Fast-Collapsing Theories

Abstract

Reinhardt's conjecture, a formalization of the statement that a truthful
knowing machine can know its own truthfulness and mechanicalness, was proved by
Carlson using sophisticated structural results about the ordinals and
transfinite induction just beyond the first epsilon number. We prove a weaker
version of the conjecture, by elementary methods and transfinite induction up
to a smaller ordinal.

Full text

arXiv:1311.3013v2 [math.LO] 14 Nov 2013
Fast-collapsing theories
Samuel A. Alexander∗
Department of Mathematics, the Ohio State University
June 22, 2018
Abstract
Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know
its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results
about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker
version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.
1 Introduction
This is a paper about idealized truthful mechanical knowing agents who know facts in a quantified arithmetic-
based language that also includes a connective for their own knowledge (K(1 + 1 = 2) is read “I (the agent)
know 1+ 1 = 2”). It is well known ([4], [6], [9], [10], [11], [12]) that such an agent cannot simultaneously know
its own truthfulness and its own code. Reinhardt conjectured that, while knowing its own truthfulness, such
a machine can know it has some code, without knowing which. This conjecture was proved by Carlson [6].
The proof uses sophisticated structural results from [5] about the ordinals, and involves transfinite induction
up to ǫ0·ω.
We will give a proof of a weaker result, but will do so in an elementary way, inducting only as far as
ω·ω. Along the way, we will develop some machinery that is interesting in its own right. Carlson’s proof of
Reinhardt’s conjecture is based on stratifying knowledge (see [8] for a gentle summary). This can be viewed
as adding operators Kαfor knowledge after time αwhere αtakes ordinal values. Under certain assumptions,
theories in such stratified language collapse at positive integer multiples of ǫ0, in the sense that if φonly
contains superscripts < ǫ0·n(na positive integer) then Kǫ0·nφholds if and only if Kǫ0·(n+1) φdoes. In this
paper, collapse occurs at positive integer multiples of ω, hence the name: Fast-collapsing theories.
Our result is weakened in the sense that the background theory of knowledge is weakened. The schema
K(ucl(K(φ→ψ)→Kφ →K ψ)) (ucl denotes universal closure) is weakened by adding the requirement that
Knot be nested deeper in φthan in ψ(the unrestricted schema ucl(K(φ→ψ)→Kφ →Kψ) is preserved,
but the knower is not required to know it); the schema ucl(Kφ →KKφ) is forfeited entirely; and a technical
axiom called Assigned Validity (made up of valid formulas with numerals plugged in to their free variables)
is added to the background theory of knowledge.
On the bright side, our result is stated in a more general way (we mention in passing how the full
unweakened result could also be so generalized, but leave those details for later work). Casually, our main
theorem has the following form:
A truthful knowing agent whose knowledge is sufficiently “generic” can be taught its own truth-
fulness and still remain truthful.
Here “generic” is a specific technical term, but it is inclusive enough to include knowledge that one has some
code, thus the statement addresses Reinhardt’s conjecture.
In Section 2 we present some preliminaries.
In Section 3 we develop stratifiers, maps from unstratified language to stratified language. These are the
key to fast collapse. They debuted in [1] and [3].
∗Email: alexander@math.ohio-state.edu
1

In Section 4 we discuss uniform stratified theories. A key advantage of stratifiers is that they turn
unstratified theories into uniform stratified theories.
In Section 5 we define some notions of genericity of an axiom schema, and establish the genericity of
some building blocks of background theories of knowledge.
In Section 6 we state our main theorem and make closing remarks.
2 Preliminaries
Definition 1. (Standard Definitions) Let LPA be the language (0, S, +,·) of Peano arithmetic and let L
be an arbitrary language.
1. For any e∈N,Weis the range of the eth partial computable function. The binary predicate • ∈ W•
is LPA-definable so we will freely act as if LPA actually contains this predicate symbol.
2. If an L-structure Mis clear from context, an assignment is a function taking variables into the
universe of M.
3. If sis an assignment, xis a variable, and a∈M,s(x|a) is the assignment that agrees with sexcept
that s(x|a)(x) = a.
4. We define LPA-terms n(n∈N), called numerals, so that 0 = 0 and n+ 1 = S(n).
5. If φis an L-formula, FV(φ) is the set of free variables of φ. If FV(φ) = ∅then φis a sentence.
6. If φis an L-formula, xis variable, and uis an L-term, φ(x|u) is the result of substituting ufor all
free occurrences of xin φ.
7. A universal closure of an L-formula φis a sentence ∀x1···∀xnφ. We write ucl(φ) to denote a universal
closure of φ.
8. We use the word theory as synonym for set of sentences.
9. If Tis an L-theory and Mis an L-structure, M|=Tmeans that M|=φfor all φ∈T.
10. If Tis an L-theory, we say T|=φif M|=φwhenever M|=T.
11. A valid L-formula is one that holds in every L-structure.
12. For any formulas φ1, φ2, φ3, we write φ1→φ2→φ3to abbreviate φ1→(φ2→φ3).
We will repeatedly use the following standard fact without explicit mention: if ψis a universal closure
of φ, then in order to prove M|=ψ, it suffices to let sbe an arbitrary assignment and show that M|=φ[s].
For quantified semantics we work in Carlson’s base logic, defined as follows.
