Concept
Independence (mathematical logic)
Overview
In mathematical logic, independence is the unprovability of some specific sentence from some specific set of other sentences. The sentences in this set are referred to as "axioms".
A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T. (This concept is unrelated to the idea of "decidability" as in a decision problem.)
A theory T is independent if no axiom in T is provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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In the relations between states, as in the relations of groups within a single state, what is to be desired is independence for each as regards internal affairs, and law rather than private force as regards external affairs. But as regards groups within a state, it is internal independence that must be emphasized, since that is what is lacking; subjection to law has been secured, on the whole, since the end of the…
Bertrand Russell, Political Ideals
The chapter “Chapter V: National Independence and Internationalism” is titled for Independence; names Independence several times.
It cannot be too strongly urged that patriotism should be only defensive, not aggressive. But with this proviso, I think a spirit of patriotism is absolutely necessary to the regeneration of China. Independence is to be sought, not as an end in itself, but as a means towards a new blend of Western skill with the traditional Chinese virtues. If this end is not achieved, political independence will have little value.
Bertrand Russell, The Problem of China
Names Independence several times; states what Independence is.
Whether objectively independent or not, they cannot be subjectively so--they cannot have independence as presented to our consciousness; and this is the only kind of independence with which we are concerned. And here, before leaving these illustrations, and especially this last one, let us not omit to notice how clearly they exhibit that increasingly active _consensus_ of the sciences which characterises their…
Herbert Spencer, Essays on education and kindred subjects
Names Independence twice; opens on Independence.
Related ideas
- Interpretation (logic)
- Zermelo–Fraenkel set theory
- Mathematical logic
- Sentence (mathematical logic)
- Continuum hypothesis
- Decision problem
- Axiom of choice
Place in the tradition
Part of Outline of logic.
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