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In Ebbinghaus' Mathematical Logic:

In this and the next section we present two simple mathematical proofs. They illustrate some of the methods of proof used by mathematicians. Guided by these examples, we raise some questions which lead us to the main topics of the book.

We begin with the proof of a theorem from group theory. We therefore require the axioms of group theory, which we now state. We use $*$ to denote the group multiplication and $e$ to denote the identity element. The axioms may then be formulated as follows:

(G1) For all $x,y,z: (x * y) * z = x * (y * z)$.

(G2) For all $x : x * e = x$.

(G3) For every $x$ there is a $y$ such that $x * y = e$.

A group is a triple $(G, *^G, e^G)$ which satisfies (Gl), (G2), and (G3). Here $G$ is a set, $e^G$ is an element of $G$, and $*^G$ is a binary function on $G$, i.e., a function defined on all ordered pairs of elements from $G$, the values of which are also elements of $G$. The variables $x, y, z$ range over elements of $G$, $*$ refers to $*^G$, and $e$ refers to $e^G$.

Is it correct that a group is a structure of a first order logic system?

In a first order logic system, the axioms are specified by its deductive system (e.g. the axioms in natural deduction).

Does "axiom" in "the axioms of group theory" mean the same as "axiom" in first order logic system? Or are they two different levels of axioms?

Are only "axioms" in first order logic system axioms?

What logical concept corresponds to "axiom" in "the axioms of group theory"?

Thanks.

Tim
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    "Does "axiom" in "the axioms of group theory" mean the same as "axiom" in first order logic system?" Yes; axioms for group can be easily formalized in first-order language. In this way, we have the first-order theory of groups. – Mauro ALLEGRANZA Jul 29 '20 at 12:13
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    A group is a mathematical structure (a collection of mathematical objects + operation(s) on them) that satisfies the group axioms. – Mauro ALLEGRANZA Jul 29 '20 at 12:14
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    Yes, a group is a first-order structure, but not all group theory concepts are expressible in first-order terms. For example, being nilpotent, I believe, cannot be. Being finite soluble, on the other hand, can be, as a product of 56 commutators is trivial and ond only if $G$ is finite soluble. – David A. Craven Jul 29 '20 at 12:17
  • I guess here and here you can find a partial answer to your first question. I'm not sure to understand what you mean when you ask "Are only "axioms" in first order logic system axioms?" and "What logical concept corresponds to "axiom" in "the axioms of group theory"?". – Taroccoesbrocco Jul 29 '20 at 12:25
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    See also the post Meaning of the word “axiom” as well as Group axioms in first order language – Mauro ALLEGRANZA Jul 29 '20 at 12:30
  • "Are only "axioms" in first order logic system axioms?" Obviously not. Spinoza's Ethics has axioms but it is not a "first order logic system". – Mauro ALLEGRANZA Jul 29 '20 at 12:35
  • @DavidA.Craven Thanks. My question is: Is it correct that different first order logic systems may have different different sets of axioms, but they all must superset a fixed set of axioms? – Tim Aug 20 '20 at 17:40
  • @MauroALLEGRANZA Thanks. My question is: Is it correct that different first order logic systems may have different different sets of axioms, but they all must superset a fixed set of axioms? – Tim Aug 20 '20 at 17:41

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Well, I think you have an axiom system of first-order logic available together with the rule modus ponens.

Built on this axiomatic system, you add the axioms of group theory. Then each model of this first-order system is a group.

Wuestenfux
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  • Thanks. Is it correct that different first order logic systems may have different different sets of axioms, but they all must superset "an axiom system of first-order logic"? – Tim Aug 20 '20 at 17:38
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    Indeed, different first-order logic systems have a core system consisting of axioms and rules (modus ponus) which can be considered to be correct and adequate. – Wuestenfux Aug 21 '20 at 06:55
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On P128 of Enderton's A Mathematical Introduction to Logic

A logic book in the bootstrap tradition might well begin with this section on a deductive calculus. Such a book would first state the logical axioms and the rules of inference and would explain that they are acceptable to reasonable people. Then it would proceed to show that many formulas were deducible (or deducible from certain nonlogical axioms, such as axioms for set theory).

The "axioms" in group theory are a nonlogical concept, just a set of formulas from logic point of view. The axioms in a proof system for a logic system are a logical conept. So they are different concepts at two different level.

Tim
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