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Single Idea 10997
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
]
Full Idea
The Von Neumann numbers have a structural isomorphism to the natural numbers - each number is the set of all its predecessors, so 2 is the set of 0 and 1. This helps proofs, but is unacceptable. 2 is not a set with two members, or a member of 3.
Gist of Idea
Von Neumann numbers are helpful, but don't correctly describe numbers
Source
Stephen Read (Thinking About Logic [1995], Ch.4)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.106
The
49 ideas
from Stephen Read
14187
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If logic is topic-neutral that means it delves into all subjects, rather than having a pure subject matter
[Read]
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14188
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Not all arguments are valid because of form; validity is just true premises and false conclusion being impossible
[Read]
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14182
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If the logic of 'taller of' rests just on meaning, then logic may be the study of merely formal consequence
[Read]
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14184
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In modus ponens the 'if-then' premise contributes nothing if the conclusion follows anyway
[Read]
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14186
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Logical connectives contain no information, but just record combination relations between facts
[Read]
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14185
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Conditionals are just a shorthand for some proof, leaving out the details
[Read]
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14183
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Maybe arguments are only valid when suppressed premises are all stated - but why?
[Read]
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10966
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A proposition objectifies what a sentence says, as indicative, with secure references
[Read]
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10983
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Knowledge of possible worlds is not causal, but is an ontology entailed by semantics
[Read]
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10982
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How can modal Platonists know the truth of a modal proposition?
[Read]
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10981
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A possible world is a determination of the truth-values of all propositions of a domain
[Read]
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10970
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A theory of logical consequence is a conceptual analysis, and a set of validity techniques
[Read]
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10984
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Logical consequence isn't just a matter of form; it depends on connections like round-square
[Read]
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10973
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A theory is logically closed, which means infinite premisses
[Read]
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10986
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Not all validity is captured in first-order logic
[Read]
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10972
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The non-emptiness of the domain is characteristic of classical logic
[Read]
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10985
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We should exclude second-order logic, precisely because it captures arithmetic
[Read]
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10987
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Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism'
[Read]
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10978
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In second-order logic the higher-order variables range over all the properties of the objects
[Read]
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10971
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A logical truth is the conclusion of a valid inference with no premisses
[Read]
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10988
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Any first-order theory of sets is inadequate
[Read]
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10974
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Compactness is when any consequence of infinite propositions is the consequence of a finite subset
[Read]
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10975
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Compactness does not deny that an inference can have infinitely many premisses
[Read]
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10977
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Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
[Read]
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10976
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Compactness makes consequence manageable, but restricts expressive power
[Read]
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10979
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Although second-order arithmetic is incomplete, it can fully model normal arithmetic
[Read]
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10980
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Second-order arithmetic covers all properties, ensuring categoricity
[Read]
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10992
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The point of conditionals is to show that one will accept modus ponens
[Read]
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10989
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The standard view of conditionals is that they are truth-functional
[Read]
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10995
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A haecceity is a set of individual properties, essential to each thing
[Read]
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11000
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If worlds are concrete, objects can't be present in more than one, and can only have counterparts
[Read]
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10998
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The mind abstracts ways things might be, which are nonetheless real
[Read]
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10996
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Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions)
[Read]
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11001
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Equating necessity with truth in every possible world is the S5 conception of necessity
[Read]
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10997
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Von Neumann numbers are helpful, but don't correctly describe numbers
[Read]
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11004
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Necessity is provability in S4, and true in all worlds in S5
[Read]
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11011
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Same say there are positive, negative and neuter free logics
[Read]
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11007
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Quantifiers are second-order predicates
[Read]
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11013
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Identities and the Indiscernibility of Identicals don't work with supervaluations
[Read]
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11012
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A 'supervaluation' gives a proposition consistent truth-value for classical assignments
[Read]
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11005
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Negative existentials with compositionality make the whole sentence meaningless
[Read]
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11014
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Self-reference paradoxes seem to arise only when falsity is involved
[Read]
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11016
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Would a language without vagueness be usable at all?
[Read]
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11019
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Supervaluations say there is a cut-off somewhere, but at no particular place
[Read]
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11018
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There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers
[Read]
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11017
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Some people even claim that conditionals do not express propositions
[Read]
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11020
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Realisms like the full Comprehension Principle, that all good concepts determine sets
[Read]
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11025
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Infinite cuts and successors seems to suggest an actual infinity there waiting for us
[Read]
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11024
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Semantics must precede proof in higher-order logics, since they are incomplete
[Read]
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