In this section we present some open questions which are of particular interest in the area of Bolzano-Weierstraß theorem and non-archimedean real closed fields.
3.4.1 Trees and non-archimedean fields
As we have seen in §3.2, properties of fields such as the Bolzano-Weierstraß theorem are strongly connected to existence of certain trees with particular properties. In [23] Cowles and LaGrange started a systematic study of these connections.
Definition 3.32. LetK be ordered field andκ be a cardinal. Then F ⊂ K+ is said to be separated if there is r ∈ K+ such that for all x, y ∈ F we have |x−y| > r. We say that K is κ-archimedean iff it has a separated family of size κ and if no such family is bounded.
Note that according to the previous definition K is archimedean if and only if it is ℵ0-archimedean.
In the following, we will use the notions of κ-Kurepa tree, (κ, λ)-Kurepa tree, and κ-Aronzajn tree from §1.3.1.
In [23] Cowles and LaGrange proved the following results:
Theorem 3.33 (Cowles and LaGrange). If there is κ+-archimedean ordered field of size at least κ+ then there is a κ+-Kurepa tree.
Proof. See [23, p. 138].
Corollary 3.34 (Cowles and LaGrange). For each infinite cardinal κ, it is consistent with ZFC+“there is an inaccessible cardinal” that κ+-archimedean fields of cardinality larger than κ+ do not exist.
Proof. See [23, p. 139].
In [90], Sikorski asked whether a totally ordered fieldK of size >λsuch thatλ-BWTK
holds exists or not. In [87], Schmerl proved the following:
Theorem 3.35 (Schmerl). Suppose λ≥κ > ℵ0, are cardinals with κ regular. Then the following are equivalent:
1. There is an ordered field K of cardinality λ such that κ-BWTK. 2. There is a (κ, λ)-Kurepa with no κ-Aronszajn subtrees.
Proof. See, [87, p. 145].
This result reduces Sikorski’s question to the existence of (κ, λ)-Kurepa with no κ-Aronszajn subtrees with λ > κ > ℵ0. We list some known facts about (κ, λ)-Kurepa trees.
1. If V = L then for every successor cardinal κ there is a (κ, κ+)-Kurepa tree; see, e.g., [27, Theorem 3.3];
2. ifZFC+“there is a proper class of inaccessible cardinals” is consistent so isZFC+“there are no κ+-Kurepa trees for every regular κ”; see, e.g., [86, Theorem 2.20].
L¨ucke (personal communication) reports that the techniques of [66, Theorem 4.5] were used for results on Kurepa trees (e.g., [92, Theorem 3.5]) and that this method yields the following:
Definition 3.36. A cardinal κ is indestructibly weakly compact if it is weakly compact and in any forcing extension obtained by a<κ-closed notion of forcing κ is weakly com-pact.
Theorem 3.37 (L¨ucke, private communication). If ZFC+“there is an indestructibly weakly compact cardinal” is consistent so is ZFC+“there are κ < λ such that there are (κ, λ)-Kurepa trees with no κ-Aronszajn subtrees”.
Corollary 3.38. If there is an indestructibly weakly compact cardinal then it is consistent with ZFC that there is a totally ordered field K of size larger than λ such that λ-BWTK holds.
Proof. Follows from Theorems 3.37 & 3.35.
It is then natural to ask the following questions:
Question 3.39. Is the indestructibly weakly compact cardinal needed in Corollary 3.38?
Question 3.40. Let f ∈ 2ω be a countable binary sequence. Is it consistent, relative to large cardinals, to have κ with 2κ ≥ κ+ω and for all n ∈ ω, f(n) = 1 iff there is a (κ, κ+n)-Kurepa Tn tree which has no κ-Aronszajn subtrees?
We do not even know the answer to the simplified version of Question 3.40 in which we do not require that Tn does not have κ-Aronszajn subtrees.
3.4.2 The Bolzano-Weierstraß theorem at successor cardinals
As we have seen in §3.2.4, our proof of the existence of fields where the κ-wBWT holds strongly depends on the assumption thatκ is a strong limit cardinal. It is therefore very natural to ask if this assumption can be removed.
Note the following facts:
Theorem 3.41 (Specker [96]). If 2<λ =λ then λ+ does not have the tree property.
Theorem 3.42(Jensen [26, Theorem 5.2]). IfV =Lthen for every uncountable cardinal λ we have that λ+ does not have the tree property.
From the previous facts and our results in §3.2.4 it is easy to see the following:
Theorem 3.43. For every infinite cardinalλ, if2<λ =λthen for every Cauchy complete, λ+-spherically complete totally ordered field(K,+,·,0,1,≤)withbn(K) = λthe weakλ+ -Bolzano-Weierstraß theorem λ+-wBWTK fails.
Proof. The claim follows from Theorems 3.17 & 3.41.
In particular:
Corollary 3.44. If GCH holds then for every cardinal λ which is successor of a regu-lar cardinal and for every Cauchy complete, λ-spherically complete totally ordered field (K,+,·,0,1,≤)withbn(K) =λthe weakλ-Bolzano-Weierstraß theoremλ-wBWTK fails.
Theorem 3.45. Assume GCH. Let (K,+,·,0,1,≤) be a Cauchy complete, λ-spherically complete totally ordered field with bn(K) = λ. Then we have that each element of the following list implies the subsequent:
1. λ is weakly compact;
2. λ-wBWTK;
3. λ is weakly compact or λ = κ+ for some singular cardinal κ and has the tree property.
