CONCEPT CALCULUS : universes by Harvey

This is the second paper on Concept Calculus, following the first paper that treats Better Than and Much Better Than. Concept Calculus investigates basic formal systems arising from informal commonsense thinking, establishing interpretability relations with a range of formal systems from mathematical logic. Here we focus on commonsense notions of universe, plenitude, and explosion. Interpretations with PA, Z2, Z3, ..., ZF, and various large cardinal axioms are presented. 1. Concept calculus. 2. Universes and explosions. 3. FOU, SOU. 4. FOU, SOU explosions. 5. Conservative FOU, SOU explosions. 6. FOU, SOU explosion series. 7. Super FOU, SOU explosions. 1. Concept calculus. The initial publication on concept calculus has appeared in [Fr11]. This is the second publication on concept calculus. Concept calculus is an open ended investigation that seeks to connect commonsense thinking and mathematical thinking in a rigorous way. In concept calculus, we focus on informally described notions of general familiarity. We isolate simple fundamental principles that generate simple formal systems which are then shown to be mutually interpretable with familiar formal systems of mathematical logic. For a discussion of interpretations, following Alfred Tarski, see [Fr07]. In [Fr11], we investigated a number of systems based on the two commonsense partial orderings of better than and much better than, capitalization on subtle interactions. We 2 showed that these systems are mutually interpretable with the systems Z (Zermelo set theory) and ZF (Zermelo Frankel set theory). Here we consider the informal idea of a physical universe. We use two driving commonsense ideas. One is a principal of plenitude, asserting that if you go out far enough, any situation will be found. The second is that the universe can explode, suddenly creating more space, not only at the outer end, but possibly in the middle or in the front. We shall see that the simple idea of plenitude generates comprehension axioms, whereas the simple idea of an exploding universe of various kinds generates power sets, replacement, and elementary embeddings. The axiom of choice turns out to be foreign in this context, as it was in [Fr11]. Therefore, the corresponding formal set theories are missing AxC. This creates some unresolved difficulties with the systems involving elementary embeddings. It is not clear how strong there are, although it is well known that they are considerably stronger than the existence of measurable cardinals. See section 7. We have based the investigation thus far on the particularly naïve but particularly manageable structure of a linear ordering. Thus a universe is taken to be a triple (S,<,F), where (S,<) is a linear ordering and F:S → S. We require a plenitude principle, and that S have at least three elements (to avoid trivialities). It follows that S is infinite. The triples (S,<,F) are a particularly simple and convenient representation of universes. We can view F as responsive to the following informal idea. We use the points of S as points in space, but also as magnitudes in two senses. One corresponds to the "distance" from the beginning (although we do not assume that there is an actual first point). The other corresponds to a measurement of a physical quantity at any point in space. Thus F(x) is the, say, "temperature" at x ∈ S. As in [Fr11], we emphasize a minimalist approach. Generally, we seek the weakest and simplest axiomatizations that generate the greatest logical strength. Accordingly, we use only one plenitude axiom for universes (in both first order and second order formulations). It addresses 3 only one basic aspect of plenitude that is particularly easy to formulate. Specifically, any upper bounded subset of S can be realized as the set of values of F on some closed interval with endpoints from S. In the first order formulation, we use definable subsets of S, and in the second order formulation, we use all subsets of S. In this way, the first order formulation are approximations to the second order formulation. However, the second order formulation does not directly constitute a formalism for