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Single Idea 16870

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

Full Idea

The axioms are theorems, but truths for which no proof can be given in our system, and no proof is needed. It follows from this that there are no false axioms, and we cannot accept a thought as an axiom if we are in doubt about its truth.

Gist of Idea

Axioms are truths which cannot be doubted, and for which no proof is needed

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

He struggles to be as objective as possible, but has to concede that whether we can 'doubt' the axiom is one of the criteria.

Related Idea

Idea 16867 Logic not only proves things, but also reveals logical relations between them [Frege]

The 20 ideas from 'Logic in Mathematics'

11219 Frege suggested that mathematics should only accept stipulative definitions [Frege, by Gupta]
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
16863 Does some mathematical reasoning (such as mathematical induction) not belong to logic? [Frege]
16862 The closest subject to logic is mathematics, which does little apart from drawing inferences [Frege]
16865 'Theorems' are both proved, and used in proofs [Frege]
16866 Tracing inference backwards closes in on a small set of axioms and postulates [Frege]
16867 Logic not only proves things, but also reveals logical relations between them [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]
16869 To create order in mathematics we need a full system, guided by patterns of inference [Frege]
16873 Thoughts are not subjective or psychological, because some thoughts are the same for us all [Frege]
16872 A thought is the sense expressed by a sentence, and is what we prove [Frege]
16874 The parts of a thought map onto the parts of a sentence [Frege]
16876 We need definitions to cram retrievable sense into a signed receptacle [Frege]
16875 We use signs to mark receptacles for complex senses [Frege]
16877 A 'constructive' (as opposed to 'analytic') definition creates a new sign [Frege]
16878 We must be clear about every premise and every law used in a proof [Frege]
16879 A sign won't gain sense just from being used in sentences with familiar components [Frege]
9388 Every concept must have a sharp boundary; we cannot allow an indeterminate third case [Frege]