On 6/16/2013 5:47 AM, email@example.com 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.