reasoning. For that, second order formulations need to have an additional, ultimately first order apparatus, addressing "arbitrary set" or "arbitrary property", which again constitutes an approximation to the general notion. Obviously, there is more structure to F than just its range of values on intervals. There is of course the order in which these values appear, and even various interactions. These kind of features are not readily formalizable without bringing in, for example, translation structure. This is beyond the scope of the paper. We remark that the unrestricted second order plenitude principle "every subset of S is of the form F[x,y], x,y ∈ S" is impossible by cardinality reasons, since S has at least 3 elements. THEOREM 1.1. (ZFC). There is a surjective map from S onto ℘(S) if and only if S has exactly two elements. Proof: This is clear if S is finite because n = 2 has the unique solution n = 2. In the infinite case, we have |S2| = |S| < |℘(S)|. QED But can the above be proved in ZF? Coming back to the refutation of unrestricted second order plenitude, this can definitely be accomplished in ZF. In fact, even unrestricted first order plentitude is refutable formally. This is accomplished in Lemma 3.4 vi. We will use my base theory RCA0 for Reverse Mathematics, extensively as a base theory here. See [Si99], p. 23. We will also use EFA = exponential function arithmetic = IΣ0(exp), as a base theory for all claims of 4 interpretability between theories. This is convenient, since the theories are all axiomatized by schemes. See [HP98], p. 272. The system ZFWO\P is a very useful base theory for set theoretic statements. This is ZF without the power set axiom, plus "every set can be well ordered'. We use it for almost all results concerning the second order formulations. We use Z for Zermelo set theory and ZF for Zermelo Frankel set theory. We use the systems Zn, n ≥ 2, of n-th order arithmetic. By convention we set Z1 to be PA (Peano arithmetic). Zn, n ≥ 2, has n sorts, 1,...,n. We use = on each sort. We use 0,SC,+,• on sort 1 (here SC is read "successor"). We use epsilon relations ∈1,...,∈n-1, where ∈i is between sort i and sort i+1. The nonlogical axioms of Zn are as follows. i. The usual quantifier free axioms for 0,SC,+,•. ii. (∀x(x ∈i y ↔ x ∈i z) → y = z, where x is of sort i and y,z are of sort i+1. ii. Induction in the form 0 ∈1 x ∧ (∀n)(n ∈1 x → SC(n) ∈1 x) → n ∈1 x, where n is of sort 1 and x is of sort 2. iii. (∃x)(∀y)(y ∈i x ↔ φ), where x is of sort i+1, y is of sort i, and φ is a formula in the language of Zn in which x is not free. 2. Universes and explosions. We model a universe as follows. We take space to be a nonempty linear ordering. At each point x ∈ S, we have a "physical" quantity F(x) ∈ S. Thus the "physical" scale used for the quantities is the same as the scale used for space. DEFINITIOIN 2.1. Let (S,<) be a linear ordering. For A ⊆ S, we write x > A (x ≥ A, x < A, x ≤ A) if and only if for all y ∈ A, x > max(y) (x ≥ max(y), x < max(y), x ≤ max(y)). For x ∈ S, y ∈ S, we write x > y (x ≥ y, x < y, x ≤ y) for max(x) > max(y) (max(x) ≥ max(y), max(x) < max(y), max(x) ≤ max(y)). For A,B ⊆ S, we write A > B (A ≥ B, A < B, A ≤ B) if and only if for all x ∈ A, y ∈ B, max(x) > max(y) (max(x) ≥ max(y), max(x) < max(y), max(x) ≤ max(y)). We say 5 that A ⊆ S is S bounded if and only if (∃x ∈ S)(x ≥ A). For A ⊆ S, we define fld(A) to be the set of all coordinates of elements of A. We define [x,y] = [z: x ≤ z ≤ y}, (-∞,y] = {z: z ≤ y}. For F:S → S and x,y ∈ S, we write F[x,y] = {F(z): x ≤ z ≤ y}. DEFINITION 2.2. An SOU (second order universe) is a triple U = (S,<,F), where i. (S,<) is a linear ordering, F:S → S, and S has at least 3 elements. ii. every S bounded subset of S is F[x,y], for some x,y ∈ S. Condition ii is called second order plenitude. Note that second order plenitude only partially reflects the idea that if we go far enough out, any imaginable local pattern emerges. Consideration of stronger principles of plenitude are beyond the scope of this paper. We impose the condition |S| ≥ 3 in order to avoid the trivial triples (∅,∅,∅) and ({1,2},<,identity), which obey second order plenitude. This limited