A proof that FVPn does not imply FVPn+1
E.Howarth∗ and J.B.Paris†
School of Mathematics
The University of Manchester
Manchester M13 9PL
[email protected],
[email protected]
September 11, 2015
The purpose of this note is simply to sketch a proof of the fact asserted in [2]
that FVPn does not imply FVPn+1 . We assume that the reader is already
fully familiar with [2].
This result for n = 1 is given in the main paper [2]. We shall give the construction in the case n = 2. The corresponding construction for n = 3, 4, . . . ,
is analogous but with co-linear replaced by co-planar etc.. An example for
n = 0 follows from the example given for n = 2 using the method described
below (or by an easy direct construction).
Let L be the language with a single ternary relation symbol R. Let M +
be the structure with universe the Cartesian plane and R interpreted as
holding between three points just if they are distinct and co-linear. Let M
be an elementary substructure of M + with countable universe {ei | i ∈ N+ }.
Notice that we can define equality in M + and M .
∗
Supported by a UK Engineering and Physical Sciences Research Council Studentship.
Supported by a UK Engineering and Physical Sciences Research Council Research
Grant.
†
1
For any points ei , ej , ek , er with i = j just if k = r, there is a translation plus
a rotation of the plane plus a scaling which sends ei to ek and ej to er . Since
this mapping is co-linearity preserving it gives an isomorphism of M + and
hence for any θ(a1 , a2 ) ∈ SL(2) ,
M + (or M ) |= θ(ei , ej ) ⇐⇒ M + (or M ) |= θ(ek , er ).
(1)
Also for any ei , ej , ek , er which form a proper quadrilateral with no parallel
sides we can define in M the point which is the intersection of the lines formed
by ei , ej and by ek , er and so on for the other disjoint pairs from ei , ej , ek , er .
In turn we can define the intersection points of the lines determined by the
original ei , ej , ek , er and the first intersection points to form some new ‘second’
intersection points, and so on. In general then we can find for each m ∈ N a
sentence θ(a1 , a2 , . . . , a5 ) ∈ SL(5) such that
M |= θ(ei , ej , ek , er , es ) ⇐⇒ es is an m’th but not an n’th
intersection point for any n < m.
(2)
Now define a probability function VM on L by
(
1 if M |= θ(e1 , e2 , . . . , en ),
VM (θ(a1 , a2 , . . . , an )) =
0 otherwise,
and in turn a further function w on SL by
w(θ(a1 , a2 , . . . , an )) =
X
VM (θ(aτ (1) , aτ (2) , . . . , aτ (n) )) ·
τ
n
Y
2−τ (i)
i=1
where τ runs over all maps from {1, 2, . . . , n} into N+ . By a theorem of
Gaifman, see [1] or [3, Chapter 26], w is a probability function on L satisfying
Ex. From (1) w(θ(a1 , a2 )) must be a sum of none, one or both of
X
X
2−τ (1)−τ (2) ,
2−2τ (1) ,
τ (1)6=τ (2)
τ (1)=τ (2)
so w satisfies FVP2 . However from (2) w must give non-zero probability to a5
being an m’th (but not n’th for n < m) intersection point from a1 , a2 , a3 , a4
so w does not satisfy FVP5 .
2
From this it follows that there is some 2 ≤ n ≤ 4 such that w satisfies FVPn
but not FVPn+1 . If n = 2 then we already have what we want. So suppose
n = 3 (the other case will be similar) and for notational convenience that ei
as above in (2) is e1 . Now extend the language L to L0 by adding relation
symbols R1 , . . . , R6 and form a structure M 0 for L0 with universe {ej | j > 1}
by interpreting the relation symbols as:
M 0 |= R(er , es , et ) ⇐⇒ M |= R(er , es , et ),
M 0 |= R1 (er ) ⇐⇒ M |= R(er , e1 , e1 ),
M 0 |= R2 (er ) ⇐⇒ M |= R(e1 , er , e1 ),
M 0 |= R3 (er ) ⇐⇒ M |= R(e1 , e1 , er ),
M 0 |= R4 (er , es ) ⇐⇒ M |= R(er , es , e1 ),
M 0 |= R5 (er , es ) ⇐⇒ M |= R(er , e1 , es ),
M 0 |= R6 (er , es ) ⇐⇒ M |= R(e1 , er , es ).
By induction on the length of θ we can now show that for θ(a1 , a2 , . . . , an+1 ) ∈
SL there is a sentence θ0 (a1 , a2 , . . . , an ) ∈ SL0 such that for any i1 , i2 , i3 , . . . , in >
1,
M |= θ(e1 , ei1 , ei2 , . . . , ein ) ⇐⇒ M 0 |= θ0 (ei1 , ei2 , . . . , ein ).
(3)
Conversely, using the fact that equality is definable in M , for any sentence
φ0 (a1 , a2 , . . . , an ) ∈ SL0 there is a sentence φ(a1 , a2 , . . . , an+1 ) ∈ SL such that
for any i1 , i2 , i3 , . . . , in > 1,
M 0 |= φ0 (ei1 , ei2 , . . . , ein ) ⇐⇒ M |= φ(e1 , ei1 , ei2 , . . . , ein ).
(4)
Now, mimicking the above construction, define the probability function VM 0
on L0 by
(
1 if M 0 |= ψ(e2 , e3 , . . . , en+1 ),
VM 0 (ψ(a1 , a2 , . . . , an )) =
0 otherwise,
and in turn a further function w0 on SL0 by
w0 (ψ(a1 , a2 , . . . , an )) =
X
VM 0 (ψ(aτ (1) , aτ (2) , . . . , aτ (n) )) ·
τ
n
Y
i=1
3
2−τ (i) .
Again w0 satisfies Ex. Also the set of values of the w0 (φ0 (a1 , a2 )) is finite by
(4) and the assumption that w satisfies FVP3 so w0 satisfies FVP2 . However
since w fails FVP4 the set of values of the w(θ(a1 , a2 , a3 , a4 )) is infinite so by
(3) the set of values of the w0 (θ0 (a1 , a2 , a3 )) is infinite and w0 fails FVP3 .
References
[1] Gaifman, H., Concerning Measures on First Order Calculi, Israel Journal of Mathematics, 1964, 2:1-18.
[2] Howarth, E. & Paris, J.B., The Finite Values Property, submitted to a
special edition of LNAI, Springer-Verlag.
[3] Paris, J.B. & Vencovská, A., Pure Inductive Logic, in the Association of
Symbolic Logic Perspectives in Mathematical Logic Series, Cambridge
University Press, April 2015.
4