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:

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

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

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, 2 = 1', 3 = 2', and so on. PA1 is built with the language of number theory and Peano's axioms, as explained.

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:

Let PA1_∞ denotes the theory built on top of PA1 and all axioms κ. Taking only the first k axioms we obtain subtheories PA1_k. 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 PA1_k. The bottom line is that each PA1_k, 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 PA1_k 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 PA1_k, 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 κ + 1 and a predecessor κ - 1. The latter is also infinite (and so is greater than any natural number) because otherwise, from the properties of addition in PA1 (and hence in PA1_∞) (κ - 1) + 1 = κ would be finite.

So there is infinitude of infinite integers, i.e., not finite elements of PA1_∞:

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 n < κ, which is true for any natural n, we get n^-1 > κ^-1, which says that the reciprocal κ^-1 is less than any fraction 1/n. Since it's not zero, it satisfies

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.

1. P. J. Davis & R. Hersh, The Mathematical Experience, Houghton Mifflin Co, 1981, 237-254
2. J. M. Henle, E. M. Kleinberg, Infinitesimal Calculus, Dover, 2003
3. D. Marker, Logic and Model Theory, in T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 635-646
4. A. Robinson, Non-standard Analysis, Princeton University Press (Rev Sub edition), 1996
5. J. E. Rubin, The Compactness Theorem in Mathematical Logic, Mathematics Magazine, Vol. 46, No. 5 (Nov., 1973), pp. 261-265
6. I. Stewart, Nonstandard Analysis, in R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996, pp. 518-524
7. I. Stewart, Non-Standard Analysis, in From Here to Infinity: A Guide to Today's Mathematics, Oxford University Press, 1996, pp. 80-81.

• Infinitesimals. Non-standard Analysis
• Formal Languages
• Theories and Proofs
  • Formal Systems - an Example
• Models and Metamathematics
  • Godel's Incompleteness Theorem
• Hyperintegers and Hyperreal Numbers
• Structure of Hyperreal Numbers
• The Transfer Principle
• Common Concepts - A Non-standard View
• Is .999... = 1? A Non-standard View

Back to What Is Infinity?

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