second order formulation does have a suitably powerful first order formulation, which we present now. Throughout the paper, we follow the usual convention that definability always allows parameters, and 0-definability does not. DEFINITIONI 2.3. An FOU (first order universe) is a triple U = (S,<,F), where i. (S,<) is a linear ordering, F:S → S, and S has at least 3 elements. ii. every U definable S bounded subset of S is F[x,y], for some x,y ∈ S. Condition ii is called first order plenitude. We also use FOU (SOU) to refer to the corresponding first (second) order theory in <,F. The focus of this investigation is on SOU and FOU explosions, as well as SOU and FOU explosion sequences. These "explosions" are loosely motivated by informal accounts of the big bang and cosmological inflation, whereby additional space is created. DEFINITION 2.4. Let triples U = (S,<,F), U' = (S',<',F') be given. We define U ⊆ U' if and only if S ⊆ S', < ⊆ <', F ⊆ 6 F'. We say that U,U' is an SOU explosion if and only if U,U' are SOU's with U ⊆≠ U' and (∃x ∈ S')(x > S). FOU explosions are required to obey an additional first order plenitude condition. DEFINITION 2.5. We say that U,U' is an FOU explosion if and only if U,U' are FOU's with U ⊆ U', (∃x ∈ S')(x > S), and where two strengthened plenitude conditions hold. i. every (S',<',F',S) definable S bounded subset of S is F[x,y], for some x,y ∈ S. ii. every (S',<',F',S) definable S' bounded subset of S' is F'[x,y], for some x,y ∈ S'. DEFINITION 2.6. We say that U,U' is an outer SOU (FOU) explosion if and only if U,U' is an SOU (FOU) explosion, where S < S'\S. Thus in an explosion, new space is added at the end. In an outer explosion, new space is added at the end only. We write FOU[exp], FOU[out-exp] for the first order theories corresponding to an FOU explosion, outer FOU explosion, respectively. How do things change for points in U, after U explodes to U'? DEFINITION 2.7. We say that U,U' is a conservative SOU (FOU) explosion if and only if U,U' is an SOU (FOU) explosion such that the following holds. Let φ be a second order (first order) formula in the language of universes, with free variables among v1,...,vk. For all x1,...,xk ∈ S, U |= φ[x1,...,xk] ↔ U' |= φ[x1,...,xk]. The above equivalence is the so called second order (first order) elementary substructure condition from model theory. Thus in a conservative explosion, properties are conserved. We write FOU[con-exp] for the first order theory corresponding to a conservative FOU explosion. We also consider outer conservative explosions, and write FOU(outcon-exp] for the corresponding first order theory. We also consider explosion series. DEFINITION 2.8. An SOU (outer SOU, conservative SOU, outer conservative SOU) explosion series is a sequence of SOU's 7 U1,...,Un, n ≥ 1, such that for all 1 ≤ i < n, Ui,Ui+1 is an SOU (outer SOU, conservative SOU, outer conservative SOU) explosion. An FOU (outer FOU, conservative FOU, outer conservative FOU) explosion series is a sequence of FOU's U1,...,Un, n ≥ 1, such that for all 1 ≤ i < n, Ui,Ui+1 is an FOU (outer FOU, conservative FOU, outer conservative FOU) explosion. We write FOU[n-exp], FOU[n-out-exp], FOU[n-con-exp], FOU[nout-con-exp], for the first order theories corresponding to an FOU explosion of length n, outer FOU explosion of length n, conservative FOU explosion of length n, outer conservative FOU explosion of length n, respectively. We now introduce a more powerful kind of (two stage) explosion involving three universes. This is the most powerful kind of explosion considered in this paper. DEFINITION 2.9. We say that U1,U2,U3 is a super SOU (FOU) explosion if and only if i. U1,U2,U3 is a conservative SOU (FOU) explosion series. ii. S1 < min(S3/S1) < S2\S1. It follows easily that U1,U2, and U1,U3 are outer SOU (FOU) explosions, and U2,U3 is not an outer SOU (FOU) explosion. We write FOU[sup-exp] for the first order theory corresponding to a super FOU explosion. As promised in section 1, we refute unrestricted plenitude in first order form. THEOREM 2.1. There is no (S,<,F), where (S,<) is a linear ordering with at least 3 elements, F:S → S, and every definable A ⊆ S is of the form F[x,y], x,y ∈ S.