So we have seen that our constructedLF P model hQ, <, νi can be extended to a new model
hR, <, νi while preserving the truth ofLF P formulas on hQ, <, νi. Now let us consider this
procedure forhQ, τ, νiand hR, τ, νi in the languages L andL2F P.
Let us first consider extendinghQ, τ, νi tohR, τ, νi inL. Unlike the temporal case, we know thatS4is the logic of bothhQ, τiand hR, τi, so we only need to fill in hQ, τi withS4 maxi-
mally sets. Otherwise, let us proceed as above.
Assume ∆ 0S4 χ and let hQ, τ, νi be our constructed topological model which satisfies ∆∪ {¬χ}. Extend hQ, <i to a structure hR, <i ∼= hR, <i. Now associate each x ∈ R\Q with Γx, an S4maximally consistent extension of:
Γ ={2ψ| ∃q1, q2 ∈Q such thatq1 < x < q2 and ∀q ∈Qsuch that q1 < q < q2,ψ∈Γq} ∪
{3ϕ| ∀q3, q4 ∈Q such thatq3 < x < q4,∃q0 ∈Q such thatq3 < q0 < q4 and ϕ∈Γq0} Lemma 3.28 hR, τ, νi, r|=ϕ⇔ϕ∈Γr.
Proof. By induction on the complexity ofϕ. We will only treat the caseϕ=2ψ.
[⇐] Assume 2ψ ∈ Γr. We claim that ∃q1, q2 ∈ Q such that q1 < r < q2 and ∀q ∈ Q,
q1 < q < q2,ψ∈Γq. Ifr ∈Q, this holds by the construction of hQ, τi . And ifr ∈R\Q this
follows from the definition of Γr (since3¬ψ /∈Γr).
Now assume towards a contradiction that there is some r0 ∈ R\Q such that q1 < r0 < q2
and¬ψ∈Γr0. Then by maximal consistency 2ψ6∈Γr0. It then follows from the definition of Γr0 that for all q3, q4 ∈Q such thatq3< r0 < q4, there exists some q0 ∈Q , q3< q0 < q4, and
¬ψ∈Γq0. But this means there is aq0∈Qsuch thatq1 < q0< q2 and¬ψ∈Γq0, contradicting the previous paragraph.
So for all x ∈ R such that q1 < x < q2, ψ ∈ Γx. By letting U = {y | q1 < x < q2}
and the inductive hypothesis we get thathR, τ, νi, r|=2ψ.
[⇒] Assume2ψ6∈Γr. We claim that∀q3, q4 ∈Qsuch thatq3 < r < q4,∃q0∈Q,q3 < q0 < q4,
and ¬ψ ∈ Γq0. If r ∈ Q this holds by the construction of hQ, <i and if r ∈ R\Q then this holds by definition of Γr (since 2ψ 6∈Γr). So it follows from the inductive hypothesis that
hR, τ, νi, r6|=2ψ. qed
In order to show our constructed modelhQ, τ, νican be extended tohR, τ, νiwhile preserving the truth ofLformulas, it only remains to be shown that Γ is consistent. Unfortunately, this can not be done. For unlike the previous cases, our natural syntactic candidate to associate with a new point may be inconsistent. For example, assume x ∈ R\Q and hQ, τi has the valuation ν(p) = {q | q < x}. Then according to the definition of Γx, 3p and 3¬p are in
Γx. Furthermore, for every q ∈ Q, either q |= 2p or q |= 2¬p, so 2(2p∨2¬p) ∈ Γx. But
`S42(2p∨2¬p)→ ¬(3p∧3¬p) and this entails that ¬(3p∧3¬p)∧(3p∧3¬p)∈Γx
So already things are worse than the temporal case, but one might still hope that they are better than using the model theoretic approach. As mentioned before, the completeness ofS4with respect tohR, τican be established model theoretically, but with tricky diversions
through finite transitive and reflexive trees and subsets of the Cantor Space. As we have seen, using the construction method we have some control over how formulas are satisfied onhQ, τi
and perhaps we can make use this control when jumping tohR, τi.
Looking at the failure of Γ, it is clear what property we want our constructed modelhQ, τ, νi
to possess: if there are rationalϕ1, .., ϕn sequences approaching an irrational point x, then
there is a rational sequenceϕ1∧...∧ϕn approaching x. We can then show Γ is consistent.
For assume not. Then there are formulas2ψ,3ϕ1, ...,3ϕn such that
- ∃q1, q2 ∈Q,q1< x < q2 and for allq0 ∈Q,q1< q0 < q2,ψ∈Γq0
- ∀q3, q4 ∈Q,q3 < x < q4, there exists some qi0 ∈Qsuch that q3 < qi0 < q4 and ϕi ∈Γq0
i,
for 1≤i≤n
- `S4 2ψ→ ¬(3ϕ1∧...∧3ϕn)
But as there are rational sequencesϕ1...ϕn approachingx, by assumption there is a rational
sequence 3(ϕ1∧...∧ϕn) approaching x. This implies there is some rational point q0 in be-
tweenq1 and q2 such that: 2ψ,3ϕ1, ...,3ϕn∈Γq0.
However, after some reflection, it is clear that the desired property of hQ, τ, νi is (at least) very difficult to ensure. For while we have control on how formulas are satisfied on the ra- tionals, this tells us very little about how they will line up on the irrationals. There is no natural way to force a construction ofhQ, τ, νi to tell us what kinds of sequences are being created towards irrational points.
The underlying difficulty of extending hQ, τ, νi to hR, τ, νi in L is that there is no way of
expressing any topological relation between the points in the language. We can see this de- fect clearly when comparing our failed attempt in hR, τi to the successful temporal proof for hR, <i. In the the temporal construction, each state of affairs expressible in RFP (i.e.
each maximal consistent set) is related to every other state of affairs in a way that respects order structure onhQ, <i. Furthermore, each description of an RFP state of affairs existing inhQ, <i says that if there is a boundedϕsequence, then this ϕsequence has a least upper
bound respecting the ≺ ordering. Thus when extending hQ, <i to hR, <i, we know we can associate an irrational point x with its natural syntactic candidate Γx, since this state of
affairs has already been described to exist inhQ, <i.
The temporal construction for hR, <i goes so smoothly because there is a kind of expres- sive harmony between what can be said to occur in the language and the structure hR, <i.
Clearly this is not the case inhR, τi, where we are working with a fairly rich topological struc- ture and a language that can express almost no topological properties of the structure. In this sense, one might think that the seemingly more difficult construction of hR, τiinL2F P would
fare better, since additional topological properties such as connectedness are expressible in
L2F P. However, L2F P is still to weak of a language for things to go smoothly. In order to
successfully extend hQ, τi to hR, τi we need to be able to express in the language when a
particular 2 stretch ends, which is a much stronger property than connectedness.