The theory of the generalised real numbers and other topics in logic
Lorenzo Galeotti
Abstract:
In Chapter 2 we briefly introduce the two generalised versions of the real line studied in this thesis. Then, we use these spaces in the context of generalised metrisability theory and generalised descriptive set theory. In particular, we use generalised metrisability theory to define a generalised notion of Polish spaces which we will compare and combine with the game theoretical notion introduced by Coskey and Schlicht. The main results of this chapter are illustrated in the following diagram which shows that a partial generalisation of the classical equivalence between Polish spaces, G_δ spaces, and strongly Choquet spaces can be proved in the generalised context:
Y is strongly λ-Polish ----> Y is strongly λ-Choquet < - - > Y is6 λ-G_δ in X
^ ^ ^
\\ | /
X\ X /
\\ | /
v | v
Y is λ-Polish
In the previous diagram an arrow from A to B means that A implies B; a crossed arrow from A to B means that A does not imply B; and dotted arrows are used to emphasise the fact that further assumptions on Y or λ are needed. See p. 25 for a complete explanation of these results.
In Chapter 3 we study generalisations of the Bolzano-Weierstraß and Heine-Borel theorems. We consider various versions of these theorems and we fully characterise them in terms of large cardinal properties of the cardinal underlining the generalised real line. In particular we prove the following:
Corollary (Corollary 3.23, p. 53). Let κ be an uncountable strongly inaccessible cardinal and let (K, +, ·, 0, 1, ≤) be a Cauchy complete and κ-spherically complete totally ordered field with bn(K) = κ. Then the following are equivalent:
1. κ has the tree property and
2. κ-wBWT_K holds.
In particular κ has the tree property if and only κ-wBWT_{ℝ_κ} holds.
In Chapter 4 we use the generalised real line to develop two new models of transfinite computability, one generalising the so called type two Turing machines and one generalising Blum, Shub and Smale machines, i.e, a model of computation introduced by Blum, Shub and Smale in order to define notions of computation over arbitrary fields. Moreover, we use the generalised version of type two Turing machines to begin the development of a generalised version of the classical theory of Weihrauch degrees. In Chapter 4 we prove the following generalised version of a classical result in the theory of Weihrauch degrees:
Theorem (Theorem 4.24, p. 68).
1. If there exists an effective enumeration of a dense subset of ℝ_κ, then IVT_κ ≤_{sW} B^κ_I .
2. We have B^κ_I I ≤_{sW} IVT_κ .
3. We have IVT_κ ≤^t_{sW} B^κ_I , and therefore IVT_κ ≡^t_{sW} B^κ_I .
The last two chapters of this thesis are the result of the work of the author on topics in logic which are not directly related to generalisations of the real number continuum.
In Chapter 5 we study the possible order types of models of syntactic fragments of Peano arithmetic. The main result of this chapter is that the following arrow diagram between fragments of PA is complete with respect to order types of their models. By this we mean that an arrow from the theory T to the theory T' means that every order type occurring in a model of T also occurs in a model of T' and a missing arrow means that there is a model of T of an order type that cannot be an order type of a model of T' .
SA
↗ ↖
Pr‒ ⟵ Pr
↑ ↑
PA‒ ⟵ PA
In Chapter 6 we study Löwenheim-Skolem theorems for logics extending first order logic. In particular, we extend the work done by Bagaria and Väänänen relating upward Löwenheim-Skolem theorems for strong logics to reflection principles in set theory. Our main result in this area is the following theorem:
Theorem (Theorem 6.49, p. 123). Let L^∗ be a logic and R be a predicate in the language of set theory such that L^∗ and R are bounded symbiotic and L^∗ has dep(L^∗) = ω and is ∆^B_1(R)-finitely-definable. Moreover, let λ be a cardinal such that there is a sequence (δ_n)_{n∈ω} of ∆^B_1(R)-definable cardinals such that U_{n∈ω} δ_n = λ. Then the following are equivalent:
1. ULST_λ(L∗ ) = κ and
2. USR_λ (R) = κ.
In particular, the statement holds for λ = ω and in general for all the logics in [5, Proposition 4].
Finally, we apply the previous result to the study of the large cardinal strength of the upward Löwenheim-Skolem theorem for second order logic; we provide both upper and lower bounds.