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Single Idea 10492
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
]
Full Idea
Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement.
Gist of Idea
A few axioms of set theory 'force themselves on us', but most of them don't
Source
George Boolos (Must We Believe in Set Theory? [1997], p.130)
Book Ref
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.130
A Reaction
Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture.
The
31 ideas
from George Boolos
10482
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The logic of ZF is classical first-order predicate logic with identity
[Boolos]
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10483
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Mathematics and science do not require very high orders of infinity
[Boolos]
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10484
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The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first
[Boolos]
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10485
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Naďve sets are inconsistent: there is no set for things that do not belong to themselves
[Boolos]
|
10488
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It is lunacy to think we only see ink-marks, and not word-types
[Boolos]
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10487
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I am a fan of abstract objects, and confident of their existence
[Boolos]
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10489
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We deal with abstract objects all the time: software, poems, mistakes, triangles..
[Boolos]
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10491
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Infinite natural numbers is as obvious as infinite sentences in English
[Boolos]
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10490
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Mathematics isn't surprising, given that we experience many objects as abstract
[Boolos]
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10492
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A few axioms of set theory 'force themselves on us', but most of them don't
[Boolos]
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8693
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An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect
[Boolos]
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13547
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Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects)
[Boolos, by Potter]
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18192
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Do the Replacement Axioms exceed the iterative conception of sets?
[Boolos, by Maddy]
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14249
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Boolos reinterprets second-order logic as plural logic
[Boolos, by Oliver/Smiley]
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13841
|
Why should compactness be definitive of logic?
[Boolos, by Hacking]
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10829
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A sentence can't be a truth of logic if it asserts the existence of certain sets
[Boolos]
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10830
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Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems
[Boolos]
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10832
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'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
[Boolos]
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10833
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Many concepts can only be expressed by second-order logic
[Boolos]
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10834
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Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
[Boolos]
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7806
|
Boolos invented plural quantification
[Boolos, by Benardete,JA]
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13671
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Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology
[Boolos, by Shapiro]
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10267
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We should understand second-order existential quantifiers as plural quantifiers
[Boolos, by Shapiro]
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10225
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Monadic second-order logic might be understood in terms of plural quantifiers
[Boolos, by Shapiro]
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7785
|
The use of plurals doesn't commit us to sets; there do not exist individuals and collections
[Boolos]
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10736
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Boolos showed how plural quantifiers can interpret monadic second-order logic
[Boolos, by Linnebo]
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10780
|
Any sentence of monadic second-order logic can be translated into plural first-order logic
[Boolos, by Linnebo]
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10697
|
Identity is clearly a logical concept, and greatly enhances predicate calculus
[Boolos]
|
10698
|
Plural forms have no more ontological commitment than to first-order objects
[Boolos]
|
10699
|
Does a bowl of Cheerios contain all its sets and subsets?
[Boolos]
|
10700
|
First- and second-order quantifiers are two ways of referring to the same things
[Boolos]
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