Appendix 1: Technical Notes on Set Theory

Home , Axiom of determinacy, Paradoxes of set theory, Reinhardt cardinal

Cohen’s Technique of Forcing

Cohen’s technique of forcing involves constructing an outer model L[G] of L. Th e aim of this technique is to extend L by adding a new set G which is forced to have certain desired properties not possessed by the sets in L (e.g. the property of violating the continuum hypothesis). Th is is done in such a way that L[G] (i.e. the result of adjoining to L the set G and everything constructible from G together with the elements of L) is consistent with the axioms of ZFC. Using the work of Gödel, Cohen (1963: 1143) assumed the existence of a model M which satisfi es the axioms of ZF + V = L (i.e. ZF with the Axiom of Constructability). He took M to be a countable standard transitive 1 model. Countability is convenient and it follows from the Lowenheim–Skolem theorem that if a fi rst-order set theory has a model at all, then it has a countable model. Cohen’s focus on M being standard

1 A set X is countable if its elements can be put in a one-to-one relation with the elements of ω. A set X is transitive if every element of X is a subset of X. A model M of a set theory is standard if the elements of M are well-founded sets ordered with the set membership relation.

© Th e Author(s) 2017 331 J. McDonnell, Th e Pythagorean World, DOI 10.1007/978-3-319-40976-4 332 Appendix 1: Technical Notes on Set Theory and transitive also made his proof easier, but these are not essential criteria for the technique of forcing in general. For every object X which can be proved to exist in ZF + V = L, M contains an analogue object Y, which is often equal to X (Chow 2008: ℵ 6). For example, M contains all the ordinals and 0 . However, the analogue Y is not always equal to X. For example, since M is countable, ℵ the set y which is the analogue of the power set of 0 in M will also be ℵ countable, so it cannot be the same as the power set of 0 in ZF + V = L (which Cantor showed to be uncountable). Many subsets are missing from y. We say that y is ‘uncountable in M’ even though it is countable in ZF + V = L. Th e notions of ‘countability’ and ‘powerset’ are relative to the model. Notions, such as ‘the empty set’, which do not depend on the model, are said to be absolute . Cohen’s aim (1963: 1144) was to extend M by adjoining a subset U ℵ of 0 that is missing from M and that violates the continuum hypothesis ℵ (e.g. contains 2 or more reals). Th e resulting model M[U] would be a model for ZF and the Axiom of Choice and the negation of the continuum hypothesis. He was concerned that, if U were chosen indiscrimi- nately, then the set that plays the role of a particular cardinal number in ℵ M might not play the same role in M[U]. For example, the set 2 in M might become countable in M[U].2 However he argued that, if U were chosen in a generic manner from M,3 then no new information would be extracted from it in M[U] which was not already contained in M, so sets such as the cardinals would be preserved. Cohen wrote:

we do not wish U to contain “special” information about M, which can only be seen from the outside… Th e U which we construct will be referred to as a “generic” set relative to M. Th e idea is that all the properties of U must be “forced” to hold merely on the basis that U behaves like a “generic” set in M. Th is concept of deciding when a statement about U is “forced” to hold is the key point of the construction. (Cohen 1966: 111)

Cohen formalised the notion of a set U being M-generic and of the forcing relation, through which satisfaction for the extension of M could be

2 Th is behaviour is called cardinal collapse. 3 For example, to be generic, U should contain infi nitely many primes. Appendix 1: Technical Notes on Set Theory 333 approached from inside M. He showed how to build the new set U one step at a time, tracking what new properties of M[U] would be “forced” to hold at each step (Chow 2008: 8). Th e result was a model which, together with Gödel’s earlier work on L, proved the independence of the continuum hypothesis in ZFC.

Generating Large Cardinal Axioms Using Elementary Embeddings

Th e most general method for generating large cardinal axioms uses elementary embeddings (Woodin 2011b: 97–99). An elementary embedding j of V into a transitive class M is a mapping j: V → M which preserves all relations defi nable in the language of set theory (i.e. if arbitrary a 0 ,… Ф ,a n in V satisfy an arbitrary fi rst-order formula in the language of set Ф theory, then their mappings j(a0 ),…,j(an ) satisfy in M, and conversely). A basic template for large cardinal axioms asserts that there exists such an embedding j which is not the identity. If j is not the identity then there will be a least ordinal which is not mapped to itself (i.e. such that j(α) ≠ α) and this is called the critical point of j. Th en the critical point of j is a measurable cardinal and the existence of the transitive class M and the elementary embedding j are witnesses for this. By placing restrictions on M, by requiring it to be closer and closer to V (in a manner analogous to width refl ection for inner models, but precisely defi ned in terms of the closure properties of M (Koellner 2011a: 10–12), we can generate stronger and stronger large cardinal axioms. By continuing this process to its natural limit; that is, by taking M = V, the axiom generated asserts the existence of a Reinhardt cardinal which is the critical point of a non-trivial elementary embedding j: V → V. However, this axiom has been shown to be inconsistent with ZFC by a theorem of Kunen (Kanamori 1994: 319). Th e strongest large cardinal axiom which evades Kunen’s proof, and so is not known to be inconsistent with ZFC, is one asserting the existence of an ω -huge cardinal:

A cardinal κ is an ω-huge cardinal if there exists λ > κ and an elementary → κ embedding j: V λ+1 Vλ +1 such that is the critical point of j. (Woodin 2011c: 456) 334 Appendix 1: Technical Notes on Set Theory

Th e outlined procedure for generating large cardinal axioms does not show us how to build up to the large cardinal from below and so it is not intrinsic. Also, to be very clear, it does not guarantee the existence of the large cardinal or the relative consistency of the associated axiom with ZFC. It simply provides us with a means of pushing the level of cardinals as high as it can go, right up to the point where we bump into Kunen’s theorem. A further sobering fact is that even the largest of large cardinal axioms is not enough to determine the continuum hypothesis. Levy and Solovay (1967) showed that all such large cardinal assumptions are rela- tively consistent with both the continuum hypothesis and its negation.

Large Cardinal Axioms and Deﬁ nable Determinacy

In the early development of set theory, there was a focus on studying the properties of the hierarchy of defi nable sets of real numbers L(R)4 — what is now called descriptive set theory (Koellner 2011b: 16–24). Many important results were proved. Bendixson (1883) and Cantor (1884) showed that all closed sets of reals have the perfect set property5 and so satisfy the continuum hypothesis. Th is result was later extended to the fi rst-order in a hierarchy of certain defi nable open sets. Another property of interest is Lesbesgue measurability. It was known that there are sets of reals which are not Lesbesgue measurable (giving rise to the Banach–Tarski paradox discussed in Sect. 4.2.3) but it was not known whether or not any defi nable sets of reals lacked this property. Again, proofs of Lesbesgue measurability were limited to the fi rst-order in a hierarchy of certain defi nable open sets. Th e story was much the same

4 See the discussion of Gödel’s constructible universe L in Sect. 4.2.4 for a defi nition of defi nable sets. By starting with the real numbers R and iterating the defi nable power set operation along the ordinals, we get the hierarchy L(R) of defi nable sets of reals. 5 See Koellner (2011b: 17–18) for a defi nition of the hierarchy of defi nable open and closed sets of reals. A set of reals is perfect if it is non-empty, closed and contains no isolated points. A set of reals has the perfect set property if it is either countable or contains a perfect subset. If a set of reals contains a perfect subset, it must have the same size as the continuum. Th erefore, if a set of reals has the perfect set property, it must satisfy the continuum hypothesis (Koellner 2011b: 20). Appendix 1: Technical Notes on Set Theory 335 for other regularity and structural properties of interest.6 Beyond this, independence results developed in the 1960s (following the work of Gödel and Cohen) showed that many questions about the properties of defi nable sets of reals are not decidable in ZFC. Th e situation was greatly clarifi ed in the 1970s and 1980s with the development of new axioms for ZFC (Koellner 2011b: 24–30). One strand of research studied the implications of large cardinal axioms. Th e other strand, to do with axioms of determinacy , had its origins in a quite diff erent area of mathematics—game theory. Th e Axiom of Determinacy , AD, is an assertion about a two-person infi nite game. It says that for every set of reals A, there is a winning strategy for one of the players in the game associated with A.7 AD is known to be inconsistent with the Axiom of Choice so, in the search for new axioms for ZFC, it is usual to consider restrictions of AD to certain defi nable sets of reals, giving rise to axioms of defi nable determinacy which are consistent with Choice. For example, by restricting AD to the hierarchy of defi nable sets of real numbers L(R), one gets the new axiom candidate ADL(R) . Th e underlying connection between determinacy and the regularity properties of sets of reals is not immediately obvious to outsiders, but to some set theorists “determinacy lies at the heart of the regularity properties and may be considered their true source” (Koellner 2011b: 26). What is obviously remarkable is the intricate web of results interweaving large cardinals, determinacy and regularity properties; as outlined by the examples below:

Th eorem A.1.3.1 Assume ZFC + AD L(R) . Th en all the sets of reals in L(R) have the perfect set property, are Lesbesgue measurable and have the property of Baire.

Th eorem A.1.3.2. (Woodin) Assume ZFC + “there exist ω-many Woodin cardinals with a measurable cardinal above them all”. Th en AD L(R) .

6 Note that there are many such properties other than the ones mentioned in the text; for example, the property of Baire and the uniformisation property (Koellner 2011b). 7 See Woodin (2010a: 4) for details. 336 Appendix 1: Technical Notes on Set Theory

Th eorem A.1.3.3. (Woodin) Assume AD L(R) . Th en there is an inner model N of the theory ZFC + “there are ω-many Woodin cardinals”.

