Math Forum Discussions

fom

Posts: 1,968
Registered: 12/4/12

Re: Cantor's absurdity, once again, why not?
Posted: Mar 15, 2013 7:09 PM

Plain Text

On 3/15/2013 2:12 AM, WM wrote:
> On 14 Mrz., 23:36, fom <fomJ...@nyms.net> wrote:
>> On 3/14/2013 5:15 PM, WM wrote:
>> >
>>
>>> distinguishable, that means definable by finite words
>>
>> How does a definition "distinguish"?
>
> A definition is a name.

Ok. But, then I would have to ask
what you mean by name. What you have
replied with is "definitory" (keeping
with Aristotle), but it is definitory
in way that even made Aristotle uncomfortable.

Let me save some time. There are Millian
names and descriptive names. And, for the
purposes of mathematics, one should distinguish
between descriptive names and descriptively-defined
names because one shall never ostensively indicate
a physically material object as the referent of
a description used in mathematics.

So, say that what you mean is a Millian name.
I am making this choice based on your prior
statements that would even put you at odds
with the positions of Abraham Robinson in
his criticism of Russellian description theory.

The sense in which you are using it is probably
inadmissible because one shall never ostensively
indicate a physically material object as
the referent for most mathematical objects.
Consequently, it has no semantics unless you
are following some free logic. And, it has
already been deduced that that is one of the
possible ways of thinking about your claims.

But, we shall ignore this possible
inadmissibility since a similar situation
exists when one applies Tarski semantics to
mathematics generally.

As far as names go, the modern criticism
that would best seem to fit your stance is
Kripke's criticism of Russellian description
theory. The causal theory of naming that
he gave to explain his criticism in "Naming
and Necessity" would characterize your use
of a name in such a way so as to say:

A definition is a authorial dubbing of an
intension with a meaningful expression of
the language.

So, we have intensions...
http://en.wikipedia.org/wiki/Intension

...and the semiotics corresponding with
observable sign vehicles.
http://en.wikipedia.org/wiki/Sign_%28semiotics%29#Dyadic_signs

There is the consequent philosophical form arising
from the work of Saussere and subsequent responses.
http://en.wikipedia.org/wiki/Structuralism
http://en.wikipedia.org/wiki/Post-structuralism
http://en.wikipedia.org/wiki/Deconstruction

You probably cannot involve yourself with
deconstruction.
http://en.wikipedia.org/wiki/Authorial_intentionality#Post-structuralism

You probably cannot involve yourself with
post-structuralism.
http://en.wikipedia.org/wiki/Post-structuralism#Destabilized_meaning

So, you are left with...
http://en.wikipedia.org/wiki/Structuralism

...because you certainly wish to preserve authorial
intentionality...
http://en.wikipedia.org/wiki/Authorial_intentionality

...as you "know reality" and others should learn it.

However, there can be problems with intentionality
http://en.wikipedia.org/wiki/Intentionality#Is_Intentionality_discourse_a_problem_for_science.3F

so that you will probably end up at the Quinean double
standard
http://en.wikipedia.org/wiki/Indeterminacy_of_translation

Now, modern mathematics -- in so far as it accepts
various versions of set theory as foundational -- attempts
to circumvent many of these issues by claiming an
extensional rather than intensional foundation.

It does not really succeed in this because it
has not fully dealt with the issue of definability
in terms of definite descriptions. Historically,
mathematicians do not like to think about the
issue of definability. But, that is a denial in
the sense that the modern logical foundations
will eventually force them to come to terms with
those issues.

Woodin's ideas involving truth persistence under
forcing (see van Frassen for truth persistence
generally) are not classical ideas of truth.
They are perfectly legitimate philosophically and
are probably the best that any set theory using
extensionality as an axiom can do.

But, your statement:

"A definition is a name"

does not quite satisfy the expectations of
my question concerning how a name manages
to inform "distinguishability".

> Extensionality says that sets with same elements are identical.

There are several interpretations of this statement.

