Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

My question is the title. I would be glad if someone could supply a proof if true, or a counterexample if false.

share|improve this question

2 Answers 2

up vote 6 down vote accepted

We assume that we are working in the predicate calculus with equality. Let $L$ be any language, and $M$ any finite $L$-structure, say with $n$ elements. Let $\Sigma$ be the set of sentences that says that there exist exactly $n$ elements, and that describes the full diagram of $M$. Then any model of $\{\sigma\}$ is isomorphic to $M$. If the language is finite, instead of $\Sigma$ we can use a single sentence.

Remark: We consider the special case of groups, with language that has a single binary function symbol $\times$. Suppose that $M$ is a group, with elements $a_1,\dots,a_n$. In addition to the sentence that says there are exactly $n$ elements, we need as axioms that there exist $x_1,\dots,x_n$, all distinct, such that $x_i\times x_j=x_k$, for all triples $i,j,k$ such that $m_im_j=m_k$. The full diagram is unnecessary, the multiplication table is enough.

share|improve this answer
    
Did you assume that the language $L$ has only a finite number of non-logical symbols? The OP's question still has an affirmative answer for languages with infinitely many relations, but it no longer suffices to consider a single sentence. –  bof Jul 13 at 5:40
    
Thank you, I was half general language half group theory. Replaced $\sigma$ with $\Sigma$. –  André Nicolas Jul 13 at 5:45
    
Concerning your Remark: What are $x_1,...,x_n$ (e.g. are they "words" in the alphabet $\{a_1,...,a_n\}$)? And what are $m_i$, $m_j$, and $m_k$? Also, do the $x_i$ need to be distinct? For example, could we specify a group with two elements $e$ and $a$ by the rules $a\times a=e$, $e\times a=a$, etc.? –  John Bentin Jul 13 at 6:19
    
The $x_i$ are what are usually called variable symbols, every first-order language has a countable infinity of them. Yes, they are distinct, we need $n$ of them to make the sentence that says there are exactly $n$ distinct objects. –  André Nicolas Jul 13 at 6:25

Let us spell out an example of Andre's answer (for concreteness).

Suppose a group $G$ satisfies the same first-order sentences as $C_2$, the cyclic group of order $2$.

  1. Then $G$ satisfies the following sentence, asserting that there are at most $2$ elements in the group. $$\forall xyz(x=y \vee y=z \vee x=z)$$

  2. And $G$ satisfies the following sentence, which encodes the diagram of $C_2$ (also known as its Cayley table) as a first-order formula.

$$\exists xy(x \neq y \wedge xx=x \wedge xy=y \wedge yx=y \wedge yy=x)$$

But this means that $G$ is isomorphic to $C_2$.

share|improve this answer
    
But this is kind of trivial in the first place, since there is only one group of order 2. But is does show that if G is any mathematical object satisfying the same F0 sentences as $C_2$ in the language of group theory, then $G\cong C_2$. –  Kyle Gannon Jul 13 at 8:48
    
@KyleGannon, sure. All I wanted was to "unpack" Andre's answer for the OP; concrete examples are good. I could have done it for $C_4$ instead, but that would make the whole thing exponentially more painful to write out. –  goblin Jul 13 at 8:52
    
Understandable, I'm never against concrete examples. –  Kyle Gannon Jul 13 at 8:55

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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