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Topic: Matheology § 288
Replies: 160   Last Post: Jun 21, 2013 8:42 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Matheology § 288
Posted: Jun 16, 2013 11:38 AM
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On 6/16/2013 5:47 AM, mueckenh@rz.fh-augsburg.de wrote:
> On Sunday, 16 June 2013 11:25:27 UTC+2, fom wrote:
>> But, you respect neither mathematics based upon axioms nor logic based upon
>
> contradictions. Like the undefinability of elements and extensionality:
>
> "Eventually, most mathematicians came to accept that definability should
> not be required, partly because the axiom of choice leads to nice

results,
> but mostly because of the difficulties that arise when one tries
> to make notion of definability precise." (Andreas Blass)
>
> That is a real surprise to me. Which mathematicians accepted that and when?


I must concede to you on this one. Tarski wrote a paper
on definability in which he made the observation that
mathematicians are not too keen on the subject.

> Was there a public meeting with voting like in meta or like
> in the astronomy scene when Pluto has been degraded?
>


I spent a great deal of time trying to discern the origins
of "undefined language primitives" in the literature. Of
course, I have only limited means and a handful of translations.

The evidence I have been able to gather directs attention,
primarily, to Bolzano and his search for a definition of
simple substance.

The Aristotelian class hierarchy has two directions. Aristotle
asserts that genera are prior to species. Hence, his view of
class organization is a downward-directed view:

genus -> species -> individual

But, when Aristotle speaks of substance, he asserts that primary
substance is associated with individuals, secondary substance is
associated with species, and so forth. Hence, the notion of
substance is an upward-directed view:

individual -> species -> genus

Based on Leibniz' remarks, it seems that Aquinas asserted that
there are enough "properties" so that God can know every
individual. The generalization of this is Leibniz' principle
of identity of indiscernibles. But, in discussing his views
on logic, Leibniz contrasts himself with the Scholastic
tradition. Leibniz associates his views with the
downward-directed view of Aristotelian origin:

genus -> species -> individual

Thus, one must surmise that the Scholastic view is an
upward-directed view:

individual -> species -> genus

Let me call the upward-directed view "extensional" and
the downward-directed view "intensional". These are standard
terms to the best of my knowledge.

It is important to remember that Bolzano is historically
prior to the modern compositional logical systems. The
kind of definitions that Bolzano may have wished to consider
would be of the form,

"A rational man is a man"

This definition segregates the genus 'man' into the 'rational men'
and the 'irrational men'. Such is the general problem for this
syllogistic logic. In order to satisfy the general Aristotelian
requirement that "truth is the result of division and combination",
one is confronted by the fact that individuals cannot be divided
and combined.

To make matters worse, the notion of priority with regard to
language terms in definition appears to have already been
established. Thus,

"A rational man is a man"

had been admissible to Leibniz. But, for Bolzano it could not
have been because of the circular use of the term 'man'. Rather,
something along the lines of

"A bachelor is an unmarried man"

would have been more like what he considered a definition.

This kind of logic had been inappropriate for the definition of
individuals in relation to the extensional, Scholastic view that
Bolzano had been trying to implement.

Bolzano then goes on to argue for undefined language terms.

There is a second aspect to this that is discussed in the
work of De Morgan.

Specifically, the introduction of novel number systems such
as the complex numbers and the quaternions forced mathematicians
to accept the fact that arithmetical operations could be
applied more generally -- or, at least, more abstractly -- than
had been previously considered. Thus, 'number systems' begin
to be understood with respect to stipulations rather than some
intrinsic metaphysical explanation of number.

De Morgan recognized what we would now call semantic indeterminacy.
So, even familiar operations between numbers presuppose the
interpretation of abstract symbols. It is a simple step to
correlate this with Bolzano's arguments.


> Wouldn't a set with undefined elements contradict the Axiom of Extensionality:
>
> If every element of X is an element of Y and every element of Y is an element of X, then X = Y.
>
> How could that be decided for undefined elements?
>


For that I have no answer. This is, in fact, where I have non-standard
views. In my version of foundations, definability is fundamental. Of
course, I am using this notion differently from you. But, for
example, my theory begins with

AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

where the transitive, irreflexive order relation of 'proper part' is
prior to the 'membership' relation because the latter depends upon
the former for its definition.

In a modern interpretation, my symbols are still undefined and the
sentences above are axioms. In this view of things, any set of
axioms constitute "definitions-in-use" as opposed to the traditional
expectation whereby a defined symbol (definiendum) is related to
its definition (definiens) by a substitutivity criterion. When the
non-circularity criterion is applied (as with Bolzano) this traditional
expectation permits elimination of defined language symbols until the
only expressions remaining have undefined language symbols as
constituents.

When I say my view is different from yours, I do not care if
definitions are merely recognized in principle. So, I am
not restricting the idea to some countable set of terms. What
I consider important is to understand that the semiotics of
naming imposes a well-ordering criterion on the admissibility
of models -- to be a unique identifier, each name is restricted
from being the same as a prior name.

Logic uses names rather than numbers. Because of this, one
cannot distinguish between ordinal numbers and a unique system
of names. So, how can one speak of an inner model that cannot
be put into correspondence with the ordinal numbers?

> But my actual question is this: I have heard (but don't remember where) that there is another
> solution: The set of finite definitions is countable. That cannot be

explained away, can
> it?

Actually, it can.

