Re: Richard's antinomy

rupertmccallum_at_yahoo.com
Date: 02/09/05 

Date: 9 Feb 2005 13:48:40 -0800

examachine@gmail.com wrote:
> Is there anybody familiar with Richard's antinomy, and how it was
> resolved by logicians?
>
> I noticed that there are alternative ways to formalize the argument,
> making it even more interesting from a metamathematics point of view.
I
> think this paradox was omitted in almost every resource that I have
> read. There is a rigorous description of the paradoxes in a book I've
> been reading but I didn't find a good enough web resource. All the
> sites I found either misrepresent the paradox or pull it under the
rug.
> I want to see a decisive solution or non-solution to it.
>
> Basically, the paradox is a generalization of Cantor's diagonal
> argument which seeks to show the absurdity of the method. Let me try
to
> tell it from memory with a few minor improvements to refurbish its
> precision. This paradox was contrived by Richard a hundred years ago.
>
> Let us think of a completed countable list L of _all_ real number
> definitions in (mathematical) English. The criterion for an
admissible
> definition is that it consists in a finite English text,
comprehensible
> to a mathematician (let's please assume mathematician in the genetic
> sense, not necessarily limited to limited human brains) that uniquely
> defines a real number. [*]
>
> A diagonal argument defines the following number and argues that it
> does not exist in L.
> S "Let number r be the diagonal of the list L such that .... " [where
> ... proceeds as in Cantor's diagonal argument]
>

Your list L consists of all the real numbers definable in some
language, say the language T. Then the definition of your real number r
will contain the phrase "the n-th digit of the n-th real number
definable in T". This phrase, however, will not be definable in T. We
can see that because the assumption otherwise leads us to a
contradiction.

This solution is perfectly mathematically rigorous. For example, take
the list of all reals definable in the language of second-order
arithmetic, call this language T. The notion "the n-th real number
definable in T" is not definable in T. The reductio ad absurdum of
Richard's antinomy shows this.

Relevant Pages

  • Re: Richards antinomy
    ... > think this paradox was omitted in almost every resource that I have ... > to a mathematician (let's please assume mathematician in the genetic ... language, say the language T. Then the definition of your real number r ... the list of all reals definable in the language of second-order ...
    (sci.logic)
  • Richards antinomy
    ... sites I found either misrepresent the paradox or pull it under the rug. ... to a mathematician (let's please assume mathematician in the genetic ... is impossible to loop out of a completed list L of reals. ... cannot be expressed in _any_ finite English sentence whatsoever. ...
    (sci.math)
  • Richards antinomy
    ... sites I found either misrepresent the paradox or pull it under the rug. ... to a mathematician (let's please assume mathematician in the genetic ... is impossible to loop out of a completed list L of reals. ... cannot be expressed in _any_ finite English sentence whatsoever. ...
    (sci.logic)
  • Re: Richards antinomy
    ... > think this paradox was omitted in almost every resource that I have ... > definitions in English. ... > to a mathematician (let's please assume mathematician in the genetic ... > is impossible to loop out of a completed list L of reals. ...
    (sci.logic)
  • Re: Richards antinomy
    ... > think this paradox was omitted in almost every resource that I have ... > definitions in English. ... > to a mathematician (let's please assume mathematician in the genetic ... > is impossible to loop out of a completed list L of reals. ...
    (sci.math)