Abstract
In this paper we intend to argue that: (1) the question ‘True V or not True V’ is central to both the philosophical and mathematical investigations of the foundations of mathematics; (2) when posed within a framework in which set theory is seen as a science of objects, the question ‘True V or not True V’ generates a dilemma each horn of which turns out to be unacceptable; (3) a plausible way out of the dilemma mentioned at (2) is provided by an approach to set theory according to which this is considered to be a science of structures.
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- 1.
The former position on the nature of mathematics is known as metaphysical realism whereas the latter as metaphysical anti-realism. However, in the literature, besides being there a debate between metaphysical realists and anti-realists, there also is a much more recent controversy which sees realists about truth opposing anti-realists about truth. For a realist about truth, the truth of a mathematical statement S transcends verification, whereas for the anti-realist a mathematical statement S is true if and only if we can produce a constructive proof of S. In other words, for the anti-realist about truth, the concept of mathematical truth collapses onto that of constructive provability. The realism/anti-realism debate concerning mathematical truth was introduced by Michael Dummett (see on this Dummett 1975, 1991). For a discussion of the relationship between the metaphysical realism/anti-realism debate and the realism/anti-realism debate about truth see Oliveri (2007), Chap. 1, Sects. 7–10.
- 2.
See on this Balaguer (1998).
- 3.
Strict formalism is the position expressed by Wittgenstein in his remarks on following a rule in the Philosophical Investigations. For the strict formalist, not only mathematical theories are seen as formal games, but logic itself loses, as it were, the hardness of its ‘must’. See on this Wittgenstein (1983), Sects. 185–242.
- 4.
The translation from the original is mine:
...we are here obliged to make a fundamental distinction as we differentiate between:
- II \(^{a}\) :
-
Increasable Actual Infinity or Transfinitum.
- II \(^{b}\) :
-
Non-increasable Actual Infinity or Absolutum.
The three examples of actual infinity previously mentioned [two of which are the positive integers, and the points lying on a circle] belong all to the class II \(^{a}\) of the Transfinite. In the same way belongs to this class the smallest supra-finite ordinal number, which I call \(\omega \); then this can be augmented or increased to the next larger ordinal number \(\omega + 1,\) this again to \(\omega + 2\) and so on. But even the smallest actual-infinite power or cardinal number is a transfinite, and the same holds of the next larger cardinal number and so on.
The Transfinite with its wealth of formations and forms points necessarily at an Absolute, at the ‘true infinity’, whose size can in no way be added to or decreased and which therefore, as to quantity, is to be considered as the absolute maximum. The latter so to speak goes beyond the human power of comprehension and eludes in particular mathematical determination; whereas the transfinite not only fills the wide domain of possibilities concerning the knowledge of God, but also offers a rich and always growing field of ideal research ...(Cantor 1962, pp. 405–406).
- 5.
The symbol ‘V’ was introduced by G. Peano in Sect. IV of Peano (1889).
- 6.
- 7.
The von Neumann universe V is defined by the following equations, here On is the proper class of ordinals (proper classes are nor sets):
$$ V_{0} \,\, = \,\, \emptyset \qquad \qquad \quad {(9.1)} $$$$ V_{\alpha + 1} \,\, = \,\, \mathcal {P} (V_{\alpha }) \qquad \qquad \quad {(9.2)} $$$$ V_{\beta } \,\, = \,\, \bigcup _{\alpha < \beta } V_{\alpha } \; \; \; \; \text {if }\beta \text { is a limit ordinal}\qquad \qquad \quad {(9.3)} $$$$ V \,\, = \,\, \bigcup _{\alpha \in On} V_{\alpha }. \qquad \qquad \quad {(9.4)} $$ - 8.