Definition 2. (The Base Logic) A language Lin the base logic is a first-order language L0together with a
set of symbols called operators. Formulas of Lare defined in the usual way, with the clause that whenever
φis an L-formula and Kis an L-operator, Kφ is also an L-formula (and FV(Kφ) = FV(φ)). Syntactic
parts of Definition 1 extend to the base logic in obvious ways. Given such an L, an L-structure Mis
a first-order L0-structure M0together with a function that takes one L-formula φ, one L-operator K,
and one assignment s, and outputs True or False—in which case we write M|=K φ[s] or M6|=Kφ[s],
respectively—satisfying the following three conditions (where φranges over L-formulas and Kranges over
operators):
1. Whether or not M|=K φ[s] is independent of s(x) if x6∈ FV(φ).
2. (Alphabetic Invariance) If ψis an alphabetic variant of φ, meaning that it is obtained from φby
renaming bound variables while respecting binding of the quantifiers, then M|=K(φ)[s] if and only
if M|=K(ψ)[s].
2

3. (Weak Substitution)1If the variable yis substitutable for the variable xin φ, then M|=K φ(x|y)[s]
if and only if M|=Kφ[s(x|s(y))].
Theorem 3. (Completeness and compactness) Let Lbe an r.e. language in the base logic.
1. The set of valid L-formulas is r.e.
2. For any r.e. L-theory T,{φ:T|=φ}is r.e.
3. There is an effective algorithm, given (a G¨odel number for) an r.e. L-theory T, to find (a G¨odel number
for) {φ:T|=φ}.
4. If Tis an L-theory and T|=φ(φany L-formula), there are τ1,...,τn∈Tsuch that (Viτi)→φis
valid.
Proof. By interpreting the base logic in first-order logic. For details, see [1].
Definition 4. Let LEA be the language of Epistemic Arithmetic from [13], so LEA extends LPA by a
unary operator K. An LEA-structure (more generally an L-structure where Lextends LPA) has standard
first-order part if its first-order part has universe Nand interprets 0, S, +,·in the intended ways.
Definition 5. Suppose Lextends LPA and φis an L-formula with FV(φ)⊆ {x1,...,xn}. For any
assignment sinto N, we define
φs≡φ(x1|s(x1)) ···(xn|s(xn)),
the sentence obtained by replacing all free variables in φby numerals according to s.
Definition 6. For any LEA-theory T, the intended structure for Tis the LEA -structure NTthat has
standard first-order part and interprets Kso that for any LEA-formula φand assignment s,
NT|=K φ[s] if and only if T|=φs.
We say Tis true if NT|=T.
It is easy to check that the structures NTof Definition 6 really are LEA -structures (they satisfy Conditions
1–3 of Definition 2). The following lemma shows that they accurately interpret quantified formulas in the
way one would expect.
Lemma 7. For any LEA-theory T,LEA-formula φand assignment s,
NT|=φ[s] if and only if NT|=φs.
Proof. Straightforward induction.
Armed with these definitions, we can make more precise some things we suggested in the introduction.
Let TSMT be the following LEA -theory (φand ψrange over LEA-formulas):
1. (E1) ucl(Kφ) whenever φis valid.
2. (E2) ucl(K(φ→ψ)→Kφ →K ψ).
3. (E3) ucl(Kφ →φ).
4. (E4) ucl(Kφ →KKφ).
5. The axioms of Epistemic Arithmetic, by which we mean the axioms of Peano Arithmetic with the
induction schema extended to LEA.
6. (Mechanicalness) ucl(∃e∀x(Kφ ↔x∈We)) provided e6∈ FV(φ).
7. Kφ whenever φis an instance of lines 1–6 or (recursively) 7.
1Note that the general substitution law, where yis replaced by an arbitrary term, is not valid in modal logic.
3

Combining lines 6 and 7 yields the Strong Mechanistic Thesis,K(ucl(∃e∀x(K φ ↔x∈We))). One of the
main results of [6] is that TSMT is true, that is, NTSMT |=TSMT. To establish NTSMT |=E3, Carlson uses
transfinite recursion up to ǫ0·ω, as well as deep structural properties (from [5]) about the ordinals. That
NTSMT satisfies lines 2, 5, 6, and 7, is trivial; that it satisfies line 4 follows from the fact that it satisfies lines
1–2. Line 1 would be trivial if we added the following line to TSMT :
1b. (Assigned Validity) φs, whenever φis valid and sis any assignment.
Theorems from [6] imply Assigned Validity is already a consequence of TSMT, so this addition is not necessary,
however it becomes necessary if (say) line 2 is weakened.
The main result in this paper is that by weakening E2, removing E4, and adding Assigned Validity, we
remove the need to induct up to ǫ0·ω. Induction up to ω·ωsuffices, and the computations from [5] can also
be avoided. This is surprising because we do not weaken E3, the lone schema for which such sophisticated
methods were used before.
Definition 8. For any LEA-formula φ, let depth(φ) denote the depth to which Koperators are nested in
φ, more formally:
•If φis an LPA-formula then depth(φ) = 0.
•If φ≡K(φ0) then depth(φ) = depth(φ0) + 1.
•If φ≡(ρ→σ) then depth(φ) = max{depth(ρ),depth(σ)}.
•If φ∈ {(¬φ0),(∀xφ0)}then depth(φ) = depth(φ0).
