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Single Idea 10674
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
]
Full Idea
A language with plurals is better for arithmetic. Instead of a first-order fragment expressible by an induction schema, we have the complete truth with a plural induction axiom, beginning 'If there are some numbers...'.
Gist of Idea
A plural language gives a single comprehensive induction axiom for arithmetic
Source
Keith Hossack (Plurals and Complexes [2000], 4)
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.420
The
21 ideas
from 'Plurals and Complexes'
10666
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Plural reference will refer to complex facts without postulating complex things
[Hossack]
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10664
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Complex particulars are either masses, or composites, or sets
[Hossack]
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10665
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Leibniz's Law argues against atomism - water is wet, unlike water molecules
[Hossack]
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10663
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A thought can refer to many things, but only predicate a universal and affirm a state of affairs
[Hossack]
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10668
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We are committed to a 'group' of children, if they are sitting in a circle
[Hossack]
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10669
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Plural reference is just an abbreviation when properties are distributive, but not otherwise
[Hossack]
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10671
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Plural definite descriptions pick out the largest class of things that fit the description
[Hossack]
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10673
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Plural language can discuss without inconsistency things that are not members of themselves
[Hossack]
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10675
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A plural comprehension principle says there are some things one of which meets some condition
[Hossack]
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10674
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A plural language gives a single comprehensive induction axiom for arithmetic
[Hossack]
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10676
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The Axiom of Choice is a non-logical principle of set-theory
[Hossack]
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10677
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Extensional mereology needs two definitions and two axioms
[Hossack]
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10678
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The relation of composition is indispensable to the part-whole relation for individuals
[Hossack]
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10682
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The fusion of five rectangles can decompose into more than five parts that are rectangles
[Hossack]
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10681
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In arithmetic singularists need sets as the instantiator of numeric properties
[Hossack]
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10680
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The theory of the transfinite needs the ordinal numbers
[Hossack]
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10684
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I take the real numbers to be just lengths
[Hossack]
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10683
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We could ignore space, and just talk of the shape of matter
[Hossack]
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10685
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Set theory is the science of infinity
[Hossack]
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10686
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The Axiom of Choice guarantees a one-one correspondence from sets to ordinals
[Hossack]
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10687
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Maybe we reduce sets to ordinals, rather than the other way round
[Hossack]
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