Are you assuming that identity is non-eliminable as on
page 5 of Thomas Jech's "Set Theory" where the "standard
acccount of identity" is being presumed?

http://books.google.com/books?id=pLxq0myANiEC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Kunen also assumes a prior standard account of
identity

http://books.google.com/books?id=S_ASNtsaynYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

(The standard account of identity can be found here)
http://plato.stanford.edu/entries/identity-relative/#1

Are you assuming that identity is eliminable as described
by Willard Quine on pages 14 and 15 of "Set Theory and
Its Logic" where identity is eliminated by virtue of
syntax?

http://books.google.com/books?id=S_ASNtsaynYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

If you are making this assumption, what justifies
your presumption that the singular terms of your
language satisfy the purport of singular reference,
and, how do you justify any form of informative
identity such as illustrated by "0.999...=1.000..."?

Are you assuming a meta-linguistic interpretation
of identity as in Zermelo's original paper?

"If two symbols, a and b, denote the same
object, we write a=b, otherwise -(a=b)."

Observe, however, that this does not get you
out of any argument leading to "dubbings"
in the sense of Kripke. Zermelo goes on
to clarify the exact nature of his usage here
with respect to his assertion that a
singleton exists for every domain element

"..., there exists a set {a} containing
a and only a as its element;"

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

Furthermore, observe that if you try to
choose Zermelo's option here, then any naive
semantics you apply will force you to
address my other observations involving
Leibniz' law and Cantor's intuitions
concerning a nested sequence of closed
subsets having vanishing diameters converging
to a non-empty set having only a singleton
as element. That interpretation involves
an implicit use of completed infinities -- one
which Cantor merely made explicit.

> Therefore it must be possible, in ZF, to fix whether elements are same
> or not. Therefore it must be possible to compare and to recognize
> elements. For that sake you need names, unless the elements are
> material objects. That's part of ZF.

In his 1908 paper, Zermelo does not say that every
object of his domain has a denotation. He merely gives
necessary conditions for dealing with two given
denotations purporting reference to objects of the
domain.

He does speak of "definiteness" and had been criticized
for not being clear on this matter.

In that regard, he says,

"A question or assertion F is said to be definite if
the fundamental relations of the domain, by means of
the axioms and the universally valid laws of logic,
determine without arbitrariness whether it holds or
not."

Modern set theory has followed Skolem in clarifying
the statement concerning "definiteness" (although
the choices made along these lines may not necessarily
be attributable to Skolem given what van Heijenoort
says in his introductory remarks).

Skolem writes:

"By a definite proposition we now mean a
finite expression constructed from elementary
propositions of the form aeb or a=b by means
of the five operations mentioned"

Those five operations to which he refers are
conjunction, disjunction, negation, universal
quantification, and existential quantification.

Thus, your claim concerning ZF -- in so far as
it corresponds with any information available
to me -- is inaccurate. It leads to standard
forms of logic of which you and others have been
critical.

By "standard forms of logic" I mean the compositionality
of formal statements involving a set of primitive
logical connectives necessary to conform with the
grammatical requirements of a formal deductive
calculus. If you wish to disagee with this,
we can toss Skolem out with the rest. But, both
Jech and Kunen formulate their set theory on
the basis of what Skolem described.

In Jech it is on page 2

http://books.google.com/books?id=pLxq0myANiEC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

In Kunen, it is on page 3

http://www.amazon.com/Introduction-Independence-Studies-Foundations-Mathematics/dp/0444868399#reader_0444868399

One can still debate the interpretation of
the quantifiers. However, it is simply irresponsible
to say that something has been done incorrectly and
should be ignored without providing an alternative.

An example of the appropriate way to proceed is
to be found in the Markov excerpts in

news://news.giganews.com:119/qtmdnXtM3_iSVqPMnZ2dnUVZ_sadnZ2d@giganews.com

Or, one can argue concerning what it means
specifically for the objects of a domain to
be "definable in principle" as a way to address
the relationship of names to the model theory
of sets. I did this in

news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdnZ2d@giganews.com

But, like most critics, you do not believe
some aspect. So, you engage in debate along
the lines of an agenda rather than along
the lines of a search for foundational principles.