Now let Tw
SMT be the LEA-theory containing the following schemas:
1. E1and E3.
2. Assigned Validity: φswhenever φis valid and sis any assignment.
3. (E′
2) ucl(K(φ→ψ)→Kφ →K ψ) provided depth(φ)≤depth(ψ).
4. The axioms of Epistemic Arithmetic.
5. Mechanicalness.
6. Kφ whenever φis an instance of lines 1–5 or (recursively) 6.
Our main result (obtained by inducting only up to ω·ω) will imply Tw
SMT is true.
3 Stratifiers
Definition 9. Let Lω·ωbe the language obtained from LPA by adding operators Kαfor all α∈ω·ω. For
any Lω·ω-formula φ, let
On(φ) = {α∈ω·ω:Kαoccurs in φ}.
An example of an Lω·ω-formula: ∀x(KωKω·7+2K53 K0(x= 0) →Kω·7+3(x= 0)).
Definition 10. (Stratifiers) For any infinite subset X⊆ω·ω, the stratifier given by Xis the function •+
that takes LEA-formulas to Lω·ω-formulas in the following way.
1. If φis atomic, φ+≡φ.
2. If φis φ1→φ2,¬φ1, or ∀xφ1, then φ+is φ+
1→φ+
2,¬φ+
1, or ∀xφ+
1, respectively.
3. If φis Kφ0, then φ+≡Kαφ+
0where αis the smallest ordinal in X\On(φ+
0).
By a stratifier, we mean a stratifier given by some X. By the veristratifier, we mean the stratifier given by
X={ω·1, ω ·2,...}. If •+is a stratifier and Tis an LEA-theory, T+denotes {φ+:φ∈T}.
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For example, if •+is the veristratifier, then
(K(1 = 0) →KK(1 = 0))+≡Kω(1 = 0) →Kω·2Kω(1 = 0).
Lemma 11. Suppose φis an LEA -formula, sis an assignment into N, and •+is a stratifier. If α, β ∈ω·ω
are such that (Kφ)+≡Kαφ+and (Kφs)+≡Kβ(φs)+, then α=β.
Proof. By induction.
Lemma 12. Suppose φand ψare LEA-formulas and •+is a stratifier. Let α, β ∈ω·ωbe such that
(Kφ)+≡Kαφ+and (Kψ)+≡Kβψ+. Then depth(φ)<depth(ψ) if and only if α < β.
Proof. By induction.
Definition 13. For any Lω·ω-structure Mand stratifier •+, let M+be the LEA-structure that has the
same universe and interpretation of LPA as M, and that interprets Kso that for any LEA -formula φand
assignment s,
M+|=K φ[s] if and only if M|= (Kφ)+[s].
It is easy to check that if Mis an Lω·ω-structure then M+really is an LEA-structure (it satisfies
Conditions 1–3 of Definition 2). From now on we will suppress this remark when defining new structures.
Lemma 14. Let Mbe an Lω·ω-structure, •+a stratifier. For any LEA -formula φand assignment s,
M+|=φ[s] if and only if M|=φ+[s].
Proof. A straightforward induction.
Definition 15. For any Lω·ω-formula φ,φ−is the LEA-formula obtained by changing every operator of
the form Kαin φinto K. If Tis an Lω·ω-theory, T−={φ−:φ∈T}.
Example 16. Kω·8+3∀xK 17(x=y)−≡K∀xK(x=y).
Lemma 17. Let •+be a stratifier. For any LEA-formula φ, (φ+)−≡φ.
Proof. Straightforward.
Definition 18. If Mis an LEA-structure, let M−be the Lω·ω-structure that has the same universe as M,
agrees with Mon LPA, and interprets each Kαso that for any Lω·ω-formula φand assignment s,
M−|=Kαφ[s] if and only if M|=K φ−[s].
In [6] (Definition 5.4), M−is the stratification of Mover ω·ω.
Lemma 19. For any LEA -structure M,Lω·ω-formula φand assignment s,
M−|=φ[s] if and only if M|=φ−[s].
Proof. A straightforward induction.
Theorem 20.
1. For any valid Lω·ω-formula φ,φ−is valid.
2. For any LEA-formula φand stratifier •+,φis valid if and only if φ+is valid.
Proof.
(1) Let φbe a valid Lω·ω-formula. For any LEA-structure Mand assignment s, since φis valid, M−|=φ[s]
and so by Lemma 19, M|=φ−[s]. By arbitrariness of Mand s,φ−is valid.
(2, ⇒) Assume φis a valid LEA-formula. For any Lω·ω-structure Mand assignment s, since φis valid,
M+|=φ[s], and M|=φ+[s] by Lemma 14. By arbitrariness of Mand s, this shows φ+is valid.
5

(2, ⇐) Assume φis an LEA-formula and φ+is valid. For any LEA-structure Mand assignment s, since φ+
is valid, M−|=φ+[s], and M|= (φ+)−[s] by Lemma 19. By Lemma 17, M|=φ[s]. By arbitrariness of M
and s,φis valid.
Definition 21. For any LEA -theory T, let
T⊕={φ+:φ∈Tand •+is a stratifier}.
Example 22. Suppose Tis the LEA -theory consisting of K φ →KKφ for all LPA-sentences φ. Then T⊕
is the Lω·ω-theory consisting of Kαφ→KβKαφfor all LPA-sentences φand ordinals α < β < ω ·ω.
Theorem 23. (Upward proof stratification) For any LEA-theory T,LEA-sentence φ, and stratifier •+, the
following are equivalent.
