Principles of the standard predicate

Statement

The following principles follow from the above intuitive motivation and so should be deducible from the formal axioms. For the moment we take the domain of discussion as being the familiar set of whole numbers.

• Any mathematical expression that does not use the new predicate standard explicitly or implicitly is an internal formula.
• Any definition that does so is an external formula.
• Any number uniquely specified by an internal formula is standard (by definition).
• Nonstandard numbers are precisely those that cannot be uniquely specified (due to limitations of time and space) by an internal formula.
• Nonstandard numbers are elusive: each one is too enormous to be manageable in decimal notation or any other representation, explicit or implicit, no matter how ingenious your notation. Whatever you succeed in producing is by definition merely another standard number.
• Nevertheless, there are (many) nonstandard whole numbers in any infinite subset of N.
• Nonstandard numbers are completely ordinary numbers, having decimal representations, prime factorizations, etc. Every classical theorem that applies to the natural numbers applies to the nonstandard natural numbers. We have created, not new numbers, but a new method of discriminating between existing numbers.
• Moreover, any classical theorem that is true for all standard numbers is necessarily true for all natural numbers. Otherwise the formulation "the smallest number that fails to satisfy the theorem" would be an internal formula that uniquely defined a nonstandard number.
• The predicate "nonstandard" is a logically consistent method for distinguishing large numbers—the usual term will be illimited. Reciprocals of these illimited numbers will necessarily be extremely small real numbers—infinitesimals. To avoid confusion with other interpretations of these words, in newer articles on IST those words are replaced with the constructs "i-large" and "i-small".
• There are necessarily only finitely many standard numbers—but caution is required: we cannot gather them together and hold that the result is a well-defined mathematical set. This will not be supported by the formalism (the intuitive justification being that the precise bounds of this set vary with time and history). In particular we will not be able to talk about the largest standard number, or the smallest nonstandard number. It will be valid to talk about some finite set that contains all standard numbers—but this non-classical formulation could only apply to a nonstandard set.