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Single Idea 18183
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
]
Full Idea
Set theoretic foundations bring all mathematical objects and structures into one arena, allowing relations and interactions between them to be clearly displayed and investigated.
Gist of Idea
Set theory brings mathematics into one arena, where interrelations become clearer
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.2)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.26
A Reaction
The first of three benefits of set theory which Maddy lists. The advantages of the one arena seem to be indisputable.
Related Ideas
Idea 18184
Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
Idea 18185
Unified set theory gives a final court of appeal for mathematics [Maddy]
The
26 ideas
from 'Naturalism in Mathematics'
18182
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The extension of concepts is not important to me
[Maddy]
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18163
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Mathematics rests on the logic of proofs, and on the set theoretic axioms
[Maddy]
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18169
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Axiom of Reducibility: propositional functions are extensionally predicative
[Maddy]
|
18168
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'Propositional functions' are propositions with a variable as subject or predicate
[Maddy]
|
18172
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Infinity has degrees, and large cardinals are the heart of set theory
[Maddy]
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18175
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For any cardinal there is always a larger one (so there is no set of all sets)
[Maddy]
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18164
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Frege solves the Caesar problem by explicitly defining each number
[Maddy]
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18167
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We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
[Maddy]
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18171
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Cantor and Dedekind brought completed infinities into mathematics
[Maddy]
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18184
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Making set theory foundational to mathematics leads to very fruitful axioms
[Maddy]
|
18185
|
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]
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18186
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Identifying geometric points with real numbers revealed the power of set theory
[Maddy]
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18188
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The line of rationals has gaps, but set theory provided an ordered continuum
[Maddy]
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18177
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In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
[Maddy]
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18187
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Theorems about limits could only be proved once the real numbers were understood
[Maddy]
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18190
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Completed infinities resulted from giving foundations to calculus
[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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18193
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The Axiom of Foundation says every set exists at a level in the set hierarchy
[Maddy]
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18194
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'Forcing' can produce new models of ZFC from old models
[Maddy]
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18195
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A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
[Maddy]
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18196
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An 'inaccessible' cardinal cannot be reached by union sets or power sets
[Maddy]
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18204
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Scientists posit as few entities as possible, but set theorist posit as many as possible
[Maddy]
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18205
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The theoretical indispensability of atoms did not at first convince scientists that they were real
[Maddy]
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18206
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Science idealises the earth's surface, the oceans, continuities, and liquids
[Maddy]
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18207
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Maybe applications of continuum mathematics are all idealisations
[Maddy]
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