I gave Safe use of infinitesimals. The idea being expressed was that the phrases “for any” and “there exists” can only be used in phrases like “for any real number x” and “there exists a real number ysuch that. . . ” The transfer principle does not apply to statements like “there exists an integer x such that. . . ” or even “there exists a subset of the real numbers such that. . . ”
The way to state the transfer principle more rigorously is to get rid of the ambiguities of the English language by restricting ourselves to a well- defined language of mathematical symbols. This language has symbols ∀ and ∃, meaning” for all” and” there exists,” and these are called quantifiers. A quantifier is always immediately followed by a variable, and then by a statement involving that variable. For example, suppose we want to say that a number greater than 1 exists. We can write the statement ∃xx>1, read as “there exists a number xsuch that xis greater than 1.” We don’t actually need to say “there exists a number x in the set of real numbers such that . . . ,” because our intention here is to make statements that can be translated back and forth between the reals and the hyper reals. In fact, we forbid this type of explicit reference to the domain to which the quantifiers apply. This restriction is described technically by saying that we’re only allowing first-order logic.
Quantifiers can be nested. For example, I can state the commutativity of addition as ∀x∀yx+ y= y+ x, and the existence of additive inverses as ∀x∃yx+ y= 0.
After the quantifier and the variable, we have some mathematical assertion, in which we’re allowed to use the symbols =, >, ×and + for the basic operations of arithmetic, and also parentheses and the logical operators ¬, ∧and ∨for “not,” “and,” and “or.” Although we will often find it convenient to use other symbols, such as 0, 1, −, /, ≤, =, etc., these are not strictly necessary. We use them only as a way of making the formulas more readable, with the understanding that they could be translated into the more basic symbols. For instance, I can restate ∃xx>1 as ∃x∃y∀zyz= z∧x>y. The number yends up just being a name for 1, because it’s the only number that will always satisfy yz= z.
Finally, these statements need to satisfy certain syntactic rules. For example, we can’t have a string of symbols like x+ ×y, because the operators + and ×are supposed to have numbers on both sides.
A finite string of symbols satisfying all the above rules is called a well- formed formula (wff ) in first-order logic.
The transfer principle states that a wff is true on the real numbers if and only if it is true on the hyperreal numbers.
If you look in an elementary algebra textbook at the statement of all the elementary axioms of the real number system, such as commutativity of multiplication, associativity of addition, and so on, you’ll see that they can all be expressed in terms of first-order logic, and therefore you can use them when manipulating hyperreal numbers. However, it’s not possible to fully characterize the real number system without giving at least some further axioms that cannot be expressed in first order. There is more than one way to set up these additional axioms, but for example one common axiom to use is the Archimedean principle, which states that there is no number that is greater than 1, greater than 1 + 1, greater than 1 + 1 + 1, and so on. If we try to express this as a well-formed formula in first order logic, one attempt would be ¬∃xx>1 ∧ x >1 + 1 ∧ x >1 + 1 + 1 ..., where the ...indicates that the string of symbols would have to go on forever. This doesn’t work because a well-formed formula has to be a finite string of symbols. Another attempt would be ∃x∀n∈N x >n, where N means the set of integers. This one also fails to be a wff in first-order logic, because in first-order logic we’re not allowed to explicitly refer to the domain of a quantifier. We conclude that the transfer principle does not necessarily apply to the Archimedean principle, and in fact the Archimedean principle is not true on the hyperreals, because they include numbers that are infinite.
Now that we have a thorough and rigorous understanding of what the transfer principle says, the next obvious question is why we should believe that it’s true. This is discussed in the following section.
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