Collapsed models

Paraconsistent collapsed models of arithmetic are essentially structures obtained by quotienting classical models of arithmetic such that their elements are collapsed through an equivalence relation that is also a congruence relation on the operations of successor, addition and multiplication. The main difference here is that, unlike clas- sical quotient algebras of_N, the elements in a collapsed model gain the non-identity of its members: if [x] ={x, y}andx6=y, then [x]6= [x]. Though the techniques stem originally from the work of Meyer(1976), by far the most well-studied constructions have been elaborated by Priest(1997) and Priest(2000) using the logicLP.

For ease of symbolism, we let I denote the interpretation of the non-logical vocab- ulary in a model. Let ∼ be an equivalence relation on the domain M of a con- sistent LP-model M of _pa, such that it is also a congruence relation on the in- terpretation of the function symbols; that is, for n-ary function symbol f and ele- ments d1, ..., dn, e1, ..., en ∈ M, if di ∼ ei (for 1 ≤ i ≤ n) then I(f)(d1, ..., dn) ∼

I(f)(e1, ..., en). Given M and ∼, we call M/ ∼ the collapsed model of M under

∼. In order to construe a collapsed model, we will then have to define the domain of the new model and the denotation of the vocabulary through I∼.6 For this, we let the domainM/∼ ofM/ ∼to be the set of equivalence classes obtained by the partition defined onM under∼. That is,M/∼ = {[m]|m∈M}.7 We can define a collapsed interpretationI∼ _as:

• For every constantc, I∼(c) = [I(c)]

• For every n-place functionf,I∼(f)([d1], ...,[dn]) = [I(f)(d1, ..., dn)] 5_{Throughout we will mostly write|= for|=}

LP letting context disambiguate.

6_I∼_{corresponds to the interpretation in the collapsed model.}

• For every n-ary predicateP,h[d1], ...,[dn]i ∈I∼+[−](R) iff

for some e1∼d1, ...,en∼dn it followshe1, ..., eni ∈I+[−](R)

• I∼_{([x] = [y]) =I}∼_(x∼y) • I∼([x]6= [y]) =I(x6=y) As a consequence:

• I∼(0) = [I(0)]

• I∼(S([x])) = [I(S(x))] • I∼([x] + [y]) = [I(x+y)] • I∼([x]×[y]) = [I(x×y)]

We quickly note that the collapsed interpretation for function symbols is well-defined since∼is by assumption a congruence relation. Otherwise, there would be no natural way to define the collapsed interpretation; for instance, without congruence, we could not letI∼_{(f)([d]) = [I(f)([d])] (for u-naryf) sinced∼eand [d] = [e] need not imply} [I(f)([d])] = [I(f)([e])]. Now, the important point is thatI∼_{identifies all members in} the same equivalence class, producing a composite element (that is, the equivalence class itself) inheriting all the properties of its members (even if these properties are inconsistent). From the way the collapsed model is defined, the crucial Lemma follows:

Collapsing Lemma: For anyLP-interpretationI and arbitraryϕ:

vI(ϕ)⊆vI∼(ϕ)

vI, vI∼ are valuation functions under their respective interpretations.

Proof. The proof is by induction on the complexity ofϕ. See Priest(1991).

The Lemma guarantees that in collapsing a model no truths are lost. If in the original interpretation 1 ∈vI(ϕ) then 1 ∈vI∼(ϕ) (and similarly with 0), though there may

be formulasψsuch that{1,0} ∈vI∼(ψ).

Linear Models

In order to better understand the significance of the above result we want to introduce a special class of collapsed models calledlinear models. The following Definition and Lemma from classicalpa will prove themselves very useful.

DefinitionFor a classical modelMof pa, we callS⊆M a slice iffS is an initial section ofM(i.e. ifx < y andy∈S, thenx∈S) closed under successor, addition and multiplication. We say S isproper iffS6=_Nand S 6=M.

Lemma For every non-standard classical model M and slice S ⊂ M, there is a proper slice that extendsS.

