Originally posted by chronoaug
thread needs more logic and online genius certification
<----
Also,
In first-order languages, there are some things we can say, and some that we cannot. Suppose, for example, that we want to express facts about the arithmetic of the natural numbers. That is, we want to express facts about the structure (N; 0, S, <, +, ~) consisting of the set N = {0, 1, ...} of natural numbers, together with the common arithmetical operations and relations. And we want to use a first-order language with quantifiers interpreted as gfor every natural numberh and interpreted as gfor some natural number.h Moreover, we include in the language a constant symbol 0 for the number zero, a one-place function symbol S for the successor operation (which applied to a natural number gives the next one), a two place predicate symbol < for the ordering relation < on N, and two-place function symbols + and ~ for addition and multiplication, respectively.
With this language, we can now symbolize many of the facts we know to be true about the natural numbers. We can form the sentence x(x < Sx) expressing the fact that each number is smaller than the next one, for example. But a difficulty arises if we want to express the gwell-ordering propertyh that any non-empty set of natural numbers has a smallest member. If P is a new one-place predicate symbol, then
x Px x(Px & y(Py (y = x v x < y)))
thread needs more logic and online genius certification
<----
Also,
In first-order languages, there are some things we can say, and some that we cannot. Suppose, for example, that we want to express facts about the arithmetic of the natural numbers. That is, we want to express facts about the structure (N; 0, S, <, +, ~) consisting of the set N = {0, 1, ...} of natural numbers, together with the common arithmetical operations and relations. And we want to use a first-order language with quantifiers interpreted as gfor every natural numberh and interpreted as gfor some natural number.h Moreover, we include in the language a constant symbol 0 for the number zero, a one-place function symbol S for the successor operation (which applied to a natural number gives the next one), a two place predicate symbol < for the ordering relation < on N, and two-place function symbols + and ~ for addition and multiplication, respectively.
With this language, we can now symbolize many of the facts we know to be true about the natural numbers. We can form the sentence x(x < Sx) expressing the fact that each number is smaller than the next one, for example. But a difficulty arises if we want to express the gwell-ordering propertyh that any non-empty set of natural numbers has a smallest member. If P is a new one-place predicate symbol, then
x Px x(Px & y(Py (y = x v x < y)))
