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I've read the definitions, but they are not very clear to me.

Looks like both are a premisse so evident to be accepted as true without controversy.

But, what about the axioms on the set theory?? Many of them are not evident, which contradicts the definition i've read about axiom.

Thank you!

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An axiom is expressible as a complete sentence, with a subject and a predicate.

A concept, on the other hand, is not expressible as a sentence, but as a word or a phrase, usually a noun or a noun phrase. A primitive concept is one that is not defined in terms of other concepts.

Thus "point" and "line" are primitive concepts in some axiomatizations of geometry, and "For any two points, there is exactly one line incident to both" is an axiom in some such systems.

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  • $\begingroup$ Axioms in propositional calculi do NOT have a subject-predicate structure. $\endgroup$ – Doug Spoonwood Oct 30 '13 at 1:43
  • $\begingroup$ @DougSpoonwood : Suppose I write $A \vee \sim A$. I think $A$ itself is a predicate, so the whole thing is a complete sentence. Alright, some sentences don't need a subject, so maybe I exaggerated there. In some languages there's a verb that means "It's raining", and like any verb it may have various tenses, and even in English the impersonal "It" isn't really a subject. At any rate, an axiom always contains a verb. $\endgroup$ – Michael Hardy Oct 30 '13 at 1:54
  • $\begingroup$ Michael could you provide more examples of axioms and concepts? $\endgroup$ – Trismegistos Oct 30 '13 at 8:14
  • $\begingroup$ Thank you, michael! Could you provide also more examples of concepts and primitive concepts? $\endgroup$ – Voyager Oct 30 '13 at 20:52
  • $\begingroup$ A predicate describes a property of a subject. The sentence "either this sentence exists or this sentence does not exist.", does NOT have "sentence" predicated or describing any subject. If I say "I went to the store. It was a large store." "It" does qualify as a subject in the second sentence. Of course "it" requires context, but what doesn't? I think you're right that every axiom contains a verb. Concepts need not contain verbs. $\endgroup$ – Doug Spoonwood Oct 31 '13 at 0:46
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Every field of knowledge involves concepts. Sciences, and especially mathematics, require high precision through carefully formulated concepts. The dictionary phenomenon is that concepts are explained by simpler concepts and hence the simplest (so-called primitive) concepts have to be left unexplained. In set theory, we have primitive concepts like "set" and "membership". In geometry we have primitive concepts like "point", "line" and "incidence", etc.

The axiomatic method of mathematics is a wonderful and pretty unique way of getting around this problem: the basic idea is not attempting at defining primitive concepts but prescribing how they (inter)relate. In practice, this is exactly what you need to deal with them. For instance, projective geometry requires (among other things) that every two distinct lines have a unique common coincident point and every two distinct points have a unique coincident line. Set theory requires (among many other things) a set with no members (empty set) and with every set A and every statement P(x), a set with members: all members x of A satsifying P(x).

In this way, a beautiful method obtains to express primitive concepts or mutually dependent concepts (like set and membership) accurately and elegantly. Axioms are also the starting point of logical deduction. In a way, mathematics is information processing with logic, the information being about concepts.

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  • $\begingroup$ The first two paragraphs seem like a full answer. I don't understand the purpose of the third. $\endgroup$ – Jacob Maibach Apr 22 '18 at 15:16

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