Re: Significance of antinomies

From: Owen (oorionus_at_yahoo.com)
Date: 02/09/05 

Date: Wed, 9 Feb 2005 08:22:07 -0500

  <examachine@gmail.com> wrote in message
news:1107945326.696128.42880@z14g2000cwz.googlegroups.com...
> Recently Chapman "dismissed" Cantor's antinomy by saying that it only
> shows that such a thing as "the set of all sets" is an incoherent
> notion and should not be pursued, while it may be considered a valid
> objection to say that "the set of all sets does not exist" I think this
> is undermining this antinomy as well as Russell's antinomy which is
> connected.

  Nonsense, the set of all sets cearly exists, simply because it contains
some set!
  Fx -> E!x, ExFx -> E!x. AxE!x. !!

>
> This is not a good case to make of antinomies in the history of set
> theory. How do we decide that its use should be restricted?

  It is always the case that: Ey(Ax(x e y <-> Fx)) if we assume a theory of
types.

   (e.g. that
> is an invalid set definition) That is the very question that paradoxes
> like Cantor's and Russell's paradoxes bring by presenting us
> *contradictions* if they were allowed.

  What do you mean by 'allowed'?
  Of course, E!{x:~(x e x)} is contradictory.
  Contradictions, paradoxes, show that we have made a logical mistake.

  Logic abhors contradiction, I think Russell said that.
  That our reasoning brings us to a contradiction, allows us to say
  it is contradictory, i.e. (p -> F) <-> ~p.
  That is, there cannot be a situation that is contradictory.

   So, we cannot out of hand
> "dismiss" those notions, but have to restructure the set theory such
> that *none* of these paradoxes appear. That's why we have various
> foundational theories, or otherwise Cantor's naive set theory would
> have been sufficient.

  Ey(Ax(x e y <-> Fx)) is valid, true for all F, in a theory of Types,
including Quines's Stratification.

  Clearly, ~EyAx(x e y <-> ~(x e x)) ..is tautologous, with or without
Types.
  (p <-> ~p) is false.

  It follows from: ~Ey(Ax(xRy) <-> ~(xRx)), for the R (e).
  And for every other R.
   |- ~EREyAx(xRy <-> ~(xRx)), resolves these paradoxes.

>
> Here is what Fraenkel, Bar-Hillel and Levy have to say in "Foundations
> of Set Theory" about the proper attitude towards the paradoxes. Section
> 1.1, Historical Introduction. I think it's worth reading, at least for
> the beautiful wording.
>
> -------------------------------------------------------------------------------------------
>
> ....
>
> Russell's antinomy came as a veritable shock to those few thinkers who
> occupied themselves with foundational problems at the turn of the
> century.Dedekind, in his profound essay on the nature and purpose of
> the numbers, had based number theory on the membership relation ... and
> had utilized the notion of a set in its full Cantorian sense for the
> proof of the existence of an infinite set. Under the impact of
> Russell's antinomy, he stopped for some time the publication of his
> essay, the fundaments of which he regarded as shattered. Still more
> tragic was Frege's fate: had just put the final touches on his chief
> work, after decades of tiresome effort, when Russell wrote him about
> his discovery. In the first sentence of the appendix, Frege admits that
> one of the foundations of his edifice had been shaken by Russell.
>
> It is not surprising that many mathematicians who had just begun to
> accept set theory as a full-fledged member of the community of
> mathematical disciplines reversed their attitude...
>
> Cantor himself, to be sure, did not for a moment lose faith in his
> theory in its full "naive" extent though he was unable to meet the
> challenge of Russell's antinomy. [Which is in fact connected to
> Cantor's antinomy -- Eray] Other scholars professed not to be
> especially disturbed by this and other antinomies and, distinguishing
> between "Cantorism" and "Russellism" warned against attributing to the
> "artificially constructed" antinomies any decisive significance. It is
> however difficult to defend this attitude. Even if Burali-Forti's
> antinomy does not appear as long as one restricts himself to the
> ordinals of a few number-classes, this cannot release the serious
> thinker from the obligation of scrutinizing the theorems that involve
> the *general* concept of an ordinal; and the contemptuous reference to
> the "artificial" character of many antinomies should be no more
> convincing than the claim, say, that every continuous function has a
> derivative since continuous functions without derivatives are
> "artificial". I may safely be stated that, on the contrary, throughout
> mathematics -- and other disciplines -- the investigation of the most
> general notions, in all their unrestricted generality, has often proved
> to be of extreme value for the advancement of research. To think that
> difficulties can be overcome simply by disregarding the general case is
> somewhat naive. Finally, to draw a sharp line between mathematics
> (which is fine) and logic (which a self-conscious mathematician should
> shun for the benefit of his soul) is less than useless: logic is
> constantly applied in mathematics,though this use is not often brought
> into the open and explicitly taken account, and if one wishes to put
> restrictions on this appliction, as some intuitionists do, it is better
> to formulate these restrictions openly and clearly rather than leaving
> them in dark.
>
> ---------------------------------------------------------------------------------------
>
> Also in 1.2.1 Russell's Antinomy
>
> ---------------------------------------------------------------------------------------------
> Let it be very clearly stated at the outset that there was absolutely
> nothing in the traditional treatments of logic and mathematics that
> could serve as a basis for the elimination of this antinomy. We think
> that all attempts to handle the situation without any departure from
> traditional, i.e. pre-20th century, ways of thinking have completely
> failed so far and are misguided as to their aim. Some departure from
> the customary ways of thinking is definitely indicated, though it is by
> no means clearly determined where this departure should take place.
> Indeed, 20th century research into the foundations of logic and
> mathematics can be fruitfully classified in terms of the place of
> departure from the Cantorian approach.
> --------------------------------------------------------------------------------------------
>

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