Proof. Consider a non-standard model Mand let S be a slice such that S ⊂M. We want to show there is a proper slice that extendsS. First, we note thatNis the smallest slice ofMand soN⊆S. Takea6∈S; then, a is non-standard. Define

aN_{={m∈M | ∃n∈}

N:M |=m < an}

It is easy to see that aN _{is closed under successor, addition and multipli-}

cation. For the successor case consider x∈ aN_{; then, there is ann ∈}

N such thatx < an_{. Then, either (a)x+ 1 < a}n_{in which casex+ 1∈a}N_,

or (b) x+ 1 =an _{in which casex+ 1 < a}n+1 _{andx+ 1∈a}N_.8 _Similar,

for the other cases. So that aN _{is a slice.}

Define ϕ(x, a) = ∃x(x < a). It is clear that for every n ∈ N : M |= ϕ(n, a). From this latter fact and from Overspill, we know that there is a non-standardc∈M withM |=ϕ(c, a) so thatc∈aN_{. Hence, sincec6∈}

N we have N6=aN.

Also, it is clear that aN _{⊆ M. But, since for every n ∈}

N we have an _{< a}a _{it follows thata}a _6∈aN_andaN_{6=M. Hence, a}N_{is a proper slice}

that extendsS.

Consider a classical modelM |=_pa, standard or non-standard. Define forMa chain {Si | i ≤µwith 0 ≤µ ≤ ω} of strictly initial segments of M such that: Sµ = M

and Sj a slice (for 0 < j ≤ µ). It is important to note that given the way it is

defined S0 is not necessarily a slice, but only an initial segment of the model. Now,

for 0 < j ≤µ, defineCj =Sj−Sj−1. Also, for 0 < j≤µ, we let p1 be a non-zero

(possibly non-standard) number withp1∈S1 and if j < k, thenpj is a multiple of

pk. We define a relation∼such that:

DefinitionWe say that∼is a linear relation if it is of the form: x∼y iff

(x, y∈S0∧x=y)∨(for somei > 0 andx, y∈Ci:x=y (modpi))

Lemma ∼is an equivalence relation and a congruence relation with re- spect to successor, addition and multiplication.

Proof. That ∼is an equivalence relation follows easily from inspection of cases together with the fact that equality and congruence modulo n are themselves equivalence relations. For congruence:

• Successor: Assumex∼y. Then, either (a) x, y∈S0 or (b) for some

i > 0,x, y∈Ci. Suppose (a). Then,x=y. Hence,S(x) =S(y). If

S(x), S(y)∈S0, then S(x)∼S(y). If for some i > 0,S(x), S(y)∈

Ci, then S(x) = S(y) (mod pi), and S(x) ∼ S(y). Suppose (b).

Then,x=y (mod pi), from where it follows S(x) =S(y) (modpi).

Also, from the fact thatCi is closed under successor, it follows that

S(x), S(y)∈Ci. Hence,S(x)∼S(y).

• Addition: Assume x1 ∼y1 and x2 ∼y2. Then, either (a) x1, x2 ∈

S0∨y1, y2 ∈S0, or (b) for somei, j > 0 (not necessarily distinct), 8_{A better notion here for the successor ofxwould beS(x). We choose it to write it asx+ 1 to}

x1, x2 ∈ Ci ∧y1, y2 ∈ Cj. Suppose (a). Assume, without loss of

generality that the first disjunct holds; that is, x1, x2 ∈ S0. Then,

x1=x2. Now, ify1, y2∈C0theny1=y2, and thenx1+y1=x2+y2,

from where it follows, x1+y1 ∼ x2+y2, irrespectively of the Ci

or S0 that has them as elements. If, for some i > 0, y1, y2 ∈ Ci,

then x1+y1, x2+y2 ∈ Ci, for Ci is closed under addition. Then,

since modulo operation is congruent on addition, it followsx1+y1=

x2+y2 (mod pi) and, therefore, x1+y1 ∼ x2+y2. Suppose (b).

Then x1 = x2 (mod pi) and y1 = y2 (mod pj). Assume, without

loss of generality, i < j. Then, pi is by construction a multiple of

pj. Hence, x1 =x2 (mod pj). Hence,x1+y1=x2+y2 (mod pj).

Also, sinceCjis closed under addition,x1+y1, x2+y2∈Cj. Hence,

x1+y1∼x2+y2.

• Multiplication: (Same as Addition).