1. T|=φ. 2. T+|=φ+. 3. T⊕|=φ+.
This theorem is so-named because it is an upside-down version of a harder theorem that we called [1]
proof stratification. In non-upward proof stratification, Tand φare taken in the stratified language and the
theorem states that T|=φif and only if T−|=φ−. This uses complicated hypotheses on Tand φ. Versions
of these hypotheses could be stated in an elementary way, but a priori they might imply Tis inconsistent
(in which case Theorem 23 is trivial). The only way we know to exhibit consistent theories that satisfy such
hypotheses is to exploit the machinery from [5] on the Σ1-structure of the ordinals.
Proof of Theorem 23. Let T,φand •+be as in Theorem 23.
(1 ⇒2) Assume T|=φ. By Theorem 3, there are τ1,...,τn∈Tsuch that (Viτi)→φis valid. By Theorem
20, Viτ+
i→φ+is valid, showing T+|=φ+.
(2 ⇒3) Trivial: T+⊆T⊕.
(3 ⇒1) Assume T⊕|=φ+. By Theorem 3 there are τ1,...,τn∈T⊕such that (Viτi)→φ+is valid. By
definition of T⊕there are σ1, . . . , σn∈Tand stratifiers •1,...,•nsuch that each τi≡σi
i. By Lemma 17
Viσi
i→φ+−≡(Viσi)→φ,
so Theorem 20 guarantees (Viσi)→φis valid, and T|=φ.
4 Uniform Theories and Collapsing Knowledge
Definition 24. Suppose X⊆ω·ωand h:X→ω·ω. For any Lω·ω-formula φ, we define h(φ) to be the
Lω·ω-formula obtained by replacing Kαby Kh(α)everywhere Kαoccurs in φ(α∈X). (If α6∈ X, we do
not change occurrences of Kαin φ.)
Example 25. Suppose α1<··· < α4are distinct ordinals in ω·ω. Let X={α2, α3}, let h(α2) = α3,
h(α3) = α4. Then
h(Kα3Kα2Kα1(1 = 1)) ≡Kα4Kα3Kα1(1 = 1).
Definition 26. An Lω·ω-theory Tis uniform if the following statement holds. For all X⊆ω·ω, for all
order-preserving h:X→ω·ω, for all φ∈T, if On(φ)⊆Xthen h(φ)∈T.
Example 27. If Tcontains K1K0(1 = 0) and Tis uniform, then Tmust contain KβKα(1 = 0) for all
α < β ∈ω·ω.
Lemma 28. Suppose •+is a stratifier, X⊆ω·ω,h:X→ω·ωis order preserving, and φis an LEA -formula
with On(φ+)⊆X. There is a stratifier •∗such that φ∗≡h(φ+).
Proof. Let Y0={h(α) : α∈On(φ+)},Y=Y0∪ {β∈ω·ω:β > Y0}, and let •∗be the stratifier given by
Y. By induction, for every subformula φ0of φ,φ∗
0≡h(φ+
0).
Lemma 29. (Uniformity lemma) For any LEA-theory T,T⊕is uniform.
6

Proof. Let X⊆ω·ω, let h:X→ω·ωbe order preserving, let φ∈T⊕, and assume On(φ)⊆X. By
definition of T⊕,φ≡φ+
0for some φ0∈Tand some stratifier •+. By Lemma 28 there is a stratifier •∗such
that h(φ+
0)≡φ∗
0. This shows h(φ)∈T⊕.
Unfortunately, the range of ⊕does not include every uniform Lω·ω-theory. For example, suppose Tis
the Lω·ω-theory consisting of
Kα(φ+→ψ+)→Kαφ+→Kαψ+
for all LEA-sentences φand ψand stratifiers •+with On(φ+),On(ψ+)< α ∈ω·ω. The reader may check
that despite being uniform, Tis not T⊕
0for any LEA-theory T0.
Definition 30. If Mis an Lω·ω-structure, X⊆ω·ω, and h:X→ω·ω, we define an Lω·ω-structure h(M)
that has the same universe as M, agrees with Mon the interpretation of LPA, and interprets Kαso that
for any Lω·ω-formula φand assignment s,
h(M)|=Kαφ[s] if and only if M|=h(Kαφ)[s].
Lemma 31. Suppose M,X, and hare as in Definition 30. For any Lω·ω-formula φand assignment s,
h(M)|=φ[s] if and only if M|=h(φ)[s].
Proof. By induction.
We will only need part 1 of the next lemma, we state part 2 for completeness.
Lemma 32. Suppose M,X, and hare as in Definition 30 and φis an Lω·ω-formula.
1. If φis valid then h(φ) is valid.
2. Assume his injective. If On(φ)⊆Xand h(φ) is valid, then φis valid.
Proof.
(1) Similar to Theorem 20.
(2) If h(φ) is valid then h−1(h(φ)) is valid by part 1. Since On(φ)⊆X,h−1(h(φ)) ≡φ.
Definition 33. For any Lω·ω-theory Tand α∈ω·ω, let T∩α={φ∈T: On(φ)⊆α}be the subset of T
where all superscripts are strictly bounded by α.
Example 34.
•For any Lω·ω-theory T,T∩0 = {φ∈T:φis an LPA-sentence}.
•For any Lω·ω-theory T,T∩1 = {φ∈T:φis an LPA ∪ {K0}-sentence}.
