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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 ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Surprise at my failure to resolve an issue in an elementary paper

Posted: Nov 3, 2013 11:54 PM

On 11/3/2013 10:43 PM, fom wrote:

<snip>

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 )

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 ) )

) )

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