Definition We call anLP-model linear if it is obtained by collapsing a classical model under a linear relation.

Linear models have a tail comprising an initial segment of the original model followed by µ-many cycles such that for each cycle Ci (where i>0) the period is pi. For

instance, consider M a classical non-standard model of _pa and a chain of strictly initial segments S0 ⊆S1 ⊆S2 withS0={m|m < n} (for some finite n),S1=N andS2=M. Lettingp1, p2stand for the (finite) period of the cycles (withp1multiple

ofp2) andc non-standard, we consider the linear relation:

x∼y iff

• (x, y < n∧x=y) or

• (n ≤x, y < ω∧x=y (mod p1)) or • (x, y > ω∧x=y (modp2))

The relation produces a collapsed model M/ ∼with a tail isomorphic to an initial segment of _N of length n, followed by two cycles of period p1 and p2. Letting the

arrows represent the successor operation, the successor graph of the collapsed model is then9_: [0] [1] [2] [...] [n] [n+ 1] [...] [n+p1−1] [c] [c+ 1] [...] [c+p2−1]

Since the model has finitely many elements it is obviously finite and with a different order type than the standard model. Since by assumptionM |=_paby the Collapsing Lemma we have that M/ ∼ |= _pa. We then have an example of a finite model of arithmetic.

9_{We assume, for illustration, thatnis greater than 2. Of course, ifn= 2 orn= 1, the tail would}

For the rest of the paper we will be mostly interested in linear structures obtained by collapsing the standard model N. In building collapsed models of N, µ must either be 0 or 1, for the standard model only has one slice (i.e.Nitself). Ifµ= 0 we consider a single strict initial segment of the model, corresponding to its entire domain Nand buildN/∼by collapsing the initial structure with x∼y iffx, y∈N:x=y. The resulting structure will produce an isomorphic copy of_N, where each integer is raised to its type lift (that is, ifn∈_N, then [n] ={n} ∈_N/∼). A more interesting construction is lettingµ= 1. Here, there are two possibilities. IfS06=∅, the collapsed

model consists of a tail of finite length followed by a cycle also of finite period. The linear relation will be:

x∼y iff (x, y < n∧x=y)∨(x, y_>n∧x=y (mod p1))

And the successor graph will be:

[0] [1] [...] [n] [n+ 1]

[...] [n+p1−1]

The relation puts each integer up tonin its own equivalence class, producing a cycle of periodp1. It is easy to see that the model is inconsistent for

N/∼ |= [n] = [n+p1]∧[n]6= [n+p1]

The first conjunct follows fromn∼n+p1, whereas the second fromn6=N n+p1. The

collapsed model is non-trivial since it can be easily checked that N/ ∼ 6|= [0] = [1]. But again by the Collapsing Lemma, N/ ∼ |= pa. We will call such models with finite tail followed by a cycle, aheap model.

If S0 =∅, the collapsed model will only be formed by a cycle also of finite period.

Again as an example, we consider collapsingN under: x∼y iff (x=y (mod p1))

The successor graph ofN/∼may be depicted as:

[0] [1]

[...] [p1−1]

It is also straightforward to see how this new N/ ∼is inconsistent but non-trivial. Further, again, N/ ∼ |=_pa. We will call such models with a single cycle, a cyclic model.10 _{In fact, heap and cyclic models are the only ‘interesting’ collapses of the}

standard model11 – any interesting equivalence relation must identify two different numbers; supposing j to be the least inconsistent number (i.e. the least number identified with another number) andkthe least number greater thanjthat is identical

10_{Interestingly, these were the first kind of inconsistent models studied in Meyer(1976).} 11_{By ‘interesting’ we mean a model not-isomorphic to the standard model.}

with it, we then have thatS0 ={m|m < j} andp1 =k−j. This means that the

collapse produces alinear model.

We then have the following structure for finite linearLP-models. The general struc- ture is a finite tail, followed by a cycle of standard numbers (or, better, their type lift) and a finite collection of cycles of non-standard numbers. Having surveyed some im- portant kinds of inconsistent models and the underlying logic, we move to a problem in comparing them.

