Semantics for ILMS
6. ARITHMETICAL COMPLETENESS FOR A SIMPLE LANGUAGE
the only?–formula inD, it is the only ?–formula whose forcing is defined in M. Mmight look as follows19:
w x: ¬(pq) y: p z: q u: q?q z’: r R Rq x Sw Sw Sw Sw
Consider the formula q?r whose forcing is initially undefined in the model. If we were to follow the strategy of the previous section in defining its truth value, the only node where we could make it true is y, as it is the only node that has
S-access to aq-node and anr-node (assuming thatqandrare only true atz and
z0respectively). However, supposing that we defineyq?r, we would need some
vwithySxvq, if axiomSwere to be valid atx(sincexRyandq?rq). However this contradicts the fact that y and all itsSx-successors are in the q-critical cone
abovex. Hence we cannot makeq?rtrue aty.
On the other hand, sincew(pq)∧(pr) andwRyp, there has to be somev
withySwvq?r(ifSis to be valid). But in order to havew(q?rr)∧(q?rq), thisvhas to furthermore have the property thatvSwzandvSwz0 (again, assuming
thatq andrare only true atz andz0 respectively). Except fory, there is no such node in the above model.
As a consequence, we cannot have an extension lemma as in the previous, where we just take an existingILMS–model restricted toDand, without making any changes in the underlying structure of the model, define the forcing of all ?–formulas. Whether an alternative route exists in the general case remains a question for future research.
6. Arithmetical Completeness for a Simple Language
In this section, we give an arithmetical completeness result forILMw.r.t. a very re- stricted set of formulas, namely the formulas constructed from a single propositional letter using only?. An important consequence of the result is that all formulas in
19We only indicate the forcing of formulas which are important for the example. We writexRq xy
6. ARITHMETICAL COMPLETENESS FOR A SIMPLE LANGUAGE 63
this simple language are arithmetically independent. To be more precise, there is an arithmetical implementation under which any two formulas of this language are even mutually inconsistent. This is already enough to establish that the arithmeti- cal suprema are in general not idempotent (w.r.t. provability) or extensional. This section presupposes knowledge of the arithmetical completeness proof ofILM. This proof was found independently by Alessandro Berarducci ([Ber90]) and Volodya Shavrukov ([Sha88]).
Let{Ai}0≤i∈ω be an enumeration of all ILMS–formulas constructed from a single
propositional letter pusing only ?. We assume that A0=p. Consider the rooted
ILM-frame F =hW, R,{S1}i withW =N\ {0}, 1Rifor all i >1, andiS1j for all
i, j > 1. SinceS1 is the onlyS-relation in F, we will from now on write just iSj instead ofiS1j. We define a forcing relation onFby letting20xAi:⇔x=i+ 2.
Thuspis true only at node 2,A1is true only at node 3, and so on. LetMbe the resulting model. Using that M has depth 2, and that all nodes R-above 1 have access to all other nodesR-above 1, it is easy to see that axiomSis valid inM. By the Berarducci-Shavrukov Arithmetical Completeness Theorem forILM,Mcan be embedded in PA. Let ∗ be the arithmetical realization that we obtain in the proof. Then we have thatp∗= (`= 2), and in general21A∗i = (`= (i+ 2)). I.o.w. the sentenceAi is represented in arithmetic as the sentence`= (i+ 2).
Since iSj for all i, j > 1, by the arithmetical completeness proof we get for all i
and j that `PA`= (i+ 2)≡`= (j+ 2). This means that all sentences ` =i for i >1 are in the same interpretability degree. As a consequence, if Ak =Ai?Aj, then the sentence`= (k+ 2) (the arithmetical representative ofAk) is in the same
degree as thereal supremum22ofA∗
i andA∗j, i.e. of`= (i+ 2) and`= (j+ 2).
Thus we can define an implementation ? of the supremum in PA as follows:
B?C:=BfC, unless B andC are of the form `= (i+ 2) and ` = (j+ 2) re- spectively for somei, j∈N. In that case, takeB?Cto be the sentence`= (k+ 2) for thekfor whichAk=Ai?Aj.
By the properties of the sentences`=i, we have that``= (i+ 2)→`6= (j+ 2) ifi6=j. It follows that ifi6=j, then 0`= (i+ 2)→`= (j+ 2), i.e.0A∗i →A∗j.
Thus all formulas in the small language are arithmetically independent. Note that the implementation ?defined above is certainly not extensional. For example, if ` A ↔ (` = i) but A 6= (` = i), and ` B ↔ (` = j) but B 6= (` = j), then it is clear that not necessarily `PA A?B ↔ (` =i)?(` =j). In order to obtain independence results for extensional implementations, a more elaborate procedure is therefore needed.
20Note that just as in Section 3.2, the
?-formulas are treated as atoms.
21The sentence`=i+ 2 can be said to “represent” the nodei+ 2 of the modelMin arithmetic. SinceAiis only true ati+ 2,Aiandi+ 2 are represented by the same arithmetical sentence.
CHAPTER 5