Potentialism about set theory Øystein Linnebo University of Oslo SotFoM III, 21–23 September 2015 Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 1 / 23
Open-endedness in set theory But [the set-theoretic paradoxes] are only apparent ‘contradictions’, and depend solely on confusing set theory itself , which is not categorically determined by its axioms, with individual models representing it. What appears as an ‘ultrafinite non- or super-set’ in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain. (Zermelo, 1930) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 2 / 23
Open-endedness in set theory But [the set-theoretic paradoxes] are only apparent ‘contradictions’, and depend solely on confusing set theory itself , which is not categorically determined by its axioms, with individual models representing it. What appears as an ‘ultrafinite non- or super-set’ in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain. (Zermelo, 1930) My goals Explore the ancient notion of potential infinity Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 2 / 23
Open-endedness in set theory But [the set-theoretic paradoxes] are only apparent ‘contradictions’, and depend solely on confusing set theory itself , which is not categorically determined by its axioms, with individual models representing it. What appears as an ‘ultrafinite non- or super-set’ in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain. (Zermelo, 1930) My goals Explore the ancient notion of potential infinity Articulate two successor concepts, which can be applied to set theory Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 2 / 23
Open-endedness in set theory But [the set-theoretic paradoxes] are only apparent ‘contradictions’, and depend solely on confusing set theory itself , which is not categorically determined by its axioms, with individual models representing it. What appears as an ‘ultrafinite non- or super-set’ in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain. (Zermelo, 1930) My goals Explore the ancient notion of potential infinity Articulate two successor concepts, which can be applied to set theory Show that both concepts have substantial explanatory value Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 2 / 23
Open-endedness in set theory But [the set-theoretic paradoxes] are only apparent ‘contradictions’, and depend solely on confusing set theory itself , which is not categorically determined by its axioms, with individual models representing it. What appears as an ‘ultrafinite non- or super-set’ in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain. (Zermelo, 1930) My goals Explore the ancient notion of potential infinity Articulate two successor concepts, which can be applied to set theory Show that both concepts have substantial explanatory value Explore logical consequences of adopting the successor concepts Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 2 / 23
Aristotle’s notion of potential infinity (I) For generally the infinite is as follows: there is always another and another to be taken. And the thing taken will always be finite, but always different (Physics, 206a27-29). Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 3 / 23
Aristotle’s notion of potential infinity (II) (1) It will always be the case that, for any natural number, we can produce a successor. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 4 / 23
Aristotle’s notion of potential infinity (II) (1) It will always be the case that, for any natural number, we can produce a successor. (2) Necessarily, for any number m , possibly there is a successor � ∀ m ♦ ∃ n Succ ( m , n ) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 4 / 23
Aristotle’s notion of potential infinity (II) (1) It will always be the case that, for any natural number, we can produce a successor. (2) Necessarily, for any number m , possibly there is a successor � ∀ m ♦ ∃ n Succ ( m , n ) Contrast the notion of actual or completed infinity: (3) For any number m , there is a successor ∀ m ∃ n Succ ( m , n ) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 4 / 23
Potentiality in set theory (2) Necessarily, for any objects xx , possibly there is their set { xx } � ∀ xx ♦ ∃ y Set ( y , xx ) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 5 / 23
Potentiality in set theory (2) Necessarily, for any objects xx , possibly there is their set { xx } � ∀ xx ♦ ∃ y Set ( y , xx ) Contrast the notion of actual or completed infinity: (3) For any objects xx , there is their set { xx } ∀ xx ∃ y Set ( y , xx ) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 5 / 23
Potentiality in set theory (2) Necessarily, for any objects xx , possibly there is their set { xx } � ∀ xx ♦ ∃ y Set ( y , xx ) Contrast the notion of actual or completed infinity: (3) For any objects xx , there is their set { xx } ∀ xx ∃ y Set ( y , xx ) Contrasts with potentialism about arithmetic While potential infinity is concerned with ω , potentialism about sets is concerned with Ω. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 5 / 23
Potentiality in set theory (2) Necessarily, for any objects xx , possibly there is their set { xx } � ∀ xx ♦ ∃ y Set ( y , xx ) Contrast the notion of actual or completed infinity: (3) For any objects xx , there is their set { xx } ∀ xx ∃ y Set ( y , xx ) Contrasts with potentialism about arithmetic While potential infinity is concerned with ω , potentialism about sets is concerned with Ω. While completion of the natural numbers is consistent, completion of the sets is not. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 5 / 23
Potentiality in set theory (2) Necessarily, for any objects xx , possibly there is their set { xx } � ∀ xx ♦ ∃ y Set ( y , xx ) Contrast the notion of actual or completed infinity: (3) For any objects xx , there is their set { xx } ∀ xx ∃ y Set ( y , xx ) Contrasts with potentialism about arithmetic While potential infinity is concerned with ω , potentialism about sets is concerned with Ω. While completion of the natural numbers is consistent, completion of the sets is not. Aristotle’s modality is metaphysical. Not so in the case of potentialism about sets. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 5 / 23
A three-way distinction Actualism : There is no use for modal notions in mathematics. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 6 / 23
A three-way distinction Actualism : There is no use for modal notions in mathematics. Liberal potentialism : Mathematical objects are generated successively. It is impossible to complete the process of generation. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 6 / 23
A three-way distinction Actualism : There is no use for modal notions in mathematics. Liberal potentialism : Mathematical objects are generated successively. It is impossible to complete the process of generation. Strict potentialism : Additionally, every truth is ‘made true’ at some stage of the generative process. Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 6 / 23
The notion of incompletability (6) Necessarily, for any φ ’s, there might have been another φ . Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 7 / 23
The notion of incompletability (6) Necessarily, for any φ ’s, there might have been another φ . To formalize (6), we use plural logic , i.e. plural variables xx , yy , . . . and ‘ x ≺ yy ’ to mean that x is one of yy : Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 7 / 23
The notion of incompletability (6) Necessarily, for any φ ’s, there might have been another φ . To formalize (6), we use plural logic , i.e. plural variables xx , yy , . . . and ‘ x ≺ yy ’ to mean that x is one of yy : � � � ∀ xx ∀ y ( y ≺ xx ↔ φ ( y )) → ♦ ∃ y ( y �≺ xx ∧ φ ( y )) Øystein Linnebo (University of Oslo) Potentialism about set theory 21–23 September 2015 7 / 23
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