Confusion with definition of $term$ in first order languages.

Question

In the book 'A course in mathematical logic' Manin defines terms as,

$Terms$ are the elements of the least subset of the expressions of the language which satisfies two conditions:

Variables and constants are (atomic)terms.

If $f$ is an operation of degree $r$ and $t_1,t_2,\dots,t_r$ are terms, then $f(t_1,\dots,t_r)$ is also a term.

What is the significance of 'The least subset' in the definition?

Answer 1

The "least subset" condition is typical of inductive definitions: it means that the defined set is the intersection of all sets containing:

(i) the "initial" elements; for $\text{Terms}$: variables and constants;

(ii) is closed with respect to new elements "generators": for $\text{Terms}$: the function symbols $f$.

The condition ensures that the set $\text{Terms}$ does not contain "unwanted" objects.

We can try a simple "mental experiment", considering the set $\text{Even}$ defined by:

(i) $2$ is in $\text{Even}$; and

(ii) if $n$ is in $\text{Even}$, then $n+2$ is in $\text{Even}$.

If we look at the set: $\{ 2,3,4,6,8,9 \ldots \}$ we can check that it satisfies the two conditions, but it is not what we want.

How can we rule out cases like this ?

Of course also $\{ 2,4,6,8 \ldots \}$ will satisfies the above condition; what is unique of the last one is that it is the intersection of all sets satisfying the conditions, i.e. it is the "least set" that satisfies them.

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The general definition applies to all specific examples in Manin's text.

For $L_1Set$, we have as initial elements the variables plus one single constant: $\emptyset$ and no function symbols.

Thus, in this case: $\text{Terms} = \{ \emptyset,x,y,z, \ldots \}$.

For $L_1Ar$, we have two constants: $0$ and $1$ and two (binary) function symbols: $+$ and $\times$.

In this case $\text{Terms}$ is more complex:

$\{ 0,1,+(0,1),\times(0,1), +(1,0), \times(1,0), \ldots, +(0,+(0,1)), +(0,\times(0,1), \ldots, x, +(0,x), \times(0,x), +(1,x), \times(1,x), \ldots,y, \ldots \}$

Answer 2

Make a list of expressions where constants and variables may be added freely and other expressions on the list are obtained from previous expressions on the list by using the generating rule involving the function symbols that you stated above .Any expressions on your list are called terms . Saying that something is a term means you somehow know it can be obtained by making such a list .Such a list is certainly what we would call finite . Manin's definition involves "modeling "the language in set theory ,he uses set theory freely ;studies using set theory and infinite sets (like the positive integers ) is called mathematical logic . If you wish to avoid set theory -for instance you might be carefully setting up a first order language to actually do (at least in principle) set theory and thus all of classical mathematics then the "list "definition is where you have to start from . Regards Stuart M.N.