Indeed, if by ‘Zermelo sequence’, \((Z_{n}),\) and ‘von Neumann sequence’, \((N_{n}),\) we mean the sequences of sets individuated by the following pairs of recursive equations:
$$ Z_{0} = \emptyset \, \, \text {and} \, \, Z_{n + 1} = \{ Z_{n} \}; $$$$ N_{0} = \emptyset \, \, \text {and} \, \, N_{n + 1} = N_{n} \cup \{N_{n}\}; $$it is well known that both \((Z_{n})\) and \((N_{n})\) are adequate set-theoretical representations of the sequence of natural numbers. Of course, the embarrassing thing here is given by the observation that \((Z_{n})\) and \((N_{n})\) give different set-theoretical representations from one another of all the natural numbers \(n \geqslant 2.\) Benacerraf discusses some of the philosophical implications of the existence of \((Z_{n})\) and \((N_{n})\) in Benacerraf (1985).
- 9.
The classical structuralist reply to Benacerraf’s problem—What is the correct set-theoretical representation of the natural numbers?—is to say that the apparent impossibility of solving Benacerraf’s problem, as a consequence of the existence of the sequences \((Z_{n})\) and \((N_{n})\) of footnote 8, is not important, because what is mathematically relevant to number theory is the number theoretical structure (see on this Sect. 9.5 especially footnote 35), and from Dedekind’s recursion theorem we know that in second-order arithmetic any two models of Peano axioms are isomorphic to one another, i.e. they have the same structure.
- 10.
Note that the lowest type of the von Neumann universe where \(\{\{\emptyset \}\}\) and \(\{\emptyset , \{\emptyset \}\}\) occur as elements is \(V_{3}.\)
- 11.
Given a mathematical theory T, Hilbert distinguishes between finitary and ideal statements belonging to the language of T. If T is number theory then a finitary number theoretical statement is a statement the truth of which can be established by a finitely long computation, e.g., \(2 + 3 = 5\). Note that the class of finitary number theoretical statements \(\mathfrak {F}\) is closed under finitely many applications of \(\lnot , \wedge , \vee .\) From this it easily follows that a clear example of ideal number theoretical statement is: for any \(m, n \in \mathbb {N}\), \(m + n = n + m\). According to Hilbert, although ideal statements ‘signify nothing’ (Hilbert 1926, p. 196), they are useful to develop mathematical theories, because they contribute to keeping the laws of logic and mathematics in their simplest form. See on this Hilbert (1926), p. 195.
- 12.
See on this Ferreiros and Gray (2006).
- 13.
- 14.
See on this Frege (1980).
- 15.
For Frege, there is no demarcation line between arithmetic and logic, but there is a demarcation line between geometry and logic: local logicism. On the other hand, for Russell, the whole of mathematics is part of logic: global logicism.
- 16.
- 17.
On this, and against the very idea of a set-theoretical form of reductionism, see E. De Giorgi’s programme concerning the foundations of mathematics in Forti et al. (1996), pp. 1–2.
- 18.
Examples of second-order mathematical statements are: The Peano axiom of full induction, Every infinite subset of a countable set is countable, Every non-empty set of reals bounded from above has a least upper bound in \(\mathbb {R}\), the Bolzano-Weierstrass theorem, etc. See on this also Isaacson (1987).
- 19.
Some categories, e.g. the category of sets Set, are too large to be expressed/defined in first-order ZFC, i.e. Set is not a set in first-order ZFC. See on this McLarty (1995), Part II, Chap. 12, Sect. 12.1, pp. 107–110.
- 20.
See on this Hamkins (2011).
- 21.
For Friedman, a multiverse/hyper-universe is simply a non-empty collection of more than one model of, say, first-order ZFC.
- 22.
- 23.
Tuller (1967), Appendix 2, pp. 182–185.
- 24.
By ‘universal property’ we mean a property common to all, say, triangles in \(\mathcal {S}_{i}\) such as: the sum of the internal angles of a triangle is \(180^{\circ }\).
- 25.
Here by ‘really basic terms’ we mean terms belonging to the language of a mathematical theory T which are not in need of an explicit definition. Within an axiomatic mathematical theory T, the relevant part of the meaning of its really basic terms is given implicitly by some of the axioms of T.
- 26.
Intuitively an inaccessible cardinal \(\kappa \) is a cardinal greater than \(\aleph _{0}\) that cannot be generated from smaller cardinals by means of the usual set-theoretic operations. For a more rigorous definition see Jech (2003), Part I, Chap. 5, p. 58. Note that inaccessible cardinals are the smallest large cardinals.