Th eorem A.1.3.4. (Woodin) Assume ZFC + “there is a proper class of Woodin cardinals”. Th en the independence of statements in L(R) cannot be established by forcing.8

Many more such linked theorems are discussed by Koellner (2011b: 26–35). Th ese theorems bring the two strands of research on new axioms together in an unexpected way. Th ey show that defi nable determinacy is necessary and suffi cient to prove the regularity properties of certain defi n- able sets of reals; and that large cardinals are necessary and suffi cient to prove defi nable determinacy. Koellner (2011b: 30–38) has used these results to support the case for defi nable determinacy and, hence, for large cardinal axioms. Some of his main points are:

1 . Th e regularity properties of structure theory are desirable properties for defi nable sets of reals (e.g. they circumvent the Banach–Tarski paradox). 2. ADL(R) lifts the regularity properties of structure theory to the level of L(R). Conversely, all theories which have this desirable consequence imply ADL(R) . 3. For suffi ciently strong large cardinal axioms, which themselves imply AD L(R), the results regarding regularity properties cannot be undone by forcing from L(R). 4. AD L(R) is implied by many theories which have proved useful in strengthening ZFC and “there is reason to believe that AD L(R) lies in the overlapping consensus of all suffi ciently strong, natural theories” (Koellner 2011b: 36). 5 . Th ere is promising research to indicate that the case can be strength- ened beyond L(R).

8 Th is is a very loose statement of an important theorem by Woodin. For a precise statement, see Th eorem 12 of Koellner (2006). Appendix 1: Technical Notes on Set Theory 337

6 . D e fi nable determinacy and large cardinal axioms are at root equivalent: “We have here a case where intrinsically plausible principles from completely diff erent domains reinforce each other” (Koellner 2009b: 111).

Although the results are remarkable, not all set theorists fi nd the case for defi nable determinacy compelling. Th ey argue that descriptive set theory is only one area and focusing on another area might produce com- peting, incompatible axioms. Part of Koellner’s argument is the claim that this is unlikely but, of course, it depends on what areas one fi nds interesting and what weight one gives to intrinsically plausible results from diff erent areas. For example, Shelah (2003: 211–213) thinks that Gödel’s Axiom of Constructibility V = L is an interesting axiom with some appealing consequences, and it is known to be incompatible with one of the axioms of defi nable determinacy.

Deﬁ ning Effective Completeness

Th eorem A.1.3.4 from the previous section off ers the hope of overcoming forcing. It says that, assuming suffi ciently strong large cardinal assumptions, the truth value of statements in L(R) cannot be changed by forcing. As stated in Koellner and Woodin (2009: 1158): “Woodin cardinals ‘seal’ or ‘freeze’ the theory of L(R)”. Furthermore, the results of descriptive set theory show that the same large cardinal assumptions eff ectively resolve all of the open problems relating to the regularity and structural properties of defi nable sets of reals. So, in this sense, we can achieve eff ective completeness at the level of L(R). Recall that L(R) is the same as L(Vω +1 ). Th is means that we can achieve eff ective completeness at a level beyond, and inclusive of, the level of second-order arithmetic, Vω +1, but below the level of third-order arithmetic, Vω +2 . It would be desirable to extend this result to V ω+2 and higher levels. However, the continuum hypothesis is a statement in Vω +2 and we know that it cannot be resolved by any standard large cardinal axioms, so something more will be needed. To defi ne eff ective completeness at higher levels of the set hierarchy, it is necessary to introduce some technical notions. 338 Appendix 1: Technical Notes on Set Theory

Firstly, there is the notion of the Levy hierarchy of formulas . A formula Σ in a language that contains the language of set theory is of type 0 (or, Π equivalently, 0) if it is built from atomic formulas through the use of logical connectives and bounded quantifi ers ∀x ∈ y and ∃x ∈ y . A formula Σ ∃ is of type 1 if it is of the form x φ where x is a list of variables and φ Π Π ∀ is of type 0 . A formula is of type 1 if it is of the form x φ where φ is Σ Σ of type 0 . In general, a formula is of type n , n>1, if it is of the form ∃ Π Π x φ where φ is of type n-1 . And a formula is of type n , n>1, if it is of ∀ Σ the form x φ where φ is of type n-1 . In set theory, an assertion is called Π α ╞ 2 if it is of the form: “For every infi nite ordinal , Vα φ ”, for some Σ ╞ 1 -sentence φ , where is the symbol for consequence in the language. Σ Similarly, an assertion is called 2 if it is of the form: “Th ere exists an infi - α ╞ Π Σ nite ordinal such that Vα φ ”, for some 1 -sentence φ . A 2 -sentence Π is expressible as the negation of a 2 -sentence. 2 Σ2 Secondly, there is the notion of a Σ n sentence . A n sentence is one of ╞ Σ the form “Vω +2 φ ”, where φ is a formula which is n in the Levy hier- Σ2 archy. Th e continuum hypothesis is equivalent to a 1 sentence. Hence, it is a sentence at the level of third-order arithmetic, V ω+2 and at the fi rst level of the Levy hierarchy. Th irdly, there is the notion of stratifying the universe of sets in terms of the cardinal partition, H( κ ), rather than the familiar ordinal partition, Vα . Here, κ is an infi nite cardinal and H(κ) is the set of all sets whose transitive closure has cardinality less than κ (i.e. the set of all sets which have cardinality less than κ, whose members have cardinality less than κ, whose members’ members have cardinality less than κ, etc.) (Koellner 2011c: 13). For example, H(ω) = Vω and the theories of the structures ω H( 1 ) and Vω +1 are mutually interpretable. If the continuum hypothesis ω ω is true, then H( 2 ) and Vω +2 are mutually interpretable, otherwise H( 2 ) is less rich than Vω +2 . So, in investigating the continuum hypothesis, it ω makes sense to consider H( 2 ) fi rst. Fourthly, and lastly, there is the notion of Ω -completeness . Th is will take a bit more explanation. We know that in order to resolve the continuum hypothesis, a new notion is needed. Part of the answer might be to introduce a logic which is stronger than fi rst-order logic and is well behaved in the sense that its consequences cannot be altered by forcing in the pres- ence of large cardinal axioms (Koellner 2011c: 15–17). Ω-logic is such a Appendix 1: Technical Notes on Set Theory 339 logic.9 It has a notion of consequence, denoted ╞Ω, which is robust to forcing under the assumption of a proper class of Woodin cardinals. It also has a quasi-syntactic proof relation, denoted ├Ω, defi ned in terms of universally Baire sets of reals which act as witnesses for the “proofs” in the logic (Woodin 2001b: 684). Ω-logic is known to be sound. Hence, if a sentence φ is Ω-provable in a set theory T, then it is necessarily an Ω-consequence of T. However, it is not known whether Ω-logic is complete; that is, whether every sentence φ which is an Ω-consequence of a set theory T is necessarily Ω-provable in T. Th e conjecture that Ω-logic is both sound and complete is called the Ω conjecture . Th is is so important that I will highlight it:

Th e Ω conjecture says that Ω-logic is both sound and complete.

A theory T is said to be Ω -complete for a collection of sentences Γ if for each φ Є Γ, either T ╞Ω φ or T ╞Ω ~φ (Koellner 2011c: 17). Hence, the Ω-completeness of a theory for some collection of sentences means that those sentences are decidable in Ω-logic.

Exploring the Effective Completability of Set Theory

Using the notions introduced in Sect. A.1.4, Th eorem A.1.3.4 asserting the invariance of the theory of L(R) under set forcing can be rephrased as:

Th eorem A.1.5.1. (Woodin) Assume ZFC and that there is a proper class of Woodin cardinals. Th en ZFC is Ω-complete for the collection of sentences of the form “L(R) ╞ φ ”. (Koellner 2011c: 17)

Because of the nature of the continuum hypothesis, we know that there is no standard large cardinal axiom LCA such that ZFC + LCA Ω Σ2 is -complete for 1 sentences. However, there is the possibility that a supplementary axiom could achieve this outcome and, surprisingly, it

9 For more information on Ω-logic and its relation to full fi rst- and second-order logic, see Koellner (2010). 340 Appendix 1: Technical Notes on Set Theory turns out that simply assuming the continuum hypothesis as an additional axiom does achieve it. Th us:

Th eorem A.1.5.2. (Woodin) Assume ZFC and that there is a proper class of measurable Woodin cardinals. Th en ZFC + CH is Ω-complete Σ2 for 1 . (Koellner 2011c: 29)

Ω Σ2 Moreover, up to -equivalence, CH is the unique 1 -statement that Ω Σ2 is -complete for 1 (Koellner and Woodin 2009: 1161). Of course, assuming the continuum hypothesis as an axiom doesn’t count as resolving it. On the contrary, for many years, Woodin promoted a parallel case in which the continuum hypothesis is negated (Woodin 2001b). Th us, assuming that there is a proper class of Woodin cardinals and that the Strong Ω Conjecture10 holds, Woodin (1999) proved that there is an axiom A such that ZFC + A is Ω-complete for the struc- ω ture H( 2 ) and that any such axiom has the feature that the continuum hypothesis is false in the associated theory. Th e axiom A is not unique, there are many such Ω-complete theories (Koellner and Woodin 2009). However, Woodin found a particular axiom, called Axiom (*) , which Π gives a “maximal” theory in the sense that it satisfi es all 2 -sentences that can possibly hold (Koellner 2011c: 14). Assuming Axiom (*), the ℵ ℵ continuum hypothesis is false and 2^ 0 = 2 . Woodin (2001b: 687) saw Axiom (*) as being a candidate for the generalisation of ADL(R) to the ω Ω ω structure H( 2 ); that is, settling in -logic the full theory of H( 2 ) just as ADL(R) had settled the open problems of descriptive set theory in L(R). So now, we have two cases for extending our Ω-complete theories: Case 1, in which the continuum hypothesis is assumed as an axiom and Case 2, which assumes Axiom (*) and has the consequence that the con- Ω Σ2 tinuum hypothesis is false. Case 1 is only -complete for 1 sentences, Ω ω whereas Case 2 is -complete for the whole of the structure H( 2 ). Case 2 uses the additional assumption that the Strong Ω Conjecture holds. Does this mean that there is a bifurcation of the universe of sets at the level of the continuum hypothesis? Th is question can’t really be answered at the local level since there is always the possibility that

10 Th e Strong Ω Conjecture is the Ω Conjecture plus a conjecture about an extended form of the AD. See Koellner (2011c: 16–17). Appendix 1: Technical Notes on Set Theory 341 further extensions of the theory will favour one path over the other. It is known that if the Strong Ω Conjecture holds, then one cannot have Ω an -complete theory of the whole of the structure Vω +2 :

