Abstract
The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First, that most set theorists were surprised by the introduction of the forcing method. Second, that most set theorists think that forcing is a natural part of contemporary set theory. Third, that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and that for most set theorists, a difference in consistency strength weighs much more than the difference between Forcing Axiom and Large Cardinal Axiom.
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Notes
- 1.
The first author is an Associate Member at the IHPST (Institut d’histoire et de philosophie des sciences et des techniques) of the Université Panthéon-Sorbonne, Paris, France. The second author is grateful for the support of the DAAD (Deutscher Akademischer Austauschdienst) and the IHPST.
- 2.
Rouse (1995, p. 2).
- 3.
But it will be considered more attentively in the PhD project of the second author.
- 4.
- 5.
Rittberg (2015).
- 6.
Large research programs have already been described by some set theorists. Consider, for example, the research programs by Hugh W. Woodin or Sy-David Friedman. For Woodin’s program see Woodin (2017), and for Friedman’s Hyperuniverse Program, see Arrigoni and Friedman (2013). Not everybody describes his/her program so specifically and we wish to discover more about these unspecified programs.
- 7.
Badiou (2005) bases ontology on set-theoretic axioms and also considers forcing.
- 8.
- 9.
See the article by Laura Fontanella in this volume.
- 10.
- 11.
Hamkins (2012).
- 12.
Thanks to Carolin Antos for emphasising this fact.
- 13.
\(0=\emptyset , n+1=n\cup \{n\}, \mathbb {N}=\{n:n<\omega \}, \mathbb {R}=\mathbb {P}(\mathbb {N})\), and so on. (These equations should not be understood as the claim that numbers are sets.)
- 14.
For example, take the coding ’∃’=8, ’∀’=9, ’¬’=5, ’(’=0, ’)’=1, ’v 0’=(2,0), ’v 1’=(2,1), then the statement ’∃v 0∀v 1 ¬(v 1 ∈v 0)’ can be coded as the sequence 〈8, (2, 0), 9, (2, 1), 5, 0, (2, 1), 4, (2, 0), 1〉.
- 15.
Of course, any stronger assumption works as well, in particular any Large Cardinal Axiom.
- 16.
- 17.
- 18.
- 19.
General Continuum Hypothesis: For every ordinal α: \(\aleph _{\alpha +1}=2^{\aleph _\alpha }\).
- 20.
Amazingly, this was not the case of Gödel, who shortly after discovering L and proving the relative consistency of GCH stated that he believes in the independence of GCH, see the beginning of the interview for this. But then, Gödel was considered a logician and a philosopher, not a mathematician, by the peers of the time.
- 21.
Gödel (1947).
- 22.
Cohen (1966).
- 23.
Kunen (1980).
- 24.
Jech (2003).
- 25.
The correct notion to distinguish between regular and singular cardinals is the notion of cofinality. The cofinality of a cardinal κ is the length of the shortest sequence of ordinals less than κ which converges to κ.
- 26.
A proper class contains all sets that satisfy a given first order formula, but is itself not a set. (Thus, a proper class is defined by unrestricted comprehension over the universe of sets.)
- 27.
König (1905).
- 28.
Easton (1970, Theorem 1, pp.140-1). It is interesting to note that this is potentially class forcing. The Forcing Theorem of Cohen only applies to special cases of class forcing, so class forcing is less widely spread in applications. All of the forcing notions to follow will be set forcings.
- 29.
“(3) Un ensemble ordonné (linéairement) sans sauts ni lacunes et tel tout ensemble de ses intervalles (contenant plus qu’un élément) n’empiétant pas les uns sur les autres est au plus dénombrable, est-il nécessairement un continu linéaire (ordinaire)?” Sierpiński et al. (1920). Translation (by the authors): “Is a (linearly) ordered set without jumps nor gaps, such that every set of its non-overlapping intervals (containing more than one element) is at most countable, necessarily a linear continuum?”
- 30.
See Jech (2003, pp.114–116).
- 31.
‘Diamond-principle’.
- 32.
A subset of ω 1 is called stationary if it intersects all closed and unbounded subsets C ⊆ω 1, where C is closed if for every sequence (a n)n<ω ⊆C the limit \(\bigcup \{a_n:n<\omega \}\) is also an element of C, and C is unbounded if for every a ∈C there is a b ∈C such that b > a.
- 33.
Jech (2003, p.191).
- 34.
Jensen actually showed a more general version of which this theorem is one instance Jensen (1972, Theorem 6.2 and Lemma 6.5, pp.292–5).
- 35.
Solovay and Tennenbaum (1971) refer to theorem 6.3 on p.228.
- 36.
G is \(\mathbb {D}\)-generic means that G ∩D ≠ ∅ for every \(D\in \mathbb {D}\).
- 37.
Solovay and Tennenbaum (1971, fn on p.232).
- 38.
Solovay and Tennenbaum (1971, Theorem 7.11 on p.242).
- 39.
Borel (1919). A strong measure zero set is a subset X of the reals such that for every sequence 〈ε n: n < ω〉 of positive real numbers there is a sequence 〈I n: n < ω〉 of intervals with length(I n) ≤ε n such that \(X\subseteq \bigcup \{I_n:n<\omega \}\).
- 40.
Sierpiński (1928).
- 41.
- 42.
Shelah (1982).
- 43.
Shelah (2017, p.90).
- 44.
[λ]ω is the set of all countable subsets of λ.
- 45.
Jech (2003, Lemma 31.2 on p. 601).
- 46.
See Shelah (2017, III. §3.).
- 47.
Assuming that there is a supercompact cardinal, one can prove that there is a model of ZFC+PFA.
- 48.
Jech (2003, Theorem 31.23 on p. 609).
- 49.
PA stands for Peano Arithmetic which is the formal theory of the natural numbers.
- 50.
Hilbert wanted to prove the consistency of mathematics and focussed on axiomatisations of number theory. His program can be transferred to set theory, as set theory counts as a foundation of mathematics. Thus, of one would prove its consistency, Hilbert’s aim would be resolved.
- 51.
Menachem Magidor certainly is an exception because he thinks that Forcing Axioms are natural axioms.
- 52.
For more on her view, see her article ‘A New Foundational Crisis in Mathematics, is It Really Happening?’ in this volume.
- 53.
His italics, Woodin (2010, p.504).
- 54.
www.math.uci.edu/node/20943 (06/05/2018)
- 55.
For every Δ 0-sentence φ (a sentence with only bounded quantifiers) and every transitive standard model M of ZFC, \(\varphi \leftrightarrow \ulcorner M\models \varphi \urcorner \) (in ZFC). Under the assumption that there is a transitive standard model of ZFC, this means that Δ 0-sentences cannot be independent. They are either provable or refutable. There are other well-known absoluteness theorems, such as the Shoenfield’s Absoluteness.
- 56.
- 57.
- 58.
For a philosophical discussion of these regions see Džamonja and Panza (2018).
- 59.
Malliaris and Shelah (2013).
- 60.
Tao (2018).
- 61.
ibid.
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Acknowledgements
The second author wants to thank the first author for her generous time to share her knowledge and views. She is also grateful to Colin Rittberg for his encouragement and discussion, to the audiences in Brussels, Berlin, and Konstanz for their feedback and questions, which helped a lot to sharpen her approach, to the anonymous referee for his/her substantial and very helpful review, and to Stefan Steins for a rigourous improvement of the language.
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Džamonja, M., Kant, D. (2019). Interview With a Set Theorist. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_1
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