Hyperintegers and Hyperreal Numbers
The ideas of Model Theory, especially the Compactness Theorem, serve as one venue for the definition of the hyperintegers and the hyperreal numbers. We would like to apply the Compactness Theorem to the arithmetic as based on Peano Axioms. However, the theorem was specifically derived for the first order language whereas the fifth of the Peano axioms - the Axiom of Induction, as formulated elsewhere at this site, admits at least two interpretations. To remind:
Let M be a (sub)set of natural numbers with the following properties:
- 1 ∈ M
- x ∈ M implies x' ∈ M
Then M = N. In other words, M contains all natural numbers.
One interpretation sees the above as an informal analogue of
|(52)||∀M⊂N ((1 ∈ M ∧ ∀x (x ∈ M ⇒ x' ∈ M)) ⇒ M = N).|
The presence of the universal quantifier ∀ binding a set variable make this form of the axiom second-order. Let's call the number theory based in the axioms 1-4, 52, the second order Peano arithmetic, PA2 for short.
Another interpretation sees in the Axiom of Induction an axiom schema wherein M is a sort of "generic element" of the theory while the axiom itself is thought of representing a long list of statements
|(5')||(1 ∈ M ∧ ∀x (x ∈ M ⇒ x' ∈ M)) ⇒ M = N.|
one for each such element. But what are these? The sets in the second interpretation are expected to be truth sets of first-order sentences, replacing (5') with
|(51)||(φ(1) ∧ ∀x (φ(x) ⇒ φ(x') ⇒ ∀x φ(x),|
where φ is a first-order sentence in the language of number theory with a single free variable. Thus to the first 4 Peano axioms (51) adds an infinitude of axioms, one for every first-order sentence of interest. Let's call the resulting theory PA1. PA1 is weaker than PA2 but it's PA1 which leads to the existence of infinite and infinitesimal numbers. This is done almost by magic.
The language of Number Theory expands the basic language of logic to accommodate the Peano Axioms. First we add the symbols of addition and the successor operator, and then with the symbols of subtraction, multiplication and other operations and new constants,
Now we expand the alphabet of the number theory with a new constant, say κ with the idea that this will stand for an infinite number, i.e. that the number greater than any natural number. We also add an infinitude of axioms:
|(κ1)||1 < κ|
|(κ2)||2 < κ|
|(κ3)||3 < κ|
Let PA1∞ denotes the theory built on top of PA1 and all axioms κ. Taking only the first k axioms we obtain subtheories PA1k. Observe that any model of PA11 is a model of PA1. The converse is also true as in any model of PA1 we can take κ to be any number greater than 1. Similarly any model of PA12 is a model of PA11 and vice versa. The same holds for other PA1k. The bottom line is that each PA1k, where k is a natural number, is as consistent as PA1 itself. The Compactness Theorem lets us conclude that the same holds of PA1∞!
Any finite subtheory of PA1∞ contains a finite number of the κ axioms and a finite number from the axioms of PA1. The latter are consistent if so is PA1, and the addition of finite number of the κ axioms does not change this fact. Put differently, any finite subtheory of PA1∞ is contained in one of PA1k and is consistent if the latter is which, as we have seen, is as consistent as PA1.
It follows that if PA1 has a model then so does PA1k, for any natural k, and then, by the Compactness Theorem, so does PA1∞.
By the κ axioms, in any model of PA1∞, κ is greater than any natural number and thus is infinite. But not only that. Like other elements of PA1∞ it is subject to the axioms of PA1. For example, it has a successor
So there is infinitude of infinite integers, i.e., not finite elements of PA1∞:
... κ - 2, κ - 1, κ, κ + 1, κ + 2, ..., 2κ, 3κ, ..., κ² - κ, ..., κ² - 1, κ², ...
which shows that our choice of κ was pretty much arbitrary. It's more common to use the symbol ω instead. However, there is certainly a danger of confusing it with the first infinite ordinal number. The latter is of course defined uniquely.
Notations PA1 and PA2 are also not standard. These have only been used provisionally to emphasize the existence of the first and second order theories.
There is a first order framework for the theory of real numbers (that naturally includes PA1). Similarly to the above it can be proved that such a theory is also as consistent as PA1, meaning that it also has a model. In a model of real numbers, any number, except 0, has a reciprocal. And, since κ belongs to that model, it also has a reciprocal κ-1 such that
κ × κ-1 = κ-1 × κ = 1, and also (κ-1)-1 = κ,
which shows that κ-1 can't be either a standard finite number (0, in particular) or a non-standard infinite. What is it then? Well, by inverting
0 < κ-1 < 1/n, or even
0 < κ-1 < a, for any real a > 0.
The numbers like that are called infinitesimals, the kind of a beast that was ridiculed by the Bishop Berkeley but whose existence was foreseen by the fathers of Calculus.
What is remarkable in the above is that the addition of κ and other infinities to the standard natural numbers does not violate the (Peano) axioms of the natural numbers. Perversely perhaps, they were obtained in such a manner as to build the compliance with the laws of arithmetic in their nature. Similarly, the infinite and the infinitesimal - purely by the construction - comply with the arithmetic of real numbers. For this reason, the models we found are known as non-standard. The elements of the non-standard models of integers are called hyperintegers and the elements of the non-standard model of the reals are known as hyperreals.
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- J. M. Henle, E. M. Kleinberg, Infinitesimal Calculus, Dover, 2003
- D. Marker, Logic and Model Theory, in T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 635-646
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