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Single Idea 18191
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
]
Full Idea
The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals.
Gist of Idea
Axiom of Infinity: completed infinite collections can be treated mathematically
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.3)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.52
The
13 ideas
with the same theme
[axiom for a vast set based on successors]:
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22288
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We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available
[Dedekind, by Potter]
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18104
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Frege, unlike Russell, has infinite individuals because numbers are individuals
[Frege, by Bostock]
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14440
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We may assume that there are infinite collections, as there is no logical reason against them
[Russell]
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14447
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Infinity says 'for any inductive cardinal, there is a class having that many terms'
[Russell]
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13430
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Infinity: there is an infinity of distinguishable individuals
[Ramsey]
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10051
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The axiom of infinity is not a truth of logic, and its adoption is an abandonment of logicism
[Kneale,W and M]
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13037
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Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x)
[Kunen]
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17800
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The misnamed Axiom of Infinity says the natural numbers are finite in size
[Mayberry]
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13021
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The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
[Maddy]
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13022
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Infinite sets are essential for giving an account of the real numbers
[Maddy]
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18191
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Axiom of Infinity: completed infinite collections can be treated mathematically
[Maddy]
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10878
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Infinity: There exists a set of the empty set and the successor of each element
[Clegg]
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13044
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Infinity: There is at least one limit level
[Potter]
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