Th eorem A.1.5.3. (Woodin) Assume ZFC and that there is a proper class of Woodin cardinals. Assume the Strong Ω Conjecture. Th en there is no recursively enumerable theory A such that ZFC + A is Ω-complete Σ2 for 3 . (Koellner and Woodin 2009: 1162)

Σ2 So this path is eff ectively closed off at the 3 level. On the other hand, the following theorem shows that the continuum hypothesis is not suf- Ω Σ2 fi cient to provide an -complete theory at the 2 level:

Th eorem A.1.5.4. (Jensen, Shelah) ZFC + CH is not Ω-complete for Σ2 2 . (Koellner and Woodin 2009: 1162)

So this path is also closed. Σ2 11 One can consider further local scenarios at the 2 level. I take it that the upshot of these is that, if the Strong Ω Conjecture holds, then it is likely that there is a “best” theory in which the continuum hypothesis has a defi nite answer, even though there is not yet a compelling case to tell us which way that answer will fall (Koellner 2011c: 33). My impression is that the local approach is nearing the end of its usefulness. Th ere were remarkable successes at the level of V ω+1 which were extended with great eff ort to fragments of Vω +2 but without any clear result for the continuum hypothesis. Cases multiply and it becomes hard to keep track of the arguments. Th e justifi cation for the “best” cases becomes less compelling. Perhaps this situation will improve with further research but, regardless, the local approach has inherent limitations. If the Strong Ω Conjecture does not hold, then there is the possibility Ω that we could have an -complete theory for V ω+2. In fact, it might be possible to achieve an Ω-complete theory for any specifi ed level V α and to piece these together into a coherent theory of the entire universe of sets. If this were achieved, then the theory would be a candidate for the eff ective

11 See, for example, Koellner (2011c: 27–33). 342 Appendix 1: Technical Notes on Set Theory completion of set theory along the lines envisaged by Gödel (1946: 151) (see Sect. 4.5.1). However, Koellner and Woodin (2009) have shown that such a theory could not be unique. If there is an Ω-complete theory of Vα for α ≥ (ω+2), then there is another, equally good but incompatible Ω-complete theory which diff ers on the truth value of the continuum hypothesis.12 Th is seriously raises the spectre of pluralism. Th e existence of two, or more, coherent but incompatible Ω-complete theories of V would undermine the universalist’s concept of truth. It would not be possible to combine such theories in the manner suggested by Martin (2001). Such a scenario is considered by Steel (2004) and discussed at length by Maddy (2005: 369–373). Th e theories would all be universal and equivalent in terms of the mathematics that would result (i.e. choosing one theory over another would not result in any behavioural or methodological diff erences). In choosing one theory, the others would be accessible as generic extensions. Th e continuum hypothesis (and other undecidable statements) would be meaningless in an absolute sense, but acquire a relative value as a matter of convention. Th is scenario is highly speculative but does serve to crystallise a plausible pluralist case. Note that a multiverse of this kind does not conform to the radical pluralist view of what a multiverse might consist of (i.e. all consistent theories). It locks in a universalist view at least to the level of ZFC + “there is a proper class of measurable Woodin cardinals” which is essential for the application of the notion of Ω-completeness. In this way, it avoids the problems which plague the radical pluralist view. Woodin (2011d) has used a global perspective to defi ne and investi- gate a multiverse of a related kind, which he calls the generic multiverse . Ultimately, he sets up the generic multiverse position as a strawman, only to show that it is untenable under fundamental principles of set theory, provided that the Ω Conjecture is true . Here, once again, we see the impor- tance of the Ω Conjecture. Woodin’s generic multiverse is generated from V by closing under generic extensions (forcing) and under generic refi nements (inner models of a universe which the given universe is a generic extension of). Th e associated generic multiverse view of truth is that a sentence is true if

12 Note that the continuum hypothesis is just a particular case; there are other suitable sentences Ω which could be forced to diff er between -complete theories of V α. Appendix 1: Technical Notes on Set Theory 343 and only if it holds in each universe of the generic multiverse. Letting the background theory be ZFC + “there is a proper class of measurable Woodin cardinals”, this view entails that the continuum hypothesis is indeterminate whilst ADL(R) is true (and, hence, all the results of descriptive set theory, described previously, are locked in). Π φ Woodin (2011d: 19–21) proves that for each 2 -sentence , the following are equivalent:

1 . φ holds across the generic multiverse; 2 . “ φ is true in Ω-logic” holds across the generic multiverse; 3 . “ φ is true in Ω-logic” holds in at least one universe of the generic multiverse.

Th is means that if φ is Ω-valid in one universe, then it holds in all the universes of the generic multiverse. Notably, a sentence asserting the existence of a proper class of Woodin cardinals (and its consequences) and a sentence asserting the truth of the Ω conjecture would be invariant across the generic multiverse. Woodin says: “the notion of truth is the same as defi ned relative to each universe of the multiverse, so from the perspective of evaluating truth all the universes of the multiverse are equivalent” (Woodin 2011d: 15). He sees this as an important point in favour of the generic multiverse view. For one thing, it implies that a statement (such as a consistency statement) can’t be indeterminate in one universe and determinate in another, as might happen under the radical pluralist view. Th us, truth is not so sensitive to the meta-universe in which the generic multiverse is being defi ned and the pluralist isn’t forced to retreat to an infi nite regression of meta-universes in order to defi ne his position. Next, Woodin (2011d: 17–18) formulates two related multiverse laws:

Π 1 . Th e set of 2 -multiverse truths cannot be recursive in the set of multiverse truths of Vα for any specifi able α; and Π 2 . Th e set of 2 -multiverse truths is not defi nable in V α across the multiverse for any specifi able α.

Th ese laws capture the idea that any reasonable concept of multiverse truth cannot be reduced to, or defi ned in, an initial fragment Vα of the universe V. Th at would lead to “a brand of formalism that denies the 344 Appendix 1: Technical Notes on Set Theory transfi nite by reducing truth about the universe of sets to truth about a simple fragment” (Woodin 2011d: 17). Woodin’s multiverse laws are in the spirit of the refl ection principles of set theory. Woodin (2011d: 23–25) goes on to show that, assuming both the Ω Conjecture and the existence of a proper class of Woodin cardinals, both multiverse laws are violated by the generic multiverse position.

Specifi cally, he proves that both laws are violated at the level of Vδ 0+1 δ where 0 is the least Woodin cardinal. Th is result can be sharpened to V ω+2 for the fi rst multiverse law, but the import is the same in any case: the generic multiverse position is untenable because the set of all its truths is reducible to the set of truths of some initial fragment of V. Th e pluralists’ best response, short of placing restrictions on the generic multiverse, is to deny the truth of the Ω Conjecture. Woodin ([2011d: 28] and [2011b: 111–112]) argues that there is evidence for the Ω Conjecture, albeit nothing conclusive. Firstly, the Ω Conjecture can be shown to be consistent with the theory ZFC + “Th ere is a proper class of Woodin cardinals”. Secondly, the assertion of its Ω-satisfi ability Σ is a 2 -assertion which is invariant across the generic multiverse and Σ there are no known examples of 2-statements that are provably absolute and not settled by large cardinal axioms. Th irdly, “recent results indicate that if [the Inner Model] program can succeed at the level of supercompact cardinals then no large cardinal hypothesis whatso- ever can refute the Ω Conjecture” [Woodin 2011d: 28]. Th ese are the results from the theory of Ultimate L (as discussed in Sect. 4.5.3 in the main text).

