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Single Idea 10881
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
]
Full Idea
The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept.
Gist of Idea
The concept of 'ordinal number' is set-theoretic, not arithmetical
Source
Leon Horsten (Philosophy of Mathematics [2007], §2.3)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.7
The
50 ideas
from Leon Horsten
10881
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The concept of 'ordinal number' is set-theoretic, not arithmetical
[Horsten]
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10882
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Predicative definitions only refer to entities outside the defined collection
[Horsten]
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10884
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A theory is 'categorical' if it has just one model up to isomorphism
[Horsten]
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10885
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Computer proofs don't provide explanations
[Horsten]
|
15323
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Truth is a property, because the truth predicate has an extension
[Horsten]
|
15324
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Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
[Horsten]
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15325
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Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
[Horsten]
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15329
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Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
[Horsten]
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15326
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Doubt is thrown on classical logic by the way it so easily produces the liar paradox
[Horsten]
|
15328
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A theory is 'non-conservative' if it facilitates new mathematical proofs
[Horsten]
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15331
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Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
[Horsten]
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15332
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'Reflexive' truth theories allow iterations (it is T that it is T that p)
[Horsten]
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15330
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Friedman-Sheard theory keeps classical logic and aims for maximum strength
[Horsten]
|
15333
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Modern correspondence is said to be with the facts, not with true propositions
[Horsten]
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15337
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The correspondence 'theory' is too vague - about both 'correspondence' and 'facts'
[Horsten]
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15334
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The coherence theory allows multiple coherent wholes, which could contradict one another
[Horsten]
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15336
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The pragmatic theory of truth is relative; useful for group A can be useless for group B
[Horsten]
|
15338
|
We may believe in atomic facts, but surely not complex disjunctive ones?
[Horsten]
|
15340
|
Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
[Horsten]
|
15341
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Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
[Horsten]
|
15344
|
Deflationism skips definitions and models, and offers just accounts of basic laws of truth
[Horsten]
|
15346
|
Axiomatic approaches to truth avoid the regress problem of semantic theories
[Horsten]
|
15345
|
Semantic theories have a regress problem in describing truth in the languages for the models
[Horsten]
|
15348
|
Propositions have sentence-like structures, so it matters little which bears the truth
[Horsten]
|
15347
|
A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding
[Horsten]
|
15349
|
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
[Horsten]
|
15350
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The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
[Horsten]
|
15351
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Axiomatic theories take truth as primitive, and propose some laws of truth as axioms
[Horsten]
|
15352
|
A definition should allow the defined term to be eliminated
[Horsten]
|
15353
|
The first incompleteness theorem means that consistency does not entail soundness
[Horsten]
|
15354
|
Tarski's hierarchy lacks uniform truth, and depends on contingent factors
[Horsten]
|
15355
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Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
[Horsten]
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15356
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Deflationism concerns the nature and role of truth, but not its laws
[Horsten]
|
15357
|
Philosophy is the most general intellectual discipline
[Horsten]
|
15358
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Deflationism says truth isn't a topic on its own - it just concerns what is true
[Horsten]
|
15359
|
Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
[Horsten]
|
15360
|
ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
[Horsten]
|
15361
|
A good theory of truth must be compositional (as well as deriving biconditionals)
[Horsten]
|
15363
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In the supervaluationist account, disjunctions are not determined by their disjuncts
[Horsten]
|
15362
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If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't
[Horsten]
|
15364
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English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
[Horsten]
|
15366
|
Satisfaction is a primitive notion, and very liable to semantical paradoxes
[Horsten]
|
15367
|
By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
[Horsten]
|
15368
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This deflationary account says truth has a role in generality, and in inference
[Horsten]
|
15369
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Set theory is substantial over first-order arithmetic, because it enables new proofs
[Horsten]
|
15370
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Predicativism says mathematical definitions must not include the thing being defined
[Horsten]
|
15371
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An axiomatic theory needs to be of maximal strength, while being natural and sound
[Horsten]
|
15372
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Some claim that indicative conditionals are believed by people, even though they are not actually held true
[Horsten]
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15373
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Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
[Horsten]
|
15374
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Truth has no 'nature', but we should try to describe its behaviour in inferences
[Horsten]
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