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Single Idea 16871

[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation ]

Full Idea

It is possible for a truth to be an axiom in one system and not in another.

Gist of Idea

A truth can be an axiom in one system and not in another

Source

Gottlob Frege (Logic in Mathematics [1914], p.205)

Book Ref

Frege,Gottlob: 'Posthumous Writings', ed/tr. Hermes/Long/White etc [Blackwell 1979], p.205

A Reaction

Frege aspired to one huge single system, so this is a begrudging concession, one which modern thinkers would probably take for granted.

Related Idea

Idea 16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]

The 31 ideas with the same theme [giving basic truths from which some system is deduced]:

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