Fending Off a Potential Threat to the Universalist View

In “Th e Realm of the Infi nite” (2011b), Woodin begins by arguing that we implicitly accept the meaningfulness of the large fi nite and that there is nothing to prevent us from accepting the meaningfulness of the universe of sets (as discussed in Sect. 4.6 in the main text). In the next part of his paper, he gives further arguments for accepting the truth of set theory, this time in opposition to a hypothetical Sceptic who denies any genuine Appendix 1: Technical Notes on Set Theory 345 meaning to a conception of uncountable sets (i.e. he accepts the meaningfulness of the large fi nite and the completed infi nity of the natural numbers, but not of uncountable sets, and certainly not of large cardinals). Woodin lays out the evidence for large cardinals and makes his prediction that the theory ZFC + “there exist infi nitely many Woodin cardinals” will never be shown to be inconsistent (even though, analogously to the large fi nite case, an inconsistency could conceivably be found in the physical realm). Th e Sceptic accepts the evidence for the consistency of the theory, and agrees with the prediction, but doesn’t accept that it is the result of the truth of the axioms (or, equivalently, the existence of the large cardinals). He suggests that the infi nite realm is so inclusive that any reasonably defi ned large cardinal axiom would turn out to be consistent, provided that there isn’t a simple proof of its inconsistency. In particular, he points to the Reinhardt cardinal axiom, which is the strongest possible large cardinal axiom generated by the elementary embedding template (as discussed in Sect. A.1.2). Kunen produced a simple proof of its inconsistency with ZFC, but that proof depended on using the Axiom of Choice, and it is still possible that it is consistent with ZF. Furthermore, if the theory ZF + “there is a weak Reinhardt cardinal” is consistent, then it implies the consistency of ZFC + “there is a proper class of ω-huge cardinals” and thereby accounts for all the consistency predictions of Woodin’s accepted large cardinal hierarchy. So this one claim—the formal consistency of the Reinhardt axiom with ZF—would account for all the claims of set theory. Th e Skeptic’s claim creates a serious problem for Woodin. Th e consistency of the Reinhardt cardinal axiom could not be grounded in the existence of a Reinhardt cardinal because then we would have an imme- diate bifurcation of set theory into a universe in which the Axiom of Choice holds and one in which it does not hold. Anyway, there is no fi ne structure theory for a Reinhardt cardinal, so no witnesses to its existence. Rather, the set theorist might try to account for its formal consistency by showing that the theory ZF + “there is a weak Reinhardt cardinal” is equiconsistent with a theory of the form ZFC + LCA, for some large cardinal axiom LCA; in the same way that ZF + AD can be shown to be equiconsistent with ZFC + “there exist infi nitely many Woodin cardinals”, even though AD itself is incompatible with the Axiom of Choice (Woodin 2011b: 100–101). However, this approach will not be possible 346 Appendix 1: Technical Notes on Set Theory because the Reinhardt cardinal axiom is stronger than any large cardinal axiom not known to refute the Axiom of Choice and, so, will not be equiconsistent with any such theory. Next, Woodin explores the idea that the generic multiverse view of truth might be able to account for the prediction that weak Reinhardt cardinals are consistent with ZF (Woodin 2011b: 107–108). Again, this approach will not be possible because, as we saw previously, the generic multiverse view of truth is untenable under the assumption of the Ω Conjecture because it violates the multiverse laws. Next, Woodin plays his trump card—Ultimate L. If a solution to Ultimate L could be found, then one corollary would be that the Reinhardt cardinal axiom is inconsistent with ZF (Woodin 2011b: 115–116). So, rather than contorting ZFC to account for the consistency of weak Reinhardt cardinals with ZF, it could be proved that they are, in fact, inconsistent. Th is would comprehensively refute the Skeptic’s claim. More than that, it would provide “for the fi rst time an example of a natural large cardinal axiom proved to be inconsistent as a result of a deep structural analysis” (Woodin 2011b: 116). Such an inconsistency result could never be achieved by Inner Model Th eory because of its incremental nature. But Ultimate L would provide us with the means to explore the boundary between possible and impossible large cardinal axioms, strengthening its claim to being the ultimate arbiter of mathematical truth and existence. Woodin’s hunch is that it would enable us to eliminate essentially all the large cardinal axioms known to contradict the Axiom of Choice, tantamount to a proof of the Axiom of Choice (Woodin 2011c: 470). Th en, according to Woodin’s view, any suggested set theory would need to be justifi ed by fi rst establishing its equiconsis- tency with a specifi c level of Ultimate L. Ulimate L, in its turn, would be justifi ed by our understanding of the hierarchy of large cardinal axioms as “true axioms about the universe of sets” (Woodin 2011b: 96). Woodin thinks that one of the main pieces of evidence supporting the truth of large cardinal axioms is our ability to construct their fi ne structure theory. Using Inner Model Th eory, evidence for each large cardinal axiom is built up from below. Starting from Gödel’s L, minimal extenders are added at each stage to witness the existence of the next large cardinal in the hierarchy. However, we know that this process cannot be extended beyond the level of an inner model with one supercompact cardinal. Th e reason is that Appendix 1: Technical Notes on Set Theory 347

Woodin’s investigation of this case has shown that such a model would automatically inherit all stronger large cardinals. Th en these stronger large cardinal axioms would require some further extrinsic justifi cation. Woodin suggests that the answer to this dilemma is to use the structural analogy of L(R) the theory of L(R) under AD to the theory of L(Vλ +1) under the assumption of an ω-huge cardinal to reveal the structure theory of ω-huge cardinals without constructing a full inner model theory (Woodin 2011c: 469–470).

References

Bendixson, L. 1883. Quelques Th eoremes de la Th eorie des Ensemble de Points. Acta Mathematica 2: 415–429. Cantor, G. 1884. De la Puissance des Ensembles Parfaits de Points. Acta Mathematica 4: 381–392. Cohen, P. 1963. Th e Independence of the Continuum Hypothesis. Proceedings of the National Academy of Sciences of the United States of America 50(6): 1143–1148. ———. 1966. Set Th eory and the Continuum Hypothesis . Menlo Park, CA: Addison-Wesley. Chow, T. 2008. A Beginner’s Guide to Forcing, pp. 1–16, downloaded from http://arxiv.org/abs/0712.1320 Gödel, K. 1946. Remarks Before the Princeton Bicentennial Conference on Problems in Mathematics. In K. Gödel Collected Works , Vol. II, edited by S. Feferman et al., 150–153. Oxford: Oxford University Press, 1990. Kanamori, A. 1994. Th e Higher Infi nite: Perspectives in Mathematical Logic . Berlin: Springer-Verlag. Koellner, P. 2006. On the Question of Absolute Undecidability. Philosophia Mathematica 14(2): 153–188. ———. 2009b. Truth in Mathematics: Th e Question of Pluralism. In New Waves in Philosophy of Mathematics, edited by O. Bueno and Ø. Linnebo, 80–116. Hampshire: Palgrave Macmillan. ———. 2011a. Independence and Large Cardinals. In Th e Stanford Encyclopedia of Philosophy (Summer ed.). Edited by E. N. Zalta, down- 348 Appendix 1: Technical Notes on Set Theory loaded from http://plato.stanford.edu/archives/sum2011/entries/ independence-large-cardinals/ ———. 2011b. Large Cardinals and Determinacy, downloaded from http://logic.harvard.edu/EFI_LCD.pdf , pp. 1–44. ———. 2011c. Th e Continuum Hypothesis, downloaded from http://logic.harvard.edu/EFI_CH.pdf , pp. 1–38. Koellner, P., and W. Woodin. 2009. Incompatible Ω-Complete Th eories. Journal of Symbolic Logic 74: 1155–1170. Levy, A., and R. Solovay. 1967. Measurable Cardinals and the Continuum Hypothesis. Israel Journal of Mathematics 5: 234–248. Maddy, P. 2005. Mathematical Existence. Th e Bulletin of Symbolic Logic 11: 351–376. Shelah, S. 2003. Logical Dreams. Bulletin (New Series) of the American Mathematical Society 40: 203–228. Steel, J. 2004. Generic Absoluteness and the Continuum Problem, downloaded from http://www. lps. uci.edu/home/conferences/Laguna- Workshops/Laguna2004.html, pp. 1–24. Woodin, W.H. 1999. Th e Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, volume 1 of de Gruyter Series in Logic and its Applications . Berlin: de Gruyter. ———. 2001b. Th e Continuum Hypothesis, Part II. Notices of the American Mathematical Society 48: 681–690. ———. 2011b. Th e Realm of the Infi nite. In Infi nity: New Research Frontiers , edited by M. Heller and W. H. Woodin, 89–118. New York: Cambridge University Press. ———. 2011c. Th e Transfi nite Universe. In Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, edited by Matthias Baaz et al., 449–474. New York: Cambridge University Press. ———. 2011d. Th e Continuum Hypothesis, the Generic Multiverse of Sets, and the Ω Conjecture. In Set Th eory, Arithmetic, and Foundations of Mathematics: Th eorems, Philosophies, edited by J. Kennedy and R. Kossak, 13–42. New York: Cambridge University Press. Appendix 2: Problems with Plenitude

Overview

Th is appendix considers the problems that arise if too much plenitude is postulated. It refers to David Lewis’ On the Plurality of Worlds (1986) for a fair statement of the problems and a defence of plenitude. Lewis’ main reason for embracing plenitude is that it allows a great benefi t in terms of unity and economy of theory. He thinks that logical space for philosophers can be developed in an analogous way to set theory for mathematicians, thus enabling philosophers to use a vast realm of possibilia to reduce the required number of primitives and premises (1986: 4). To pave the way, he has shown in Parts of Classes (1991) how the constructions of set theory can be reproduced in mereology (i.e. the study of parts and wholes).

How Much Plenitude?

Lewis defends the thesis of modal realism which he states as:

the thesis that the world we are part of is but one of a plurality of worlds, and that we who inhabit this world are only a few out of all the inhabitants of all the worlds. (1986: vii)

According to this thesis, there are many possible worlds, some just like ours and some completely alien to our understanding. A possible world has parts which are possible individuals. A world is the mereological sum of all the possible individuals that are its parts.13 Spatiotemporal relations make individuals cohere in a world. Lewis advocates a principle of recombination. Parts of any world can be duplicated, multiplied, and combined into a new world. Th ere is just one proviso:

Th e only limit on the extent to which a world can be fi lled with duplicates of possible individuals is that the parts of a world must be able to fi t together with some possible size and shape of spacetime. Apart from that, anything can coexist with anything, and anything can fail to coexist with anything. (Lewis 1986: 89–90)

Th e proviso limits the ways in which space-time may be occupied (cf. set theory, in which there is no limit to the ways in which sets can be construed and the result is Cantor’s hierarchy of infi nities). If space-time is a continuum, then it cannot accommodate more individuals than the cardinality of the real numbers. Lewis is prepared to contemplate diff erent possible space-times but he is not prepared to let the cardinality of possible individuals yield consequences about the possible size of spacetime. Th erefore, the cardinality of possible individuals is limited. Forrest and Armstrong (1984) show that, without the proviso, the principle of recombination leads to paradoxes for Lewis’ possible worlds analogous to the paradoxes of set theory. For example, consider the world which contains all the possible worlds as parts. Th is “biggest world” exists according to Lewis’ mereology of possible worlds. Suppose that it has cardinality K. Forrest and Armstrong show how to construct a world with cardinality greater than K by extracting parts of the biggest world, dupli- cating and recombining them. However, by hypothesis, no world can have cardinality greater than K. Fortunately, the proviso prevents this argument from going through: the “bigger than biggest” world is too big to fi t into the biggest world and, so, too big to fi t into any possible space-time.

