fom
Posts:
1,968
Registered:
12/4/12


Re: Surprise at my failure to resolve an issue in an elementary paper by Rado
Posted:
Nov 3, 2013 11:43 PM


On 11/3/2013 9:09 PM, fom wrote: > On 11/3/2013 8:26 PM, David Hartley wrote: >
When negating the property definition certain issues raised earlier in the other thread become apparent
> > >> But he presumably intended to define L as the >> set of numbers rho < r which have the property >> >> there exist two ordered sets of r numbers differ only at the position >> indexed by rho which do not have the same image under f. >> > > And, we define sets how? > > { x  P(x) } > > > Paul is correct about the quantifier and wrong about failing > to interpret "whenever ... then" as a conditional. > > I see that you also used an existential quantifier in your > first assessment of the 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 ) ] > > ) ) > > > > >
First, as noted in the other thread, definitions often carry an implicit biconditional. The negation of the defining property does not make sense without it. I formulated the above definition with the conditional to be faithful to the actual words in the proof.
Second, as noted in the other thread, the restriction to a single differing index appears to be too strong. In the definition above, I used
Ai( ~( i = k) <> ( a_i = b_i ) )
since everyone appears to read the statement that way.
It's negation is
Ei( ( ( i = k) /\ ( a_i = b_i ) ) \/ ( ~( i = k) /\ ~( a_i = b_i ) ) )
Compare this with
Ai( ~( i = k) > ( a_i = b_i ) )
having negation,
Ei( ~( i = k) /\ ~( a_i = b_i ) )
This is a better choice in the full context of the negated property in comparison with the definition of being Lcanonical over B.
The definition with only a conditional fails because negations of formulas such as
AmAn( p > q )
proceed as
~Am~( ~An~( ~( p > q ) ) )
EmEn( p /\ ~q )
You can see that this is a problem with:
Negation of predicate with conditional ======================================
~P(k) <> (
~( k in { 1, ..., r } )
\/
EmEn(
[ ( ( ( < 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 ) ]
) )
In contrast, consider the negated property with both changes,
Alternate negated property with biconditional ==============================================
~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 ) ]
) )
The key expressions in this alternate definition are
Ei( ~( i = k) /\ ~( a_i = b_i ) )
\/
~( a_k = b_k ) )
In other words, if k is not in L, then some index  not necessarily k  must force
f( m ) =/= f( n )
To see why this is the intended interpretation, look at the biconditional in the definition for being Lcanonical over B from the beginning of the paper.
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 Lcanonical 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 ) )
) )

