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A one to one function establishes a one to one correspondence between two sets. Other expressions for one to one correspondence are mapping relation transformation, and bijection.
If there is a one to one correspondence between two sets---in other words, if the one set can be transformed into the other---then any given statement about the interrelations of the members of the one set can be interpreted as a statement about the interrelations of the members of the other set.
If there is a one-one correspondence between two sets, then both sets are instances of the same
ordinal number.
81. To count the members of a given class K is to biject the members of K with those of an initial segment of the series of positive numbers. Thus, if K is the smallest class containing the letters A, B, and C, to count the members of K is to biject them with the smallest class containing 1, 2, and 3.
82. A recursive function is one that is defined for each of its own outputs, for example, f(x)=x+1.
83. A recursive definition is one that specifies the membership of a class in terms of a recursive function. More precisely, a recursive definition of a class k is one to the effect that, for some object αand some recursionφ,kisthesmallestclasscontainingαandcontaining φ(x)wheneveritcontainsx.
k is recursively defined ≡DF∃α ∃φ k={x: x=α⋁∃y(φ(y)=x)}
84. A recursive function generates progression. A progression is an infinitely long discrete series that has a first member. A discrete series is one each of whose non-final members has an immediate successor. An example of a progression is the series 1, 2, 3, 4, ......
85. A logic is a recursively defined statement class.
86. An analytic truth is one that holds unconditionally.
87. The expression ‘truth of logic’ is ambiguous. Sometimes, a ‘truth of logic’ is one that belongs to some particular logic, and sometimes a ‘truth of logic’ is a truth describing a condition that any given system of logic must be satisfy.