Is the formal logic in these axioms correct?

A group is a nonempty set $G$ with a binary operation $*$ that satisfies the following axioms:

  • $(a * b) \in G \; \forall \;a,b \in G$

  • $a * (b * c) = (a * b) * c \; \forall \; a,b,c \in G$

  • $\exists \; e \in G : a * e = e * a = a \; \forall \; a \in G$

  • $\forall \; a \in G \; \exists \; a^{-1} \in G: a * a^{-1} = a^{-1} * a = e$

In particular, in the last two axioms notice the difference of where the "for all" is placed.
The third axiom is meant to state that there exists an $e$ in $G$ that is a identity for all $a$ in $G$ under $*$.
While the last axiom is meant to state that for every $a$ in $G$ there exists an inverse $a^{-1}$ in $G$ under $*$.

  • $\begingroup$ correct it is.... $\endgroup$ – tattwamasi amrutam Jun 11 '14 at 5:08
  • 7
    $\begingroup$ If you are concerned about the formal logic I think you should be very careful with the placement of quantifiers. I would recommend rewriting the third statement as$$\exists \; e \in G : \; \forall \; a \in G : a * e = e * a = a$$and similarly for the first two. $\endgroup$ – David Jun 11 '14 at 5:13
  • 4
    $\begingroup$ The axioms as given are not sentences of first-order predicate calculus. In particular, the placement of the universal quantifiers is not right. There is also nothing connecting the $e$ of the fourth axiom with the $e$ of the fifth. As a very informal description, it is sort of OK. From the formal point of view, quite far from right. $\endgroup$ – André Nicolas Jun 11 '14 at 5:15
  • 1
    $\begingroup$ One can avoid the constant symbol by combining the last two sentences suitably into a single sentence. And a single binary function symbol is enough, if that's what we want. A more serious issue for formalization is the $\in$ symbol. In first-order group theory, one would omit all the $\in G$ stuff. $\endgroup$ – André Nicolas Jun 11 '14 at 5:34
  • 1
    $\begingroup$ By definition, a binary operation on a set $S$ is a function $S \times S \to S$, so the first axiom is unnecessary. $\endgroup$ – Qiaochu Yuan Jun 11 '14 at 6:01

You can see List of first-order theories under the heading : Groups.

In addition, you can check the details in every mathematical logic textbook, like Herbert Enderton, A Mathematical Introduction to Logic (2nd - 2001), page 93.


As you can see, there are different possibilities; what make the (little) differences are the symbols used as "basic".

In order to be "formal", you have to specify a language : an alphabet and the syntactical rules for building expressions (see comments above).

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.