Isomorphic LP-Models

We think that there might be a small complication for the argument we will want to put forward later that we should better tackle with now. Recall that when defining anLP-structure we must associate not only an extension but also an anti-extension to each relation symbolR. However, when presenting a classical first-order structure we only associate an extension to R, for the anti-extension may be easily obtained by taking the complement. But then, if anLP-structure imposes different conditions on the interpretation of the relation symbols, how should we capture the notion of ‘isomorphism’ when paraconsistent structures are involved? To see why this question is not at all trivial let us consider the classical case for a bit.

Classicaly, for a function to establish an isomorphism between two modelsMandN it must be the case that12

• Forn-aryL-relation symbolR, (m1, ..., mn)∈RM⇔(π(m1), ..., π(mn))∈RN

(form1, ..., mn∈M)

The clause only concerns the extension ofR so a better formulation would be: • For n-ary L-relation symbol R, (m1, ..., mn) ∈ R+

M

⇔ (π(m1), ..., π(mn)) ∈

R+N _(form

1, ..., mn ∈M)

Since the biconditional is equivalent to (m1, ..., mn)6∈R+

M

⇔(π(m1), ..., π(mn))6∈

R+N and since (assuming consistency) a ∈ R+M ⇔ a 6∈ R−M, the anti-extensions will also agree. That is, in the classical case, an isomorphism also guarantees that (m1, ..., mn)∈R−

M

⇔(π(m1), ..., π(mn))∈R−

N

. But this doesn’t need to happen in the LP-case. Now, a classical consequence of there being isomorphic models is elementary equivalence: M ∼= N ⇒ M ≡ N. But we find that with the classical notion of isomorphism in place, we are not guaranteed that two isomorphic models will satisfy the same formulas when we consider supposedly isomorphic inconsistent LP-structures.

To see why this is the case we propose a simple example. Consider the two-element models AandBwith no function symbols: A={a1, a2} andB ={b1, b2} such that ha1, a2i ∈R+A, R−A=∅ but hb1, b2i ∈R+B =R−B. Define a bijection π:A → B,

with π(a1) = b1 and π(a2) = b2. It is easy to note that the function defines an

isomorphism in the classical case. Note that the condition: • (m1, ..., mn)∈R+

A

⇔(π(m1), ..., π(mn))∈R+

B

(form1, ..., mn∈A)

It is also satisfied. Of course, we do not have • (m1, ..., mn)∈R− A ⇔(π(m1), ..., π(mn))∈R− B (form1, ..., mn∈A)

But since the classical definition of isomorphism only (explicitly) requires the agree- ment of the extension of the relation symbols, we do not need for this latter clause to come out true. So the function is indeed an elementary embedding. However, both models are not elementary equivalent. For:

A |=R(a1, a2) A 6|=¬R(a1, a2) B |=R(b1, b2) B |=¬R(b1, b2)

The philosophical problem that we found is the following. We want to make compar- isons betweenLP-models and to be able to say when two models are the sameup to isomorphism. In fact, as noted before, the goal of this chapter is to show that the argument from Tennenbaum’s Theorem does not isolate a single isomorphism type. And this because, as we will argue, there are inconsistent LP-models that are not isomorphicto the standard model. However, what is the notion of isomorphism here? We may think that one necessary condition for two models to be isomorphic is ele- mentary equivalence. Hence, if we use a notion like that of ‘classical isomorphism’, that allows for not elementary equivalent albeit isomorphic models in the LP-case, it might be suggested that the notion is not suitable to capture that which we want to capture when establishing isomorphic relations between inconsistent models. And if the above follows, we must provide a new definition to establish when these mod- els are isomorphic – one that also captures that which we want to capture with the classical notion of ‘isomorphism’. Otherwise, when claiming, as we will below, that there are inconsistent intended LP-models not isomorphic to the standard intended models, it might be counter-argued that we are using ‘isomorphism’ with a different meaning; and in this way it might not be established that those inconsistent models are ‘really’ non-isomorphic to the standard model.

A bit of further elaboration on the topic of isomorphicLP-structures is required for a better understanding of these constructions and their differences or similarities with more common classical constructions. As it will be seen below, we will go around such problems, bearing in mind that there is more to be said on exactly how two LP-models can be isomorphic.

In document MoL 2019 04: Tennenbaum’s Theorem and Non Classical Arithmetic (Page 55-61)