•For any LEA-theory T,T⊕∩ω={φ+:φ∈Tand •+is given by some X⊆ω}.
Theorem 35. (The collapse theorem) Let Tbe a uniform Lω·ω-theory. For any 0 < n ∈Nand Lω·ω-formula
φwith On(φ)⊆ω·n,T|=φif and only if T∩(ω·n)|=φ.
Proof. The ⇐direction is trivial: T∩(ω·n)⊆T. For ⇒, assume T|=φ.
By Theorem 3 there are τ1, . . . , τn∈Tsuch that
Φ≡(Viτi)→φ
is valid. Let X= On(Φ) ∩(ω·n), Y= On(Φ) ∩[ω·n, ∞), see Fig. 1.
Then |X|,|Y|<∞and X∪Y= On(Φ).
XY
Y
~
=
=h()


τ1
=
=h()
τ1
n
τ2τ3
τ2
h() τ3
h()
Figure 1: Collapse.
7

Since |X|<∞and ω·nhas no maximum element, there are infinitely many ordinals above Xin ω·n.
Thus since |Y|<∞we can find e
Y⊆ω·nsuch that X < e
Yand |e
Y|=|Y|. It follows there is an order
preserving function h:X∪Y→X∪e
Ysuch that h(x) = xfor all x∈X.
By Lemma 32, h(Φ) is valid. Since On(φ)⊆ω·n, we have On(φ)⊆Xand h(φ)≡φ. Thus
h(Φ) ≡(Vih(τi)) →h(φ)≡(Vih(τi)) →φ.
Since Tis uniform, each h(τi)∈T. In fact, since range(h)⊆ω·n, each h(τi)∈T∩(ω·n), and the validity
of (Vih(τi)) →φwitnesses T∩(ω·n)|=φ.
Definition 36. If Tis an Lω·ω-theory, its intended structure is the Lω·ω-structure MTwith standard
first-order part that interprets the operators of Lω·ωso that for every Lω·ω-formula φ, assignment s, and
α∈ω·ω,
MT|=Kαφ[s] if and only if T∩α|=φs.
Lemma 37. Suppose Tis an Lω·ω-theory. For any Lω·ω-formula φand assignment s,MT|=φ[s] if and
only if MT|=φs.
Proof. By induction.
Recall from Definition 10 that the veristratifier is the stratifier given by X={ω·1, ω ·2,...}.
Theorem 38. (The upward stratification theorem) Let •+be the veristratifier. For any LEA-theory T,
LEA-formula φ, and assignment s,NT|=φ[s] if and only if MT⊕|=φ+[s].
Again, the theorem is so-named because it is an upside-down version of a harder theorem that equates
MT|=φ[s] with NT−|=φ−[s] for stratified Tand φunder more complicated hypotheses.
Proof of Theorem 38. By induction on φ. The only nontrivial case is when φis K ψ. Then φ+≡Kαψ+for
some α. By definition of the veristratifier, α=ω·nfor some 0 < n ∈N, and On(ψ+)⊆ω·n. By Lemma
29, T⊕is uniform, so we can use the collapse theorem (Theorem 35). The following are equivalent.
NT|=K ψ[s]
T|=ψs(Definition 6)
T⊕|= (ψs)+(Upward proof stratification—Theorem 23)
T⊕∩(ω·n)|= (ψs)+(The collapse theorem—Theorem 35)
T⊕∩(ω·n)|= (ψ+)s(Clearly (ψs)+≡(ψ+)s)
MT⊕|=Kω·nψ+[s].(Definition 36)
Corollary 39. For any LEA-theory T, in order to show NT|=T, it suffices to show MT⊕|=T⊕.
Corollary 39 provides a foothold for proving truth of self-referential theories by transfinite induction up
to ω·ω: in order to prove NT|=T, one can attempt to prove MT⊕|=T⊕∩αfor all α∈ω·ωby induction
on α.
5 Upward Generic Axioms
One way to state an epistemological consistency result, for example that a truthful machine can know itself to
be true and recursively enumerable, is to show that the schemas in question are consistent with a particular
background theory of knowledge. We take a more general approach: show that the doubted schemas are
consistent with any background theory satisfying certain conditions.
We say that an LEA-theory Tis K-closed if Kφ ∈Twhenever φ∈T.
Definition 40. Suppose T0is an LEA -theory.
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1. T0is generic if NT|=T0for every LEA-theory T⊇T0.
2. T0is closed-generic if T0is K-closed and NT|=T0for every K-closed LEA-theory T⊇T0.
3. T0is r.e.-generic if T0is r.e. and NT|=T0for every r.e. LEA-theory T⊇T0.
4. T0is closed-r.e.-generic if T0is K-closed, r.e., and NT|=T0for every K-closed r.e. LEA -theory T⊇T0.
Lemma 41.
1. Generic+r.e. implies r.e.-generic.
2. Generic+K-closed implies closed-generic.
3. Closed-generic+r.e. implies closed-r.e.-generic.
4. R.e.-generic+K-closed implies closed-r.e.-generic.
Proof. Straightforward.
Lemma 42. Let T=∪i∈ITiwhere each Tiis an LEA-theory.
1. If the Tiare generic, then Tis generic.
2. If the Tiare closed-generic, then Tis closed-generic.
3. If the Tiare r.e.-generic and Tis r.e., then Tis r.e.-generic.
4. If the Tiare closed-r.e.-generic and Tis r.e., then Tis closed-r.e.-generic.
Proof. Straightforward.
