Date: Feb 14, 2013 5:42 PM Author: fom Subject: Re: distinguishability - in context, according to definitions 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.