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Single Idea 18190
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
]
Full Idea
The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals.
Gist of Idea
Completed infinities resulted from giving foundations to calculus
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.3)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.51
A Reaction
Effectively, completed infinities just are the real numbers.
The
26 ideas
from 'Naturalism in Mathematics'
18182
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The extension of concepts is not important to me
[Maddy]
|
18163
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Mathematics rests on the logic of proofs, and on the set theoretic axioms
[Maddy]
|
18169
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Axiom of Reducibility: propositional functions are extensionally predicative
[Maddy]
|
18168
|
'Propositional functions' are propositions with a variable as subject or predicate
[Maddy]
|
18172
|
Infinity has degrees, and large cardinals are the heart of set theory
[Maddy]
|
18175
|
For any cardinal there is always a larger one (so there is no set of all sets)
[Maddy]
|
18164
|
Frege solves the Caesar problem by explicitly defining each number
[Maddy]
|
18167
|
We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
[Maddy]
|
18171
|
Cantor and Dedekind brought completed infinities into mathematics
[Maddy]
|
18184
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Making set theory foundational to mathematics leads to very fruitful axioms
[Maddy]
|
18185
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Unified set theory gives a final court of appeal for mathematics
[Maddy]
|
18183
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Set theory brings mathematics into one arena, where interrelations become clearer
[Maddy]
|
18186
|
Identifying geometric points with real numbers revealed the power of set theory
[Maddy]
|
18188
|
The line of rationals has gaps, but set theory provided an ordered continuum
[Maddy]
|
18177
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In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
[Maddy]
|
18187
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Theorems about limits could only be proved once the real numbers were understood
[Maddy]
|
18190
|
Completed infinities resulted from giving foundations to calculus
[Maddy]
|
18191
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Axiom of Infinity: completed infinite collections can be treated mathematically
[Maddy]
|
18193
|
The Axiom of Foundation says every set exists at a level in the set hierarchy
[Maddy]
|
18194
|
'Forcing' can produce new models of ZFC from old models
[Maddy]
|
18195
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A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
[Maddy]
|
18196
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An 'inaccessible' cardinal cannot be reached by union sets or power sets
[Maddy]
|
18204
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Scientists posit as few entities as possible, but set theorist posit as many as possible
[Maddy]
|
18205
|
The theoretical indispensability of atoms did not at first convince scientists that they were real
[Maddy]
|
18206
|
Science idealises the earth's surface, the oceans, continuities, and liquids
[Maddy]
|
18207
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Maybe applications of continuum mathematics are all idealisations
[Maddy]
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