Lemma 43. The LEA-schema E2, consisting of ucl(K(φ→ψ)→Kφ →K ψ), is generic.
Proof. Suppose T⊇E2is arbitrary. For any LEA-formulas φand ψand assignment s, if NT|=K(φ→ψ)[s]
and NT|=K φ[s], then
T|=φs→ψs(Definition 6)
T|=φs(Definition 6)
T|=ψs(Modus Ponens)
NT|=K ψ[s],as desired. (Definition 6)
Definition 44. Suppose T0is an LEA -theory.
1. T0is upgeneric if MT⊕|=T⊕
0for every LEA-theory T⊇T0.
2. T0is closed-upgeneric if T0is K-closed and MT⊕|=T⊕
0for every K-closed LEA-theory T⊇T0.
3. T0is r.e.-upgeneric if T0is r.e. and MT⊕|=T⊕
0for every r.e. LEA-theory T⊇T0.
4. T0is closed-r.e.-upgeneric if T0is K-closed, r.e., and MT⊕|=T⊕
0for every K-closed r.e. LEA -theory
T⊇T0.
Lemma 45. (Compare Lemma 41)
1. Upgeneric+K-closed implies closed-generic.
2. Upgeneric+r.e. implies r.e.-upgeneric.
3. Closed-upgeneric+r.e. implies closed-r.e.-upgeneric.
4. R.e.-upgeneric+K-closed implies closed-r.e.-upgeneric.
9

Proof. Straightforward.
Lemma 46. Suppose T=∪i∈ITiwhere the Tiare LEA -theories.
1. If the Tiare upgeneric, then Tis upgeneric.
2. If the Tiare closed-upgeneric, then Tis closed-upgeneric.
3. If the Tiare r.e.-upgeneric and Tis r.e., then Tis r.e.-upgeneric.
4. If the Tiare closed-r.e.-upgeneric and Tis r.e., then Tis closed-r.e.-upgeneric.
Proof. Straightforward.
Lemma 47.
1. Upgeneric implies generic.
2. Closed-upgeneric implies closed-generic.
3. R.e.-upgeneric implies r.e.-generic.
4. Closed-r.e.-upgeneric implies closed-r.e.-generic.
Proof. By the upward stratification theorem (Theorem 38).
In light of Lemmas 43 and 47, the following shows that upgeneric is strictly stronger than generic.
Lemma 48. E2is not upgeneric. In fact E2is not even closed-r.e.-upgeneric.
Proof. Let Tbe the smallest K-closed LEA-theory containing the following schemata.
1. E2.
2. K(1 = 0).
3. K(1 = 0) →(1 = 0).
Since T⊇E2is closed r.e., it suffices to exhibit some θ∈E2and stratifier •+such that MT⊕6|=θ+. If •+
is the stratifier given by X={0,1,2,...}, the reader can check that
θ≡K(K(1 = 0) →(1 = 0)) →K K(1 = 0) →K(1 = 0)
works.
Lemma 48 and the following demystify our reason for weakening E2to E′
2.
Lemma 49. The schema E′
2, consisting of ucl(K(φ→ψ)→Kφ →Kψ) whenever depth(φ)≤depth(ψ)
(Definition 8), is upgeneric.
Proof. Let T⊇E′
2be arbitrary. Suppose φand ψare LEA-formulas with depth(φ)≤depth(ψ) and •+is a
stratifier, say with
(Kφ)+≡Kαφ+
(Kψ)+≡Kβψ+
(K(φ→ψ))+≡Kγ(φ+→ψ+),
we will show MT⊕satisfies
(ucl(K(φ→ψ)→Kφ →K ψ))+≡ucl(Kγ(φ+→ψ+)→Kαφ+→Kβψ+).
10

Note that by Lemma 12, α≤β=γ. Let sbe an arbitrary assignment such that MT⊕|=Kγ(φ+→ψ+)[s]
and MT⊕|=Kαφ+[s]. Then
T⊕∩γ|= (φ+)s→(ψ+)s(Definition 36)
T⊕∩α|= (φ+)s(Definition 36)
T⊕∩β|= ((φ+)s→(ψ+)s)∧(φ+)s(Since α≤β=γ)
T⊕∩β|= (ψ+)s(Modus Ponens)
MT⊕|=Kβψ+[s],as desired. (Definition 36)
Lemma 50. The Assigned Validity schema, consisting of φswhenever φis valid and sis any assignment, is
upgeneric.
Proof. Let T⊇(Assigned Validity) be arbitrary. Suppose φis valid, sis an assignment, and •+is a stratifier.
By Theorem 20, φ+is also valid. Thus MT⊕|=φ+[s], and by Lemma 37, MT⊕|= (φ+)s. By arbitrariness
of φ,s, and •+,MT⊕|= (Assigned Validity)⊕.
Lemma 51. Any set of true purely arithmetical sentences is upgeneric.
Proof. Trivial: MThas standard first-order part.
Lemma 52. The schema consisting of the axioms of Epistemic Arithmetic (Peano Arithmetic with induction
extended to LEA) is upgeneric.