That is, you are correct with regard to the limitations of
what can be expressed by "locally finite languages". But,
that is not what I am talking about.

What you are thinking of is the participation of the Lowenheim-Skolem
theorems with respect to the continuum hypothesis given by Goedel's
constructible universe (V=L).

If there is a model, then there is a countable model.

Cohen, acknowledging Shepherdson for the construction, formulates
a notion of "strongly constructible set" which, according to at
least one author, corresponds with a notion of provability concerning
the existence within the model.

I still have to look at these works more closely. My suspicion is
that this notion of provability corresponds with the notion of
provability associated with definability as discussed in Tarski's
paper mentioned above.

When I speak of what can be "explained away", I refer to the fact
that set theory ought to be logically prior to model theory. So,
I have deep reservations concerning the "model theory of set theory"
as it has been applied to prove the independence of the continuum
hypothesis.

This is why I make the distinction between set theory as a
foundational theory and set theory as "just another theory".

Since you previously cited pages from van Heijenoort, I will assume
you have it. You should look at Skolem's papers. One of them
will speak about the formability of a countable model.


> But not every finite definition has a meaning. In fact, if we
> refrain from using common sense, we cannot even define definability,

let alone the
> set of meaningful definitions.

Husserl: "What is the meaning of meaning?"

The ideas of model theory arise from the use of examples to
substantiate definitions and the use of counter-examples
to discount the universality of statements. Model theory
addresses the same questions in terms of "systems".

Unfortunately, the drive for foundations is fuel for the
skeptics of every breed.

When I finally turned to examine Aristotle and Leibniz, I understood
that the notion of definition is posterior to the deductive calculus.
Whatever linguistic analysis identifies the nature of what
transformations constitute the steps in a proof also identifies
what is admissible as a definition. In contrast to modern views,
Aristotle admits a number of notions of definition. The one
of particular relevance to my statements here are the ones he refers
to as "immediate principles".

Both of my sentences above correspond with a deductive calculus
as required by Aristotle and exemplified in Leibniz.

> Thereforethis set is not countable
> but subcountable - and if we identify subcountability with
> uncountability, we have won and can continue to enjoy the nice

results of
> the axiom of choice.
>
> Obviously to vague formulated as that matheologians could understand it - with
> their precisely defined definitions. Therefore deleted in MO
> after an hour.
>


Since I try to understand these matters with as little
deviation from classical logic and classical mathematics
as possible, I doubt that I would draw your conclusions.

But, do you have a link to where you discuss/define "subcountable"
with the intention of the interpretation you give above?

And, sorry about the long reply. I think my life would have
been easier if a committee led by Andreas Blass spoke for
all mathematicians concerning definitions and definability.

I am sure their press conference would have been on Pluto.







Date Subject Author
6/14/13
Read Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology § 288
LudovicoVan
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology § 288
LudovicoVan
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
apoorv
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
apoorv
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
apoorv
6/17/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
apoorv
6/17/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology � 288
Tanu R.
6/17/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology § 288
FredJeffries@gmail.com
6/16/13
Read Re: Matheology § 288
apoorv
6/16/13
Read Re: Matheology § 288
apoorv
6/17/13
Read Re: Matheology § 288
FredJeffries@gmail.com
6/17/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
FredJeffries@gmail.com
6/18/13
Read Re: Matheology § 288
LudovicoVan
6/18/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/18/13
Read Re: WMytheology � 288
Virgil
6/18/13
Read Re: Matheology § 288
LudovicoVan
6/18/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/18/13
Read Re: Matheology § 288
LudovicoVan
6/18/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/18/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/18/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/19/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/19/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/19/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/19/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: WMytheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/19/13
Read Re: Matheology � 288
Virgil
6/19/13
Read Re: WMytheology � 288
Virgil
6/18/13
Read Re: Matheology � 288
Virgil
6/18/13
Read Re: Matheology § 288
fom
6/18/13
Read Re: Matheology § 288
fom
6/18/13
Read Re: Matheology § 288
LudovicoVan
6/18/13
Read Re: Matheology § 288
fom
6/19/13
Read Re: Matheology § 288
LudovicoVan
6/19/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/19/13
Read Re: WMytheology � 288
Virgil
6/19/13
Read Re: Matheology § 288
LudovicoVan
6/19/13
Read Re: Matheology § 288
fom
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology § 288
FredJeffries@gmail.com
6/20/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
Ralf Bader
6/21/13
Read Re: Matheology § 288
LudovicoVan
6/21/13
Read Re: Matheology § 288
LudovicoVan
6/20/13
Read Re: Matheology § 288
FredJeffries@gmail.com
6/20/13
Read Re: Matheology � 288
Virgil
6/20/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/15/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology � 288
Tanu R.
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology § 288
LudovicoVan
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology § 288
LudovicoVan
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology � 288
Tanu R.
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology � 288
Bergholt Stuttley Johnson
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
Tucsondrew@me.com
6/14/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology � 288
Tanu R.
6/15/13
Read Re: Matheology � 288
Bergholt Stuttley Johnson
6/14/13
Read Re: Matheology � 288
Virgil
6/14/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/14/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/15/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/17/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
fom
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology � 288
Virgil
6/17/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/17/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology � 288
Virgil
6/16/13
Read Re: Matheology § 288
mueckenh@rz.fh-augsburg.de
6/16/13
Read Re: Matheology � 288
Virgil
6/15/13
Read Re: Matheology � 288
Tanu R.

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