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Single Idea 17930
[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
]
Full Idea
A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
Gist of Idea
Axioms are 'categorical' if all of their models are isomorphic
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.25
A Reaction
The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.
The
31 ideas
with the same theme
[giving basic truths from which some system is deduced]:
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13004
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Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof
[Aristotle, by Leibniz]
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13002
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It is always good to reduce the number of axioms
[Leibniz]
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17624
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To understand axioms you must grasp their logical power and priority
[Frege, by Burge]
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16866
|
Tracing inference backwards closes in on a small set of axioms and postulates
[Frege]
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16868
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The essence of mathematics is the kernel of primitive truths on which it rests
[Frege]
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16870
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Axioms are truths which cannot be doubted, and for which no proof is needed
[Frege]
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16871
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A truth can be an axiom in one system and not in another
[Frege]
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16886
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The truth of an axiom must be independently recognisable
[Frege]
|
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17963
|
The facts of geometry, arithmetic or statics order themselves into theories
[Hilbert]
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17966
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Axioms must reveal their dependence (or not), and must be consistent
[Hilbert]
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6109
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Some axioms may only become accepted when they lead to obvious conclusions
[Russell]
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17630
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The sources of a proof are the reasons why we believe its conclusion
[Russell]
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17629
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Which premises are ultimate varies with context
[Russell]
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17640
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Finding the axioms may be the only route to some new results
[Russell]
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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
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19292
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Logic doesn't split into primitive and derived propositions; they all have the same status
[Wittgenstein]
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19632
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An axiom has no more authority than a frenzy
[Cioran]
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17606
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Axioms reveal the underlying assumptions, and reveal relationships between different areas
[Kline]
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17622
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We come to believe mathematical propositions via their grounding in the structure
[Burge]
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17778
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Axiomatiation relies on isomorphic structures being essentially the same
[Mayberry]
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17779
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'Classificatory' axioms aim at revealing similarity in morphology of structures
[Mayberry]
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17780
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'Eliminatory' axioms get rid of traditional ideal and abstract objects
[Mayberry]
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13416
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Mathematics must be based on axioms, which are true because they are axioms, not vice versa
[Tait, by Parsons,C]
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17605
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Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
[Maddy]
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17625
|
If two mathematical themes coincide, that suggest a single deep truth
[Maddy]
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9649
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Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
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23443
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The axioms of group theory are not assertions, but a definition of a structure
[Linnebo]
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23444
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To investigate axiomatic theories, mathematics needs its own foundational axioms
[Linnebo]
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17761
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A compact axiomatisation makes it possible to understand a field as a whole
[Walicki]
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17763
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Axiomatic systems are purely syntactic, and do not presuppose any interpretation
[Walicki]
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17930
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Axioms are 'categorical' if all of their models are isomorphic
[Colyvan]
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