- 27.
See on this the proof of Lemma 12.13 in Jech (2003), Part I, Chap. 12, p. 167.
- 28.
We overlook here the question posed by the existence of isomorphic models of first-order ZFC, because the existence of such models would neither affect ZFC truths (not just first-order truths), nor the cardinality of the domains of the models.
- 29.
In other words, the models of the two extensions of first-order ZFC (and, a fortiori, of first-order ZFC) here mentioned are not isomorphic to one another, because they are not elementarily equivalent.
- 30.
With regard to L, the universe of Gödel’s constructible sets, see Jech (2003), Part II, Chap. 13, pp. 175–200.
- 31.
See on this Jech (2003), Part II, Chap. 14, pp. 201–224.
- 32.
The truth of a statement S transcends the verifiability of S if and only if the statement S is true/false independently of our ability to know which is the case.
- 33.
Here by ‘object’ we simply mean an element of the domain of discourse D. For a discussion of the notion of object adopted in this paper see Oliveri (2012), Sects. 3–5.
- 34.
Let \((D, \mathfrak {I})\) be a topological space in which the topology \(\mathfrak {I}\) is individuated by the interior operator i such that \(i: \mathcal {P}(D) \rightarrow \mathcal {P}(D)\) and:
-
1.
\( i(D) = D;\)
-
2.
\(i(A) \subset A\);
-
3.
\(i(A \cup B) = i(A) \cup i(B);\)
-
4.
\(i(i(A)) = i(A).\)
-
1.
- 35.
A Peano system is an ordered triple \((N, \mathfrak {0}, S)\) such that \(\mathfrak {0} \in N\), \(S: N \rightarrow N\) and:
- (i) :
-
\(\mathfrak {0} \not = S(n),\) for any \(n \in N\);
- (ii) :
-
\(S(n) = S(n') \Rightarrow n = n',\) for any \(n, n' \in N;\)
- (iii) :
-
for any \(A \subseteq N,\) if \(\mathfrak {0} \in A\) and, for all \(n \in N,\) if \(n \in A \Rightarrow S(n) \in A,\) then \(A = N.\)
The natural numbers are the elements of the domain \(\mathbb {N}\) of the Peano system \((\mathbb {N}, 0, +1)\) which, if we use second-order rather than first-order logic, as a consequence of Dedekind’s recursion theorem, is isomorphic to any Peano system \((N, \mathfrak {0}, S).\) Of course, \(+1: \mathbb {N} \rightarrow \mathbb {N}\) such that \(+1(n) \mapsto n +1,\) for any \(n \in \mathbb {N}.\) See on this Drake and Singh (1996), Chap. 6, Sect. 6.2.11.
- 36.
In first-order ZFC we can only deal with small structures.
- 37.
A class K is transitive if \(x \in \mathrm {K} \) implies \(x \subset \mathrm {K}.\)
- 38.
For Hellman, ‘[M]athematics is the free exploration of structural possibilities, pursued by (more or less) rigorous deductive means’ (Hellman 1989, Introduction, p. 6), where the notion of logical possibility ‘functions as a primitive notion, and must not be thought of as requiring a set-theoretical semantics for it to be intelligible’ (Hellman 1989, Chap. 2, p. 60).
- 39.
According to Hilbert,
[G]iven a mathematical theory T, it is possible to distinguish between results obtained within it which are of central importance—like the Fundamental Theorem of Calculus, the Fundamental Theorem of Algebra, the well-ordering theorem in set theory, the Completeness Theorem for first-order logic—and those which are not so important.
Relatively to these fundamental results, the theory can be developed either downwards or upwards. Developing the theory downwards means deriving consequences from the fundamental results. Developing the theory upwards means finding some statements of the theory from which it is possible to derive the fundamental results. Axiomatic thinking is, for Hilbert, the way of regarding a mathematical theory from the point of view of what I going to call ‘upward continuation’. [Oliveri (2005), p. 119.]
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Oliveri, G. (2016). True V or Not True V, That Is the Question. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_9
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