Proof. Let T⊇(Epistemic Arithmetic). Let σbe an axiom of Epistemic Arithmetic, •+a stratifier. If σis
not an induction instance, then MT⊕|=σ+by Lemma 51. But suppose σis an instance
ucl(φ(x|0) → ∀x(φ→φ(x|S(x))) → ∀xφ)
of induction, so that σ+is ucl(φ+(x|0) → ∀x(φ+→φ+(x|S(x))) → ∀xφ+). To show MT⊕|=σ+, let sbe
an assignment and assume MT⊕|=φ+(x|0)[s] and MT⊕|=∀x(φ+→φ+(x|S(x)))[s]. Then
MT⊕|=φ+(x|0)s(Lemma 37)
MT⊕|= (φ+)s(x|0) (Clearly ψ(x|0)s≡ψs(x|0))
∀n∈N,if MT⊕|=φ+[s(x|n)],then MT⊕|=φ+(x|S(x))[s(x|n)] (First-order semantics of ∀and →)
∀n∈N,if MT⊕|= (φ+)s(x|n),then MT⊕|= (φ+(x|S(x)))s(x|n)(Lemma 37)
∀n∈N,if MT⊕|= (φ+)s(x|n),then MT⊕|= (φ+)s(x|n+1) (Clearly ψ(x|S(x))s(x|n)≡ψs(x|n+1) )
∀n∈N,MT⊕|= (φ+)s(x|n)(Mathematical induction)
∀n∈N,MT⊕|= (φ+)[s(x|n)] (Lemma 37)
MT⊕|=∀xφ+[s], as desired. (First-order semantics of ∀)
Armed with Lemmas 42 and 46, computations such as Lemmas 43, 49, 50, 51 and 52 can be used as
building blocks for background theories of knowledge. Often, schemas we would like as building blocks are
not (up)generic in isolation, but become so when paired with other building blocks, as in the following three
lemmas.
Lemma 53. E1∪(Assigned Validity) is upgeneric (E1consists of ucl(Kφ) whenever φis valid).
Proof. Let T⊇E1∪(Assigned Validity). By Lemma 50, MT⊕|= (Assigned Validity)⊕, we need only show
MT⊕|=E⊕
1. Let φbe valid, •+any stratifier, and sany assignment. Since T⊇(Assigned Validity), T⊕
contains the instance
(φs)+≡(φ+)s
of (Assigned Validity)⊕. In fact, T⊕∩αcontains (φ+)s, where αis such that (Kφ)+≡Kαφ+. Thus by
Definition 36, MT⊕|=Kαφ+[s], that is, MT⊕|= (Kφ)+[s]. This shows MT⊕|=E⊕
1.
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Lemma 54. For any upgeneric T0,T0∪K(T0) is upgeneric, where K(T0) consists of K φ whenever φ∈
T0. Similarly with “upgeneric” replaced by “r.e.-upgeneric”, “closed-upgeneric”, “closed-r.e.-upgeneric”,
“generic”, “r.e.-generic”, “closed-generic”, or “closed-r.e.-generic” throughout.
Proof. We prove the upgeneric statement. Suppose T0is upgeneric and T⊇T0∪K(T0). Since T0is upgeneric
and T⊇T0,MT⊕|=T⊕
0. It remains to show MT⊕|= (Kφ)+for any sentence φ∈T0and stratifier •+. Let
αbe such that (K φ)+≡Kαφ+. By Definition 10, On(φ+)⊆αand thus φ+∈T⊕
0∩α⊆T⊕∩α. Since
T⊕∩α|=φ+,MT⊕|=Kαφ+as desired.
We will not use the following lemma, but it illuminates differences between this paper’s upward approach
and Carlson’s original downward approach.
Lemma 55. E1∪E2∪E4∪(Epistemic Arithmetic) is closed-generic.
Proof. Let Tbe a K-closed theory containing E1,E2,E4and (Epistemic Arithmetic).
By Lemma 43, NT|=E2. By Lemmas 52 and 47, NT|= (Epistemic Arithmetic). It remains to show
NT|=E1∪E4. We will show NT|=E4and sketch NT|=E1.
The typical sentence in E4is ucl(Kφ →KKφ). Let sbe an assignment and assume NT|=Kφ[s]. Then
T|=φs(Definition 6)
∃τ1,...,τn∈Ts.t. (∧n
i=1τi)→φsis valid (Theorem 3)
T|=K((∧n
i=1τi)→φs) (Tcontains E1)
T|= (∧n
i=1K(τi)) →K φs(Repeated applications of E2in T)
T|=∧n
i=1K(τi) (Tis K-closed)
T|=K φs(Modus Ponens)
NT|=KKφ[s].(Definition 6)
This shows NT|=E4.
Because of the lack of Assigned Validity, showing MT|=E1is tricky. We indicate a rough sketch.
Carlson’s Lemmas 5.23 and 7.1 [6] (pp. 69 & 72) imply T|= (Assigned Validity) (we invoke Lemma 7.1 with
Qa singleton). As written, Lemma 5.23 demands Talso contain E3, but it can be shown this is unnecessary.
Thus we may assume Tcontains Assigned Validity. By Lemmas 53 and 47, NT|=E1.
Lemma 55 explains why weakening E2to E′
2required two other seemingly-unrelated weakenings: adding
Assigned Validity, and removing E4altogether.
Lemma 56. The Mechanicalness schema,
ucl(∃e∀x(Kφ ↔x∈We)) (e6∈ FV(φ)),
is r.e.-upgeneric.
