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Topic: Surprise at my failure to resolve an issue in an elementary paper by Rado
Replies: 44   Last Post: Nov 10, 2013 12:23 PM

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Surprise at my failure to resolve an issue in an elementary paper

Posted: Nov 4, 2013 4:27 AM

On Monday, November 4, 2013 5:18:16 AM UTC, fom wrote:
> On 11/3/2013 10:54 PM, fom wrote:
>

> > On 11/3/2013 10:43 PM, fom wrote:
>
> >
>
> >
>
> > <snip>
>
> >
>
> >
>
> > With the following definitions, Rado's
>
> > proof is correct.
>
> >
>
> > It assumes an implicit biconditional
>
> > as is common with definitions.
>
> >
>
> > It does not assume an exclusive index
>
> > at which values may vary.
>
> >
>
> > The definition interprets the
>
> > "whenever ... then" statement
>
> > as
>
> >
>
> > Ax( P(x) -> Q )
>
> >
>
>
>
> Whoops
>
>
>
> Ax( P(x) <-> Q )
>
>
>
> I had just been thinking of the
>
> quantifier issue when I wrote that.
>
> This would be the formula with the
>
> implied biconditional of definition
>
> as well.
>
>
>

> >
>
> > Corrected property definition
>
> > =============================
>
> >
>
> > P(k) <-> (
>
> >
>
> > k in { 1, ..., r }
>
> >
>
> > /\
>
> >
>
> > AmAn(
>
> >
>
> > [ ( ( < m > = < a_1, ..., a_r > /\ < n > = < b_1, ..., b_r > )
>
> >
>
> > /\
>
> >
>
> > ( m subset B' /\ n subset B' ) )
>
> >
>
> > /\
>
> >
>
> > Ai( ~( i = k) -> ( a_i = b_i ) )
>
> >
>
> > /\
>
> >
>
> > ( a_k = b_k ) )
>
> >
>
> > <->
>
> >
>
> > f( m ) =/= f( n ) ]
>
> >
>
> > ) )
>
> >
>
> >
>
> >
>
> >
>
> > Negation of
>
> > corrected property definition
>
> > =============================
>
> >
>
> >
>
> > ~P(k) <-> (
>
> >
>
> > ~( k in { 1, ..., r } )
>
> >
>
> > \/
>
> >
>
> > EmEn(
>
> >
>
> > [ ( ( ~( < m > = < a_1, ..., a_r > /\ < n > = < b_1, ..., b_r > )
>
> >
>
> > \/
>
> >
>
> >
>
> > ~( m subset B' /\ n subset B' ) )
>
> >
>
> > \/
>
> >
>
> > Ei( ~( i = k) /\ ~( a_i = b_i ) )
>
> >
>
> > \/
>
> >
>
> > ~( a_k = b_k ) )
>
> >
>
> > <->
>
> >
>
> > f( m ) =/= f( n ) ]
>
> >
>
> > ) )
>
> >
>
> >
>
> >
>
> >
>
> >
>
> >
>
> > Defintion at beginning of paper:
>
> > ================================
>
> >
>
> > Let L subset {1, 2, ..., r} be given.
>
> >
>
> > Let A and f:[A]^r :=> F be given
>
> >
>
> > LCB(x) denotes "x is L-canonical on B"
>
> >
>
> > ===================================================
>
> >
>
> > LCB(x) <-> (
>
> >
>
> > B subset A
>
> >
>
> > /\
>
> >
>
> > AmAn(
>
> >
>
> > ( ( < m > = < a_1, ..., a_r > /\ < n > = < b_1, ..., b_r > )
>
> >
>
> > /\
>
> >
>
> > ( m subset B /\ n subset B ) )
>
> >
>
> > ->
>
> >
>
> > ( f( m ) = f( n ) <-> Ak( k in L /\ a_k = b_k ) )
>
> >
>
> >
>
> > ) )
>
> >
>
> >
>
> >
>
> >

fom,

Assuming that your definition of L is consistent with your definition on the original thread, I fully agree that it enables me to get past the blockage that I initially complained about.

Have you read the rest of the proof to ensure that your definition of L doesn't create problems further on in the paper?

[I'm absolutely not suggesting that you haven't. This is a genuine question to which I don't know the answer.]

Paul Epstein

Date Subject Author
11/3/13 Paul
11/3/13 David Hartley
11/3/13 fom
11/3/13 fom
11/3/13 fom
11/4/13 fom
11/4/13 Paul
11/4/13 Paul
11/4/13 Peter Percival
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/6/13 Paul
11/6/13 Paul
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/8/13 Paul
11/8/13 David Hartley
11/7/13 Paul
11/7/13 fom
11/8/13 Paul
11/8/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 David Hartley
11/10/13 Paul
11/4/13 Paul
11/4/13 Peter Percival