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This is an excerpt from Elements of Set theory by Enderton.(p. 11)

Our axiom system begins with two primitive notions, the concepts of "set" and "member." In terms of these concepts, we will define others, but the primitive notions remain undefined. Instead, we adopt a list of axioms concerning primitive notions. (The axioms can be thought of as divulging partial information regarding the meaning of the primitive notions.)

Having adopted a list of axioms, we will then proceed to derive sentences that are logical consequences (or theorems) of the axioms. Here a sentence $\sigma$ is said to be a logical consequence of the axioms if any assignment of meaning to the undefined notions of set and member making the axioms true also makes $\sigma$ true.

I do not understand what the highlighted sentences mean.

  1. The axioms can be thought of as divulging partial information regarding the meaning of the primitive notions: Aren't the axioms giving all the meaning to primitive terms? I mean, 'set' and 'member' are not defined, but all that they mean, is conveyed using axioms, isn't it?

  2. ...any assignment of meaning to the undefined notions of set and member making the axioms true also makes $\sigma$ true: I have no idea what this means.


Is it that we're just not interested in the exact meaning of 'set' and 'member', but there are some properties that their meanings have, and it is these properties that we specify using axioms? (hence the use of "partial information" in Enderton's book).

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    $\begingroup$ The answers and comments in this recent question may be helpful. $\endgroup$
    – blargoner
    Commented Sep 24, 2023 at 21:27

2 Answers 2

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  1. The axioms specify properties we are assuming about the undefined terms, but it is not necessarily true that they allow us to answer any question we could ask about the undefined terms. If you start with some axioms, there may be a statement for which you cannot prove true nor prove false, because the axioms haven't given enough information. So in general, axioms are only giving partial information. Consider geometry, where if you leave out the parallel postulate (or any equivalents, called neutral geometry), then you don't know whether you are in Euclidean geometry (where the parallel postulate holds) or hyperbolic geometry (where the parallel postulate fails).

  2. Because we have some undefined terms, we are free to define them how we like, provided that the axioms we have specified hold for these definitions. For example, in the neutral geometry case, we could choose the meanings of "point" and "line" to get either Euclidean or hyperbolic geometry. Therefore the parallel postulate is not a logical consequence of the axioms of neutral geometry, because in some cases it fails even though the axioms hold.

You should think of axioms as specifying properties we assuming to be true of our undefined terms.

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I suspect you are overthinking (2). Enderton is just applying the absolutely standard definition of logical consequence for sentences in some first-order language $L$ (in this case, the language of set theory).

A sentence $\sigma$ of the language is a logical consequence of set of sentences $\Gamma$ if on any interpretation of $L$'s domain of quantification and any interpretation of the non-logical vocabulary of $L$ over that domain, if all $\Gamma$ come out true on that interpretation, so does $\sigma$.

Now specialise to the current case, where $\Gamma$ are the axioms of formalised set theory, so we are dealing with a language $L$ where the intended domain of quantification comprises sets, and the one bit of non-logical vocabulary is '$\in$' intended to denote the membership relation.

Then $\Gamma$ (the axioms) will logically entail $\sigma$ (some sentence in the language of set theory $L$) if on any assignment of a domain of quantification to $L$ (if you like, whatever we take 'set' to mean) and any interpretation of '$\in$' (if you like, hatever we take 'member of' to mean), then if that interpretation makes $\Gamma$ all true, it will make $\sigma$ true. Which is what Enderton says.

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