13 Th e mereological sum of several things is the least inclusive thing that includes all of them as parts. Appendix 2: Problems with Plenitude 351

Lewis relies on the proviso to prevent the construction of “bigger than biggest” worlds. Th is forces him to contemplate a cleavage between the structure of mathematics and the structure of possible worlds. Th ere will be some set-theoretical objects that are bigger than any possible world. Nevertheless (at some cost in terms of unity and economy of theory), Lewis sticks to the principle of plenitude in listing his conception of what there is altogether: “the possible worlds, the possible individuals that are their parts, and the mathematical objects, even if those should turn out to be pure sets not made out of the parts of the worlds” (1986: 111). Lewis thinks that we should accept mathematics and he rejects the suggestion that any mathematical objects could be fi ctional. He is less concerned by Benacerraf’s dilemma than some:

If we are prepared to expand our existential beliefs for the sake of theoretical unity, and if thereby we come to believe the truth, then we attain knowledge. We can know that there exist countless objects causally isolated from us and unavailable to our inspection. (1986: 109)

According to Lewis, mathematics and possible worlds give us knowledge beyond the reach of our causal acquaintance. We arrive at this knowledge by reasoning from general principles (such as recombination). Indeed, the same principles are involved in constructing sets and possible worlds; it’s just that possible worlds have an additional constraint. Th e constraint on possible worlds derives from the intuition that they are just like our world in many relevant ways. Th ey can be described as “concrete”, just like our world and so can be distinguished from the “abstract” world of mathematics, even though Lewis cannot fi nd a principled distinction between concrete and abstract (1986: 81–86). Th e question arises as to where to draw the line. Do concrete individuals have to fi t into fi nite-dimensional space-times? Lewis thinks that that is too restrictive and he hopes that there is a natural break higher up: “high enough to make room for all the possibilities we really need to believe in” (1986: 103). Th e break must be natural because, if it were arbitrary, then we could always imagine a possible world in which the boundary is moved. It is hard to see how this natural break would be defi ned and Lewis does not off er an answer. Th e constraint on possible worlds leads to an inevitable clash of intuitions. 352 Appendix 2: Problems with Plenitude

The Convergence of Physical and Mathematical Objects

I think that Lewis’ goals of unity and simplicity would be much better served if he abandoned the concrete/abstract distinction altogether or, at least, removed the structural distinction between concrete and abstract possibilia (in other words, if he allowed the structure of all possible worlds to be isomorphic to some mathematical structure and vice versa). Of course, this is a diffi cult option for those who see possible worlds as a combination of concrete parts because it means either abandoning the higher reaches of set theory (which Lewis will not do) or fi nding some physical interpretation for mathematical objects like large cardinals (impossible for Lewis). A similar dilemma is faced by philosophers of mathematics who favour Aristotelian realism (see e.g. [Franklin 2014]). Aristotelian realists take the view that the world consists of particulars and universals and that mathematical objects are universals which are in re (i.e. in the things themselves). In their view, mathematics studies structures which are instantiated in real objects such as bathroom tiles (groups) and coastlines (fractals). Th is leaves them with the problem of what to say about unin- stantiated mathematical objects. In the modal version of Aristotelian realism, the response is that some mathematical objects can be constructed combinatorially from concepts sourced in the real world—they are possibilia similar to Lewis’ possible worlds. Still, the assumption is that they could be instantiated in concrete things and this puts a constraint on what mathematical objects are supported by this view. Some mathematical objects don’t seem to have relevance to the physical world at all. For example, it is hard to envisage any role for the inaccessible numbers of set theory in any physical process. We would not reach that sort of cardinality even in a multiverse even if an infi nite number of universes had been continually generating baby universes since past infi nity. An inaccessible number is just that: it cannot be reached by the iteration of constructive processes such as taking unions of collections of smaller sets, or taking power sets, or taking any combination of opera- tions available in set theory. If one imagines translating all mathematical Appendix 2: Problems with Plenitude 353 objects into the background language of set theory, then the subset which is relevant to known physical processes is tiny. Feferman (2006b: 446) has suggested that it can be described by the predicative part.14 One option is to restrict mathematics to the point where it can be physicalised in some sense, by claiming that some part of it is inconsistent or fi ctional. For example, one might hold (with Nelson [2011]) that only the fi nite part of mathematics is consistent. Th is could be combined with a view of the universe as a digital computer operating on a discrete system of binary information (as held by Fredkin [1992] and Wolfram [2002]). In this view, there would be a core part of mathematics—the fi nite part— which is consistent, fi rmly grounded in physical intuition and real, and physics would only be concerned with that part. Physics and mathematics would both be translatable into a language of fi nite information pro- cessing. Presumably, Aristotelian realists could adopt this ultrafi nitist position and, indeed, Franklin (2014: 133) questions whether infi nities are really needed in applied mathematics. However, he seems to prefer to adopt a version of modal realism in which some of the truths of mathematics refer to actual physical instantiations whilst others refer to possible instantiations, and the possible instantiations can be infi nite. Th is reinstates the problem of where to draw the line. Beyond the fi nite, there is no principled place to draw the line between physics and mathematics. Aristotelian realism supports an epistemology in which our knowledge of mathematical objects is explained in terms of our knowledge of everyday objects but it does not account for our knowledge of infi nity. Hartry Field is a philosopher who abolishes the distinction between concrete and abstract objects by denying the existence of abstract entities. He champions the view that mathematics is fi ctional and of only instrumental use in physical theories. As a test case, Field (1980) tries to identify the ‘physical’ part of Newtonian gravitational theory (i.e. the part which has empirical support and/or corresponds to physical structure) and separate it from the ‘surplus’ part (i.e. the part which provides logical and instrumental support). He develops a nominalised version of the theory which refers only to space-time points and regions, not to

14 According to predicativists, the natural numbers form a defi nite totality but the collection of all sets of natural numbers (i.e. the power set) does not. 354 Appendix 2: Problems with Plenitude abstract mathematical objects. He defi nes bridge laws which map statements from the nominalised theory to its mathematical counterpart (and vice versa). According to Field, the advantage of using the mathematical theory is that it allows us to use all the deductive power of mathematics to demonstrate consequences which might have been obscure or diffi cult to prove in the nominalised theory. Th e mathematics doesn’t lead to any new assertions about the physical world—all the physical consequences of the theory can be derived from the underlying nominalised theory. However, in order for the nominalised theory to retain the full power of the mathematical theory, Field is forced to build almost all the structure and complexity of the real number system into space-time itself. Eff ectively, he adds pseudo-mathematical structure to the nominalised theory until it converges towards the structure of the mathematical theory. Th us, his attempt to nominalise Newtonian gravitational theory leads to a counter-productive blurring of the concrete/abstract boundary. I argue that the concrete/abstract boundary should be abolished by embracing all objects as abstract objects. What exists are minds (abstract individuals) and thoughts (abstract classes15 ). In this view, physical objects are a subset of mathematical objects but are not essentially diff erent. Both physical and mathematical objects are the objects of intention of some mind (i.e. they are thoughts). Th e structure of the world instantiates all of mathematics, all possible forms, and physical reality condenses out by a process of self-actualisation in thought. I assume that the world is fully rational and that there are minds which are able to understand, test and validate their theories about its structure at all levels. Th us, any set is oversee-able by some mind. Gödel’s incompleteness theorems show that every level of the set-theoretical hierarchy contains true statements which can only be proved using the resources of higher levels. Th e hierarchy of eff ective fi eld theories in physics is similarly incomplete. A coherent and fully rational theory of physics and mathematics together requires a mind which embraces all of mathematics but transcends it. Th at Mind is the encompassing mind which is the mereological sum of all individual minds and their thoughts. It is a proper class.

15 Not all classes are objects, some are proper classes. Appendix 2: Problems with Plenitude 355

Comments on Lewis’ Parts of Classes

In Lewis’ metaphysical framework, reality divides exhaustively into individuals and classes (1991: 7). An individual is anything that has no members but is a member of some class. A class is either a set or a proper class. A singleton is a class that has no parts except itself—it is a mereological atom. Every individual has a singleton and so does every set. Th e only things that lack singletons are the proper classes. A class is any mereological sum of singletons. A proper class does not have a singleton but it is the mereological sum of singletons. Th e notion of a singleton is a primitive. Lewis fi nds it mysterious (1991: 29–35). Singletons have only a single member so they do not fall under the usual description of a set as being a collection of many into one. He says: “Our utter ignorance about the nature of the singletons amounts to utter ignorance about the nature of classes generally” (1991: 31). I say that a singleton is a thought: it is the result of a mind taking some individual as an object of intention. Classes are not in spacetime and they are not located where their members are, but they are not nowhere, rather, they are associated with minds. According to Lewis, any class includes the null set, but the null set is not a subclass or a part (1991: 10–11). Classes do not have the null set as part because the null set is not a class, it is an individual. Any individual will do for the null set. Lewis redefi nes the null set as the mereological sum of all individuals (1991: 14). It is everywhere rather than nowhere. Th is fi ts well with my framework. Th e null set is the mereological sum of all monads and so is the encompassing mind bare of its thoughts. It is the One which is transcendentally present in all beings and is their source (Sect. 5.4: 159).

Paradoxes of Plenitude

What about Forrest and Armstrong’s paradox which arises when we allow unrestricted plenitude for possible worlds? Th eir argument does not get off the ground in my framework because it assumes that there is a big world which contains all the possible worlds as parts. In my framework, 356 Appendix 2: Problems with Plenitude there is no world of all worlds, just as there is no set of all sets, because each individual monad is a possible world and the class of monads is a proper class. Lewis (1986: 104–108) considers a second paradox of plenitude con- cerning the set of worlds that characterises the content of somebody’s thought.16 Th ere is no natural bound on the cardinality of the set of somebody’s thoughts and the argument uses this fact to construct a possible world whose cardinality exceeds the cardinality of the set of all possible worlds. As before, this argument fails in my framework because there is no set of all possible worlds. However, some interesting points arise in Lewis’ discussion. For example, Lewis claims that:

Not just any set of worlds is a set that might possibly give the content of someone’s thought. Most sets of world, in fact all but an infi nitesimal minority of them, are not eligible contents of thought. (1986: 105)

I do not agree with this. Any set defi nes a possible thought although not all possible thoughts defi ne a set. Lewis is committed to the view that there are unthinkable contents. I hold that all content is thinkable by some mind. Lewis’ discussion of unthinkable content seems only to show that there is content which it is perverse for a mind to think about its world, given its beliefs and desires. He does not show that such content is unthinkable. I can think “I want a saucer of mud” even if I cannot put myself into a state in which I would want to turn that thought into action. Another mind in another possible world might actually want a saucer of mud!