Proof. The first implication is Corollary 3.23. For the second implication, note that by Theorem 3.17 λ must have the tree property. Moreover, it follows from GCH that λ-wBWTK implies λ limit orλ successor of a singular cardinal. Indeed, if λ is successor of a regular cardinal then by Theorem 3.41 we have thatλ does not have the tree property;
and therefore, by Theorem 3.17, λ-wBWTK must fail. Finally, if λ is limit, by GCH it is a strong limit and therefore weakly compact.
Note that the reverse of the second implication of Theorem 3.45 cannot be proved.
Indeed, it follows from [40, Theorem 1.2] that if ZFC+“there are κ+-many supercompact cardinals for a supercompact cardinal κ” is consistent so is ZFC+GCH+“ℵω+1 has the tree property”.
Corollary 3.46. If V = L then for every uncountable cardinal λ and every Cauchy complete, λ-spherically complete totally ordered field (K,+,·,0,1,≤)with bn(K) =λ the following are equivalent:
1. λ is weakly compact and 2. λ-wBWTK.
Proof. It follows from Theorems 3.45 & 3.42.
The following questions are therefore natural:
Question 3.47. Can the assumption that V = L be weakened in Corollary 3.46? In particular, let (K,+,·,0,1,≤) be a Cauchy complete, λ-spherically complete totally or-dered field with bn(K) = λ; are the weak compactness of λ and λ-wBWTK equivalent under GCH?
Chapter 4
The generalised reals: transfinite computability
Remarks on co-authorship. The results of this chapter are partially due to a collab-oration of the author with Hugo Nobrega. In particular all results in §4.2 are, unless otherwise stated, due jointly to the author and Hugo Nobrega. These results were mostly developed when the collaborators were Visiting Fellows at the Isaac Newton Institute for Mathematical Sciences for the program Mathematical, Foundational and Computational Aspects of the Higher Infinite. The outcomes of this collaboration have also been pub-lished in a joint paper [39] for which the authors won the Best Student Paper award at the conference Computability in Europe 2017 held in Turku, Finland in 2017. Lemma 4.7 and Lemma 4.25 were not included in [39] and were proved later solely by the author.
The results in§4.3 are due solely to the author and will be published in the proceedings volume of the conference Computability in Europe 2019 as an invited paper.
4.1 Introduction
In classical computability theory computations are thought as finite and discrete pro-cesses carried out by (idealised) machines. Although these assumptions are quite natural, since the beginning of the research in this area, researchers have been developing theories in which these assumptions are weakened; see, e.g., [102].
Particularly interesting for us are those notions of computability in which the finiteness of the process is relaxed. The idea is to allow computations to “go on forever”. Different formalisations of this abstract notion gave rise to different models of computability. In this chapter we will consider models of transfinite computability.
The modern approach to the study of transfinite computability began with the seminal paper [45] in which Hamkins and Lewis introduced the notion of infinite time Turing machine (ITTM). These machines are Turing machines whose clock runs over ordinal numbers rather than just natural numbers. Therefore, infinite time Turing machines have the same hardware as classical Turing machines and run classical programs; but, contrary to their classical counterpart, they can run for an amount of time corresponding to a transfinite ordinal. An ITTM behaves as a normal Turing machine at successor stages, while at limit stages the head goes to the first cell of the tape, the content of the
tape is computed by using pointwise inferior limits, and the state of the machine is set to a special limit state.
Time and space are treated asymmetrically in infinite time Turing machines. Indeed, while tapes have length ω the machine is allowed to run for an arbitrary transfinite amount of steps. This asymmetry is the source of behaviour of ITTMs that is different from that of classical Turing machines.
The theory of infinite time Turing computability is very rich and deeply connected to set theory; see, e.g., [15, 45–47, 85, 110].
In [56], Koepke started the study of ordinal Turing machines (OTMs) which are meant to repair the asymmetry between space and time introduced by ITTMs. An ordinal Turing machine is a machine with an infinite tape whose length is the supremum of all the ordinals; and which, as an ITTM, can run for a transfinite amount of time. As for ITTMs, OTMs run classical Turing machine programs and behave as standard Turing machines at successor stages. At limit stages, the content of the tape is computed by taking the point-wise inferior limit, the position of the head is set to the inferior limit of the head positions at previous stages, and the state of the machine is computed using the inferior limit of the states at previous stages.
The generalised version of many classical results from computability theory can be proved to hold for ordinal Turing machines; see, e.g., [17, 18, 25, 56, 58, 81, 88].
As we mentioned in §1.1, the fact that the construction of the generalised real line Rκ is carried out within the framework of surreal numbers gives us a natural notion of computability. Indeed, looking at the definitions of surreal numbers and surreal opera-tions in §1.3.6, it is not hard to see that they come with an intrinsically computational flavour.
In this chapter, we will exploit the computational nature of surreal numbers and of Rκ to generalise notions of computability which are based on real numbers in the classical framework.
The chapter is organised as follows: in §4.2 we will first use the generalised real line and ordinal time Turing machines to generalise the classical notion of type two Turing machines; then, we will use this new model of transfinite computability to start the generalisation of the classical theory of Weihrauch degrees; in §4.3, we will use the generalised real line to define a generalisation of Blum-Shub-Smale machines; we will compare this new model with the main models of transfinite computability; and we will show that this new notion of computability can serve as a very general type of transfinite register machines.