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An Introduction To Non-aristotelian Systems And General Semantics.

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there is nothing flexible about a definite number once it is selected. What has been already said about a variable applies, also, to a number; namely, that a 'variable' does not 'vary' in the ordinary sense; but this term applies only to the semantic processes of the mathematician. The older intensional A definition of 'number' must have led to the older identifications. The A, extensional, and non-el semantic definition of numbers- does not allow such identifications. The A term 'number' applied to a definite number, but also to an intensional definition of numbers. The A, or semantic definition of numbers, is different in the sense that it finds extensional characteristics of each number, applicable to all numbers, and so helps not to identify a definite number with the process of generating numbers, which the use of one term for two entirely different entities must involve.
Cantorian alephs, then, are the result of identification or confusion of entirely different issues and must be completely eliminated. The rejection of alephs will require a fundamental revision of those branches of mathematics and physics which utilize them; yet, as far as I know, with a very few exceptions, the alephs are not utilized or needed, although the 'name' is used, which spell-mark has become fashionable in many mathematical and physical circles. In the case of alephs, history may repeat itself and the alephs, like the 'infinitesimal', when their self-contradictory character becomes understood, will be eliminated without affecting the great body of mathematics, but only the small portions which are built on the alephs.
As to the existence of infinite processes, we know positively only about the semantic process of generating numbers and the semantic process of infinite divisibility. These processes are evident in our common experience. We cannot a priori know if such infinite processes can be found in the world which must be discovered by investigation and experimentation.
The existing terminology is still A and is based on, and leads to, identification, and so in my A presentation I cannot use it and expect to clear up some of the issues involved. The terms such as 'class', 'aggregate', 'set'., imply a definite static collection. The term 'infinite', in the meantime, can only be correctly and significantly used as applied to a dynamic semantic process. We cannot speak of 'infinite' classes, aggregates, sets., and evade the issues of identification of entirely different entities. The term 'series' has a technical meaning in connection with numbers and so, for a general discussion of processes, is a little too specific. The term 'array' is more general, yet extensional, of which 'series' would be a special case. The general term 'number*