Actuality

Lewis’ indexical analysis of actuality enables him to dodge some potential problems of plenitude, including ethical problems. According to Lewis:

16 Here, “somebody” is not considered to have human limitations (e.g. it might refer to a god) and so it should be read as “some mind”, maybe an infi nite mind. Appendix 2: Problems with Plenitude 357

every world is actual at itself, and thereby all worlds are on a par. Th is is not to say that all worlds are actual—there’s no world at which that is true. (1986: 93)

For us, our world is actual and other possible worlds are not actual. Th erefore, we should concern ourselves with evils in this world, even if there would be the same sum total of good and evil throughout the worlds no matter what we do in this one (1986: 123–128). Reasonably, it is the good and evil in our actual world which we care about. My framework also denies that other worlds are actual, so I can refer to Lewis’ arguments in defence of these problems. However, my analysis of actuality diff ers signifi cantly from Lewis’. In quantum monadology, the actual world is the one and only, necessary, best possible world. In Lewis’ modal realism, actuality is a contingent matter— it is not the case that one world alone is absolutely actual. First, Lewis asks how we would know that we are absolutely actual. Th en, he writes:

And yet we do know for certain that the world we are part of is the actual world—just as certainly as we know that the world we are part of is the very world we are part of. (1986: 93)

I take this knowledge that each of us has—of our own existence and actuality—as the most certain knowledge which we can have. It poses a problem for monadology in that it is not clear how an actual monad can know that it is actual rather than merely possible. One response is to say that it is a special property which actual monads have—to be immediately acquainted with their own actuality—but this is not satisfactory as an explanation. Another response is to invoke the concrete/abstract divide—the actual world is concrete and possible worlds are abstract— but this distinction is not available in the idealistic interpretation of monadology which I have adopted. I propose to make a distinction between actual and possible monads in terms of consciousness—actual monads are conscious whereas potential monads are like philosophical zombies. Consciousness is a property of actual monads because they follow the critical path from beginning to end and complete the neural circuit in which meaning is communicated to each individual mind by the encompassing mind. 358 Appendix 2: Problems with Plenitude

The Construction of Possible Worlds

Lewis’ view of the construction of possible worlds is a static, mereological view involving the combination of parts into wholes. I don’t fi nd this way of constructing possible worlds satisfactory. It misses the dynamic, process component of the construction of possible worlds which imposes a hierarchy of priority on thoughts; the ordinal hierarchy as opposed to the cardinal hierarchy which Lewis draws from in his imaginative experi- ments. Spatiotemporal relations are not what makes a possible world cohere. It is the laws which enable monads to have harmonious relations. Possible worlds have to be built up coherently according to local laws. Lewis seems to think that possible worlds help us to say what it means for a false theory of nature to be close to the truth: “A theory is close to the truth to the extent that our world resembles some world where that theory is exactly true” (1986: 24). Th is doesn’t seem right. Our world obeys a hierarchy of laws at diff erent scales which result in harmony. A world in which, for example, Newton’s laws were exactly true would not be like our world at all. For one thing, quantum mechanics is responsible for the stability of fundamental matter in our world. Lewis’ worlds seem to be more like the dreams (imaginings) of monads at a particular scale, more wish than possible fact, not taking total harmony into account. As he says:

imaginability is a poor criteria of possibility. We can imagine the impossible, provided we do not imagine it in perfect detail and all at once. We cannot imagine the possible in perfect detail and all at once, not if it is at all complicated. (1986: 90)

I can imagine a hundred-foot elephant walking around the earth but I know that it is not possible according to our physical laws and I doubt whether it is possible at all. It is not enough for possible worlds to be imaginable, they have to be actualisable. I think that to properly defi ne the construction of possible worlds, it will be necessary to go beyond the mereological approach of Parts of Classes to embed mathematics as a bare structure in a bigger theory of concepts which has static (cardinal, to do with size) and dynamic (ordinal, to do with priority) components. Appendix 2: Problems with Plenitude 359

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A a priori, 12, 49, 72, 79, 80, 80n4, abstract objects, 39, 300, 353, 354 109, 111, 112, 142, 207, action, 74, 150, 228, 237, 239, 262, 209, 210, 216, 235, 325, 356 310, 324 actualisation, 41, 57, 196, 270 Aristotelian realism, 352, 353 actuality, 2, 183, 220, 223–96, 305, asymptotic freedom, 19, 86, 95–7, 306, 310, 313, 325, 356–7 116. See also quantum aesthetics. See beauty chromodynamics analogy axioms. See also set theory; large formal, 56, 104 cardinals mathematical, 11, 31, 52, 54, 56, Axiom of Choice, 133, 135–8, 60, 93, 104, 114, 116, 117, 152, 158, 168, 216, 296, 120, 130 332, 335, 345, 346 physical, 92, 286n17 Axiom of Constructibility (V=L), Pythagorean, 92, 104 138, 157, 159, 218, 337 apperception, 240 Axiom of Determinacy, 158, applicability of mathematics, 3–63 163n28, 335

1 Note: Page number followed by ‘n’ refers to notes

© The Author(s) 2017 383 J. McDonnell, The Pythagorean World, DOI 10.1007/978-3-319-40976-4 384 Index axioms. See also set theory; large cardinal number, 113, 132, 133, cardinals (cont.) 135, 218, 315, 315n1, Axiom of Infinity, 134, 134n4, 317, 332 152, 154, 156, 157 Carnap, Rudolf, 9, 273 Axiom of Replacement, 152, 154 causality, 239, 255, 269 desirable characteristics, 127, 153 cellular automata, 35–7, 57. See also Empty Set Axiom, 151, 152 Wolfram, Stephen extrinsic vs. intrinsic, 127, 169 chance, 26, 33, 100, 277, 310, 323 large cardinal axioms, 127, CHQT. See consistent histories 151n15, 153–62, 164–72, quantum mechanics 333–9, 344–7 (CHQT) power set axiom, 152, 210 coarse-graining, 41, 179, 246, set theoretical, 46, 126, 128, 247n8, 255–8, 269, 272, 131–3, 161, 251 287 Azzouni, Jody, 13, 31n4, 52 Cohen, Paul, 139, 140, 158, 193, 331–3, 335. See also forcing, technique of B coherence, 180, 227, 238, 239, 285 Balaguer, Mark, 10n2, 146–9 collapse of wave function, 197, 201, Bangu, Sorin, 16, 18, 22–6, 29, 39, 257, 259, 283 42, 48 common sense, 234, 235, 248, Barrow, John, 34, 35, 36n7, 50, 54, 293, 308 59n13, 101 complementarity, 251, 252, 258 beauty, 7, 16–23, 29, 38, 39, 42, 48, completeness, 136, 137n8, 162, 164, 54, 60, 99, 105, 115, 129, 337–9, 342 301, 308, 329 complexity, 31, 35–6, 39, 50, 58, being, 193. See also Heraclitus; 73, 120, 197, 199, 210, Parmenides; Plato; Hegel 288, 292, 302, 312, 354 absolute structure of, 193 compossibility, 231, 290 Benioff, Paul, 292, 293 concept Berkeley, George, 184, 192, 192n4 abstract, 324 biology, 17, 36, 50, 102, 119, complete, 232 264–6, 274 concrete/abstract boundary, 354 broken symmetry, 103 conditional probability, 251, 254, 256, 267, 276 consciousness, 184, 188, 198, C 228, 240, 242, 247, Cantor, Georg, 30, 128, 131–5, 135n5, 256, 278, 280, 281, 137, 142, 168, 203, 204, 264, 287, 291, 294, 309, 308, 309, 332, 334, 350 328, 357 Index 385 consistency differentiation, 197, 263, 289–91, 294 in mathematics, 120, 146–8, Dirac, Paul, 16–18, 20–3, 31, 33, 147n13 42, 48, 55, 71, 113 in physics, 88 discovery, 15, 19, 41, 50, 54, 55, 61, in quantum mechanics, 241, 242 70, 71, 81, 88, 94, 117, consistent histories quantum 142, 166, 225, 235, 312 mechanics (CHQT), dualism, 184, 192, 310 223–5, 243–4, 246–61, Dyson, Freeman, 22, 24, 33 266–73, 275–8, 281–4, 288, 295, 305 constructivism, 317 E contingent, 12, 33, 38, 72, 73, 101, Eddington, Arthur, 25, 80n4, 80, 120, 147, 232, 255, 268, 82, 84 278, 357 effective completeness, 162, 337–9 continuum hypothesis, the, 128, effective field theory, 25, 73, 112–14, 132, 134, 137, 139, 141, 268, 302 144, 153–6, 155n20, 158, eigenstate, 241 159, 162, 163, 163n28, Einstein, Albert, 16, 17, 19, 24, 25, 164, 165, 171, 174, 179, 31, 33, 40, 57, 78–82, 303, 331–5, 337–43 107n15, 115 correlation, 186, 197, 199, 239, 244, electrodynamics, 11, 76 245, 255, 266, 277, 279, 285 electromagnetism, 28, 53, 75, 76, cosmology, 73, 101s, 119s, 181, 197, 79, 81, 85, 116, 196 201, 224, 242, 246, 257, 302 electroweak theory development, 23 SU(2), 116 D elementary embeddings, 157, 333–4, decoherence, 239, 244, 248, 249, 345 255, 257, 259, 269, 271, elementary particles 277, 278, 285, 288, 289 bosons, 18, 80n5, 85, 86, 100 decoherent histories, 244–6, 278. See fermions, 100, 186 also consistent histories gluons, 19, 24, 56, 69, 77, 86, quantum mechanics 92, 96, 97, 103, 113, (CHQT) 117, 188 degree of reality, 232, 268 Higgs boson, 86 Descartes, René, 184, 192 omega minus, 56, 117, 264 determinacy, 158, 159, 183, 220, quarks, 19, 24, 55, 56, 63, 69, 82, 334–7 86, 91–7, 103, 113, 117, projective, 154 118, 186, 187 determinism, 224, 242, 245, 255, 257 W and Z bosons, 18, 80n5, 86 386 Index emergence, 40, 41, 200, 244–7, 253, formalism, 1, 8, 22, 23, 53, 115, 255, 257, 284 174, 225, 260, 278, classical from quantum, 116, 220, 281n16, 282, 305, 316, 343 223, 305 framework empty set, the, 130, 132, 134, 138, coarse-grained, 40, 246, 247, 151, 152, 182, 191, 195, 254, 274 203–6, 313, 332 conceptual, 271–4, 282, 287, epistemology, 182, 202–3, 233, 353 288, 294, 319 essence, 42, 59, 63, 111, 290, 328 fine-grained, 256 existence, 2, 6, 9, 16, 19, 22, 34, 37, incompatible, 251, 256–8, 44, 45, 50, 58–60, 63, 74, 258n13, 270, 273, 276, 285 77, 78, 92, 96n11, 117, linguistic, 274 125, 127, 129, 133, 137, metaphysical, 2, 63, 129, 183, 140, 142, 148, 149, 151, 184, 195–203, 219, 225, 155–7, 163–74, 179, 182, 284, 291, 300, 303, 354 184, 186–97, 204, 205, narrative, 272, 277 211, 216, 226, 227, 230–2, quantum, 273 245, 258, 264–6, 270, freedom, 19, 49, 86, 91, 95–7, 116, 287–290, 292, 299, 301, 260, 287n18, 318, 325 303, 317, 322, 324, 325, Frege, Gottlob, 8, 43–46, 61, 329, 331, 333, 341–347 118n21, 132, 143, 315 expression, 195, 201, 226–9, 235, French, Steven, 88, 88n8, 92–3, 105, 238, 238n2, 273, 311, 318, 115, 189 324, 328 Friedman, Harvey, 130n1, 137, 137n7, 150, 168, 170