Proof. Let Tbe any r.e. LEA -theory containing the Mechanicalness schema. Let •+be a stratifier and let
αbe such that (Kφ)+≡Kαφ+. We must show
MT⊕|= ucl(∃e∀x(Kαφ+↔x∈We)).
Let sbe any assignment and note
{q∈N:MT⊕|=Kαφ+[s(x|q)]}={q∈N:T⊕∩α|= (φ+)s(x|q)}.(Definition 36)
By the Church–Turing Thesis, the latter set is r.e., so there is some p∈Nsuch that
Wp={q∈N:MT⊕|=Kαφ+[s(x|q)]}.
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For all q∈N, the following biconditionals are equivalent:
MT⊕|=Kαφ+↔x∈We[s(e|p)(x|q)]
MT⊕|=Kαφ+[s(e|p)(x|q)] iff MT⊕|=x∈We[s(e|p)(x|q)] (First-order semantics of ↔)
MT⊕|=Kαφ+[s(x|q)] iff MT⊕|=x∈We[s(e|p)(x|q)] (Since e6∈ FV(φ))
MT⊕|=Kαφ+[s(x|q)] iff q∈Wp.(Since MT⊕has standard first-order part)
The latter is true by definition of p. By arbitrariness of q,MT⊕|=∃e∀x(Kαφ+↔x∈We)[s].
Corollary 57. (Recall the definition of Tw
SMT from the end of Section 2) Let (Tw
SMT)\E3be the smallest
K-closed theory containing E1, Assigned Validity, E′
2, Epistemic Arithmetic, and Mechanicalness. (Loosely
speaking, Tw
SMT minus E3.) Then (Tw
SMT)\E3is r.e.-upgeneric.
6 The Main Result
With the machinery of Section 5, we are able to state our main result in a generalized form. Informally:
An r.e.-upgeneric theory remains true upon augmentation by knowledge of its own truthfulness.
Reinhardt’s conjecture (proved by Carlson) was that the Strong Mechanistic Thesis is consistent with a
particular background theory of knowledge. We showed (Lemma 56) that Mechanicalness is r.e.-upgeneric.
By Lemma 54, the pair consisting of Mechanicalness and the Strong Mechanistic Thesis, is r.e.-upgeneric.
Thus as long as the background theory of knowledge is r.e. and built of r.e.-generic pieces along with
truthfulness, the corresponding conjecture is a special case of this main result.
Recall (Definition 6) that an LEA-theory Tis true if NT|=T.
Theorem 58. Let T0be an r.e.-upgeneric LEA-theory. Let T1be T0∪E3, that is, T0along with all axioms
of the form ucl(Kφ →φ). Let Tbe the smallest K-closed theory containing T1. Then Tis true.
Proof. By Corollary 39 it is enough to show MT⊕|=T⊕. We will use transfinite induction up to ω·ωto
show that for all α∈ω·ω,MT⊕|=T⊕∩α.
Let σ∈T⊕∩α. Then σ≡θ+for some θ∈Tand some stratifier •+. We will show MT⊕|=θ+.
Case 1: θ∈T0. Then MT⊕|=θ+because T⊇T0is r.e. and T0is r.e.-upgeneric.
Case 2: θis K φ for some sentence φ∈T. Let α0be such that (Kφ)+≡Kα0φ+. By Definition 10,
On(φ+)⊆α0and thus φ+∈T⊕∩α0, so T⊕∩α0|=φ+, so MT⊕|=Kα0φ+.
Case 3: θis ucl(Kφ →φ) for some φ. Let α0be such that (Kφ)+≡Kα0φ+, so θ+is ucl(Kα0φ+→φ+).
Since θ+∈T⊕∩α, this forces α0< α. Let sbe any assignment and assume MT⊕|=Kα0φ+[s]. Then:
MT⊕|=Kα0φ+[s] (Assumption)
T⊕∩α0|= (φ+)s(Definition 36)
MT⊕|= (φ+)s(By ω·ω-induction, MT⊕|=T⊕∩α0)
MT⊕|=φ+[s], as desired. (Lemma 37)
Corollary 59. Tw
SMT is true.
Proof. By Theorem 58 and Corollary 57.
If one is willing to induct up to ǫ0·ωand use machinery from [5], it is possible (without the grievous
sacrifices we have made in this paper) to generalize Reinhardt’s conjecture to a statement of the form:
Any r.e. theory that is generic in a very specific sense (one that allows E2as building block)
remains true upon augmentation by knowledge of its own truthfulness. (∗)
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The specific notion of “generic” in order for this to work is somewhat complicated and hinges on [5], putting
it out of the present paper’s scope. It does admit Mechanicalness as building block, so that (∗) really is a
generalization of Reinhardt’s conjecture, and the notion also admits full E2, which in turn allows building
blocks containing E4.
The main result of [2] can also be generalized in this manner. The methods of that paper are easily
modified to prove:
For any r.e. LEA -theory Tthat is generic (in the sense of Definition 40), there is an n∈Nsuch
that T′is true, where T′is the smallest K-closed theory containing Talong with the schema
∀x(Kφ ↔ hx, pφqi ∈ Wn) (FV(φ)⊆ {x}). Less formally, any such generic knowing machine can
be taught its own code and still remain true.
One possible application of this paper is to reverse mathematics [14]. Since the results (except Lemma 52)
only use induction up to ω·ω, suitable versions (minus Lemma 52 and references to N) could be formalized
and proved in weak subsystems of arithmetic.
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