fom
Posts:
1,968
Registered:
12/4/12
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Re: Surprise at my failure to resolve an issue in an elementary paper
by Rado
Posted:
Nov 4, 2013 12:18 AM
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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 ) )
>
>
> ) )
>
>
>
>
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