F Feferman, Solomon, 150, 167, 170, G 170n30, 171, 353 gauge theories, 57, 59n13, 70, 76, Feynman, Richard, 23, 243, 271 80, 84, 87, 96, 116, 117. fibre bundles, 81, 99 See also group theory fictionalism, 1, 9, 127, 149 Yang–Mills, 18, 19, 80n5, 81, 85, Field, Hartry, 237. See also fictionalism 86, 94 fine structure methods, 160, 161, Gell-Mann, Murray, 16, 89–95, 105, 164, 167, 345, 346. See also 117, 220, 243, 243n5, set theory 244–8, 253–7, 259, 264, finitism, 152, 152n16, 309, 310 269, 272–3, 305. See also first philosophy, 180, 307 consistent histories quantum forcing, the technique of, 138, 139, mechanics (CHQT) 160, 162, 331–2 generic extension, 162, 163, 342 Index 387 generic multiverse, 164, 342–6 Hartle, James, 32, 220, 223, 243n5, geometrisation of physics, 82, 112 244–7, 248–54, 255, 269, God 271–80, 305. See also act of creation, 230, 231 consistent histories charateristics of, 74, 184, 203–4, quantum mechanics 229–33, 262, 264–71, (CHQT) 324–6, 328 Hegel, Georg, 183, 194, 208, 328 source of truth, 233, 265 Heisenberg, Werner, 16, 56, 90, 99, Gödel, Kurt 104n14, 105 Axiom of Constructibility (V=L), Heraclitus, 193–4 138, 157, 159, 218, 337 Higgs mechanism, 19, 86, 90, 102, extrinsic vs. intrinsic axioms, 127, 103, 116 141, 169 Hilbert space, 10, 12, 241, 246–51, incompleteness theorems, 2, 160, 253, 256, 259, 267, 271–3, 166, 251, 292, 354 278, 284 maximise principle, 154 Grattan-Guinness, Ivor, 13, 31n4, 50, 52 I Griffiths, Robert, 220, 223, 243, idealisation, 21, 37, 51, 61, 89, 93, 244, 247, 249, 252, 255, 104, 107, 120, 232, 236 257–9, 269, 273. See also idealism, 184, 227, 265, 300, 307–9, consistent histories 328. See also Berkeley quantum mechanics mathematical, 307–9 (CHQT) imagination, 22, 51, 233, 234, 237 Gross, David, 3, 16, 57, 77, 323 implicational opacity, 13, 52 group theory. See also symmetry incompleteness theorems. See set SU(2), 27, 90–4, 104, 105, 196 theory SU(3), 19, 27, 69, 71, 81, 86–94, independence results. See Cohen; 104, 105, 196, 264 Gödel U(1), 27, 76, 81, 89, 90, 196 individual, 4, 17, 26, 29, 30, 79, 90, 93, 94, 96, 102, 104, 152, 153, 155, 191, 204, 214, H 220, 226, 227, 232, 233, Hamkins, Joel, 140–45, 149, 162, 237, 238, 244, 261, 264, 173 268, 270, 276, 278, 281, harmony, 37, 42, 47, 79, 119, 185, 282, 293, 294, 304, 307, 227, 229, 231, 237, 239, 309, 311, 314, 316–20, 268, 284, 290, 307, 321, 326, 328, 350, 351, 317–20, 358 354–7 388 Index infinity inaccessible cardinal, 154, 156, 170 absolute, 203, 264, 282, 293 Mahlo cardinal, 154, 155n18, 156 actual, 236, 264 measurable cardinal, 157, 160, potential, 87, 121, 152, 154, 293 164, 218, 220, 333 information gathering and utilising supercompact cardinal, 160n26, systems (IGUS), 245, 254, 164, 165, 344, 346 256, 269, 276–81, 250 Woodin cardinal, 160–4, 166, innate, 73, 110, 111, 113, 236, 302 213, 335–45 inner model, 138, 144, 157, 159, 160, laws of nature, 6, 10, 11, 15, 33–5, 164–8, 333, 336, 342, 346 37, 39, 54, 58–60, 63, 112, interpretability, 166 114, 118, 170, 197, 232, interpretant, 316, 325, 326 239, 262, 268, 285, 301, isospin, 56, 90, 91, 93, 103, 105 319, 322 Leibniz, Gottfried. See monadology metaphysics, 238n2 J two kingdoms, 281, 305 Jensen, Ronald, 159, 341. See also universal language, 264 fine structure methods Lewis, David, 309, 349–52, Johnston, Mark, 327 355–8 local vs. global view, 284–5 logic. See logicism; omega logic K first-order, 137n8, 140, 143, 162, Kaluza-Klein theory. See unified field 172, 174, 213, 338 theory in quantum mechanics, 225 Kanamori, Akihiro, 139, 155n18, second-order, 45, 143, 162n27, 160n26, 205, 333 172, 212, 339n9 Kant, Immanuel, 49, 50, 142, 180, strength, 212 183, 184, 207, 317 logicism, 1, 45, 316 Koellner, Peter, 134, 145, 147, 151, Longo, Giuseppe, 49 153–163, 212, 218, 333, 335, 334n5, 336–42 M Maddy, Penelope, 135, 136, 141, L 142, 149, 151, 152, 154, language of thought, the, 60, 264, 157, 163 275, 310 Magidor, Menachem, 143, 149n14, large cardinals. See axioms 153, 154, 157 axioms, 127, 151, 153–162, Martin, Donald, 143, 144, 159, 163, 164–172, 333–7, 339, 344–7 167, 342 Index 389 mathematical intuition, 5, 16, 24, monadology 29–33, 168, 169, 208 Leibnizian, 2, 287, 324 mathematical universe hypothesis, quantum, 2, 225, 282–96 59, 187 monads. See monadology matter, 2, 15, 29, 38, 45, 52, 57, aggregations, 236 59n13, 61, 77, 80, 96, 98, complete individual concept 99, 102, 103, 114, 119, (CIC), 226, 314 139, 142, 148–50, 162, dominant, 265, 290 163, 184–6, 188, 192, 200, entelechies, 185 203, 227, 263, 279, 293, properties, 229, 266 307–10, 318, 319, 324, monist, 184 325, 329, 342, 357 multiverse Maxwell, James Clerk, 28, 53, mathematical, 2, 71, 140–51, 182 75, 214 physical, 2, 73 McAllister, James, 24, 38 meaning, 9, 60, 104, 135, 182–4, 187, 193, 198–200, 208, N 212, 214, 215, 220, 228, naturalism, 180, 211 234, 254, 271, 275, 277, necessary, 17, 38, 40, 73, 84, 93, 95, 283–7, 292, 294, 295, 300, 101, 120, 126, 139n10, 142, 306, 309, 314, 316–9, 320, 147, 159, 181, 182, 192, 324–6, 329, 357 194, 206, 233, 242, 253, measurement, 80n4, 186, 241, 248, 257, 278, 293, 304, 306, 252, 256–8, 277 319, 336, 337, 357, 358 mereology, 349, 350 Newtonian dynamics, 243, 245 metaphysical framework, 2, 63, 128, Newton, Isaac, 11, 24, 25, 28, 57, 182, 184, 195–203, 219, 71, 82, 101, 108, 112, 215, 225, 284, 291, 300, 304, 224, 243, 245, 263, 269, 355 274, 292, 353, 358 metaphysics principles, 23, 72, 110–19, 180, 182, 192, 237 O Pythagorean, 2, 6, 15, 22, 25, 28, objectivity, 78, 171, 321 38–43, 56, 59, 115, 301 observable, 97, 98, 105, 241–3 mind, 1–2, 16, 47, 73, 118, 171, observer, 74, 186, 196, 240, 242, 243, 182, 223, 354 260, 277, 278, 280, 285 model theory, 160, 166, 190, 211, omega conjecture, 162, 163n28, 215, 217, 346 164, 165, 198n5, 339, 340, modes of presentation, 312, 316, 327 340n10, 340–4, 346 390 Index omega logic, 162, 162n27, 172, 173, phenomena, 3, 20, 21, 26, 33, 36, 213, 217, 338, 339n9, 339, 37, 49, 53, 54, 58, 59, 62, 340, 343 81, 98, 101, 108, 113, Omnès, Roland, 220, 239, 243, 117n20, 136–40, 203, 224, 243n5, 246, 249, 256n12, 227, 231, 232, 236–40, 259, 293, 296, 305. See also 242, 259–61, 264, 270, consistent histories quantum 281, 287–9, 293 mechanics (CHQT) phenomenology, 8, 72, 78, 102, 115, One and the Many, the, 72, 109, 194, 202, 203, 208, 225, 110, 182, 193, 195, 201, 282, 288, 314, 321 223, 226, 238n2, 286, 302 Planck, Max, 14, 40, 53n13, 82, 98, One, the, 286, 289, 302, 316, 320, 108, 113, 120, 187, 215, 328, 355, 357 247, 260, 271 ontological relativity, 127 Plato, 25, 183, 193, 219. ontology, 139, 173, 182, 187, 189, See also Platonism 208, 211, 214, 249, 257, Timaeus, 73, 118, 219, 285, 302 260, 274 platonic divide, 300, 308 optimisation, 197, 198n5, 262, 290–5 Platonism, 1, 8, 146, 211, 316 ordinal number, 131, 133, 138 full-blooded (FBP), 146 outer model, 138, 331 plenitude principle of, 264, 350 problems with, 349–58 P Plotinus, 205 paradoxes, 121, 126. See also set pluralism, 8, 126, 140–9, 163. theory See also set theory set theoretical, 131–3 radical, 145–9 Parmenides, 110, 193, 194, 302 point of view, 70, 78, 190, 195, 229, parts and wholes, 349 267, 268, 270, 273, 282, Peano arithmetic, 45 306, 317 Peirce, Charles, 318, 322, 325, 326 possibility, 80n4, 117, 127, 128, perception 130, 137, 140, 146, 156, clear and distinct, 233, 319 163, 166, 169, 201, 225, confused, 228, 235 228, 236, 269–70, 278, perfection 281, 284, 290, 292, 300, maximisation, 287 304, 305, 316, 317, 321, principle of, 231, 232, 239, 261, 325, 339–41, 358 264 possible actual, 225, 270–4, 277, phaneron, 325, 327 282–7, 295, 305 Index 391 possible worlds Q best possible world, 231, 268, 284, QCD. See quantum 291, 294, 306, 320, 357 chromodynamics (QCD) Leibnizian, 2, 220, 223, 225, 260, quantum chromodynamics (QCD), 287, 305, 324 25, 69–71, 86–88, 91, Lewisian, 350 95–8, 107, 112, 116 potentiality, 2, 183, 220, 223–296 asymptotic freedom, 19, 86, power set, 132–35, 138, 152, 154–8, 95–7, 116 158n23, 170n31, 171, 210, development of, 301 300, 332, 334n4, 352, QCD Lite, 96–8, 107 353n14 SU(3), 19, 27, 69, 88–91 prediction, 3, 6, 11, 19, 26, 27, quantum field theory, 12, 20, 25, 34–6, 47, 48, 50, 55, 56, 53, 54, 58n12, 59n13, 61, 104, 105, 107, 108, 86, 87, 95, 186, 263. 166–7, 209, 244, 254, 274, See also quantum 291, 293, 311, 319, 345 chromodynamics; Principle of Sufficient Reason, 74 electroweak theory problem of evil, the, 198n5 relativistic, 86, 87 projection, 71, 267, 282, 286, 289, quantum mechanics. See also 314, 318, 319, 321 consistent histories properties quantum mechanics compossible, 231, 269, (CHQT) 287, 290 Copenhagen interpretation, incompatible, 231, 270, 241, 242 273, 285 quantum/classical divide, Putnam, Hilary, 213–16, 276 241, 242 Pythagoras, 129 standard interpretation, 110, 130, Pythagorean 146, 217, 240–3 analogy, 92 quasiclassical heuristic, 16, 21–3, 28, 41, 53, framework, 247, 272, 275, 278, 56, 115 280, 285 metaphysics, 2, 6, 16, 23, 26, 28, realm, 245, 246, 253, 257, 274, 38–42, 56, 59, 115, 301 277, 279, 281 theory, 39, 70, 72, 94, 106–13 variable, 245, 246, 254, 255, 257, Pythagoreanism 259, 278 Ancient, 72 Quine, Van, 32, 111n17, 127, 172, in mathematics, 2 180, 214, 215, 317 in physics, 306 Naturalism, 180, 211 392 Index

