involving
my best effort at navigating the transition across
the syntactic-semantic ladder of a formal construct.

Carnap explains it a little better than
my analysis book in college did,

"If, in a constructional system of any kind,
we carry out a step-by-step construction of
more and more object domains by proceeding
from any set of basic objects by applying in
any order the class and relation construction,
then these domains, which are all in different
spheres and of which each forms a domain of
quasi objects relative to the preceding domain
are called constructional levels. Hence,
constructional levels are object spheres which
are brought into a stratified order within the
constructional system by constructing some of
these objects on the basis of others. Here,
the relativity of the concept 'quasi object,'
which holds for any object on any constructional
level relative to the object on the preceding
level, is especially obvious."

Of course, I had to figure it out for myself
the hard way.

Now, just so we understand that I am actually
thinking about mathematics when I do these
things, let us compare Kunen, Jech, and
Zermelo 1908.

Zermelo first.

"Set theory is concerned with a domain B
of individuals, which we shall call simply
objects and among which are the sets. If
two symbols, a and b, denote the same object
we write a=b, otherwise -(a=b)"

So, Zermelo is treating the sign of equality
with respect to denotations, and, consequently,
not according to the modern received paradigm.
He did, however, recognize that more was
needed,

"The question whether a=b or not is always
definite since it is equivalent to the
question whether or not ae{b}"

Now Jech.

It takes him a little time to get to
any discussion for the sign of identity
relevant to these remarks, but he begins
with

"Intuitively, a set is a collection of
all elements that satisfy a certain
given property."

A few pages later, he simply seems to accept
whatever the received paradigm for identity
might be in his statement following the
presentation for the axiom of extensionality:

"If X and Y have the same elements,
then X=Y:

Au(ueX <-> ueY) -> X=Y

The converse, namely if X=Y then ueX <-> ueY,
is an axiom of predicate calculus. Thus we
have

X=Y <-> Au(ueX <-> ueY)

The axiom expresses the basic idea of a set:
A set is determined by its elements"

Just to be clear, Jech's last statement
does not seem quite right. That characterization
for a set corresponds with Frege's analogous
notion of identity for concepts in abstraction
of true identity for objects.

Cantor rejected that interpretation of a
set in his review of Frege:

"... fails utterly to see that quantitatively
the 'extension of a concept' is something
wholly indeterminate; only in certain cases
is the 'extension of a concept' quantitatively
determined; and then to be sure, if it is
finite it has a determinate number; if it is
infinite, a determinate power. But, for such
a quantitative determination of the 'extension
of a concept' the concepts 'number' and
'power' must previously be given from the
other side; it is a twisting or the correct
procedure if one attempts to ground the
latter concepts on the concept 'extension
of a concept.'"

Or, more succinctly in correspondence with
G.C. Young,

"In [remark 1 of Grundlagen], I expressly say
that I only call multiplicities 'sets' if
they can be conceived without contradiction
as unities."

Athough you probably do not see this, Zermelo's
language reflects Cantor's sentiments and Jech's
does not.

Here is Kunen.

"Intuitively, x=y means that x and y are the
same object. This is reflected formally in
the fact that the basic properties are logically
valid and need not be stated explicitly as
axioms of ZFC. For example,

|- x=y -> Az(zex <-> zey)

whereas the converse is not logically valid,
although it is a theorem of ZFC since its
universal closure is an axiom (Extensionality):

AxAy(Az(zex <-> zey) -> x=y)"

So, once again, there is deference to the
received paradigm.

That post was written according to the
"basic" logic that mathematicians say they
are abiding by when they say they are working
in ZFC in the sense of Jech or Kunen.

> Do you know what syntax is?

Not much different from noise with a
little geometry.

> What do Dedekind cuts have to do with
> decimal expansions, other than the trivial sense that they can be used
> to construct a model of the Reals?

Symmetry.

In the hierarchical constructions of the reals,
the identity of a real number within the system
is achieved by retaining the order relations from
each step up the ladder beginning with the simple
sequential ordering of the natural numbers. With
the Dedekind cuts, one must decide on whether the
construction shall be based upon least upper bounds
or greatest lower bounds. I recognized a possibility
to investigate the question in terms of a choice
among two equivalent names for the same object
whose indistinguishability rested on the presumption
of a completed infinity.