R Schwartz, Jack, 3, 29, 30 randomness, 242, 295 set theory. See also determinacy; stochastic process, 241, 251, 270, fine-structure methods; 284, 319 forcing, technique of rationalism, 16, 106, 111, 170, 173, independence results; inner 183, 193, 195, 196, 202, 216 model; large cardinals; realism reflection principles; ZFC mathematical, 169, 173, 211, 281 descriptive, 158, 159, 334, 337, robust, 143 340, 343 structural, 72, 83–4, 107, 188 development of, 126, 129–40 thin, 146, 149 foundational role, 149–51 record, 22, 100, 242, 256, 275, 276, interpretability, 166 279, 282–6, 288, 306, 314, maximise principle, 155 321, 323 modern, 111, 133–6 reductionism, 5, 24, 284 paradoxes, 131–3 reflection principles, 156–8, 182, pluralist, 126–8, 139, 144, 145, 200–2, 344 151, 162–4, 173, 179, 219, relational strategy, the, 70, 80–3, 111 303, 342–4 relativity universalist, 126–128, 139, 140, general theory of relativity, 70, 76 151, 163, 173–4, 179, 219, special theory of relativity, 75 303, 342, 344–7 renormalisation, 17, 25, 27, 32, Shapiro, Stewart, 8, 43n8, 45, 181, 58n12, 87, 98 189–91, 210, 212n8 representation, 14, 25, 27, 55, 57, Shelah, Saharon, 159, 166, 337, 341 62, 63, 77, 79, 89, 91, 99, signs, 12, 24, 38, 60–1, 179, 229, 111, 191, 224, 229, 263, 264, 300, 316, 325, 326 284, 310, 311, 313, 327 simplicity, 6, 16, 17, 29, 33–8, 42, Resnik, Michael, 62n15, 188, 102, 116, 205, 231, 232, 191, 208 262, 283, 301, 306, 352 Russell, Bertrand, 45, 73, 110, single framework rule, the, 252–3, 133–4, 205, 292, 302 258 Ryckman, Thomas, 57, 77–80, 84, singularity, 119, 121, 197 107n16 Smolin, Lee, 74, 78, 82, 83, 102, 113 solipsism, 228, 279, 280 spacetime, 82, 271, 309, 350 S spontaneous symmetry breaking, scaling, 40, 95, 96, 189, 302 27, 39, 71, 80n5, 102, Schrödinger’s equation, 241, 269 196, 268 Index 393 standard model, the symmetry. See also group theory cosmology, 73, 101, 181, 197, 201 dynamic, 186, 195–8, 311 particle physics, 14, 18, 27, 29, geometrical, 75, 93 32, 54, 57, 59n13, 76, 80, global, 70, 73–77, 103, 116 104, 107, 196 Leibniz’s use of, 264, 269 state Local, 57, 59, 70, 73–77, 81, 288 monadic, 296 Lorentz, 70, 75, 76, 87 physical, 241, 277, 278, 309 Poincaré, 79, 103, 283 quantum, 89, 93, 186, 242, 244, principles, 26, 38, 40, 69–71, 253, 285 75–7, 79, 84, 88, 98–106, static vs. dynamic view, 194, 311, 311 314 Steel, John, 151, 160, 161, 172 Steiner, Mark, 4, 43, 46, 47, 50, 56, T 60, 72, 88, 92, 94, Tarski, Alfred. See mathematical 104–106, 115 universe hypothesis stochastic process, 241, 251, 270, Tegmark, Mark, 2, 59, 101, 187, 284 188, 194, 291, 304, 306 string theory. See also Theory of Theory of Everything, 5, 114, 268, Everything 292, 301 problems, 14, 18, 27, 29, 71, 77 truth unified field theory, 77, 79, 81 dynamic, 322 structuralism in a framework, 322 ante rem, 189 of set theory, 127, 166–8, 344 conceptual, 171 ultimate, 20, 41, 322 in mathematics, 189–91 ontic, 72, 83, 84, 107, 187–9, 192 in physics, 185–9 U subjective, 10, 16, 26, 147, 184, Ultimate L, the theory of, 164–6 188, 197, 209, 210, 272, ultimate reality, 118, 224, 257, 278, 280–2, 286, 288, 291, 274, 300 294, 310, 313 unification, 19, 100, 102, 284 substance, 13, 184, 185, 185n1, unified field theory, 77, 79, 81 192–5, 226, 227, 237, Kaluza-Klein, 81, 85 261, 265, 290, 310 uniqueness superposition, 241–4, 247n8, in mathematics, 200, 219, 292 248, 260, 270, 283, in physics, 6, 209 285, 296, 306 unicity, 258 394 Index unity, 112, 185, 197, 198, 204, 206, Weyl, Hermann. See also gauge 209, 238, 239, 256, 265, theory 266, 281, 286, 295, 349–51 world geometry, 77–80 universalism, 140–1. See also set Wheeler, John, 268, 309, 311 theory Wigner, Eugene universal science, 230, 230n1 contributions to quantum universe of sets, the, 127, 134, 138, mechanics, 113 144, 154, 156, 163, 166, philosophy of mathematics, 168, 169, 195, 200–2, 2, 4, 7–10, 48 203–6, 210, 211n7, 213, unreasonable effectiveness 214, 219, 220, 240, 293, of mathematics, 1–4, 7–16 313, 315, 338, 341, 344 Wilczek, Frank, 16, 19, 31n4, 40, unreasonable effectiveness of 41, 50, 54, 69, 86, 95, mathematics, 1–5, 7–16. 97n12, 108, 109, 263, See also Wigner 306, 323 Wilson, Mark, 32, 107 Witten, Edward, 14, 15, 30, 99 V Wolfram, Stephen, 35, 36, 353 V. See universe of sets, the Woodin, Hugh. See omega logic; Von Neumann, John, 12–14, 195, Ultimate L; Woodin 214, 241, 242 cardinal

W Y wave function, 32, 76, 197, 200, Yang-Mills theory. See gauge theory 241, 248, 257, 259, 283, 284 Weinberg, Steven, 16, 20, 22, 25, Z 32, 36, 37, 77, 86 Zermelo-Fraenkel (ZFC), 133–40, well-founded 150–61, 163–6, 170, 180, phenomena, 231, 232, 244 213, 218, 331, 333–6, sets, 135n6 339–46