ON Till': NOTION OV INFINITY |
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is multiordinal and intensional and so, in the
A extensional system, rj-valued, and must be used in the plural; namely, 'number/. The term 'number' in the singular will be used to indicate a definite number. The term 'denumerable' has been introduced by Cantor and means any extensional array of terms, facts, states, observables., which can be put in one-to-one correspondence with the infinite array of positive integers.Let me repeat once more: the semantic process may be carried on without limits, and the infinite series of positive integers is an extensional, technical, and verbal expression of this semantic process and the only infinite array of which existence we are certain.
We shall be able to explain, and to give a better definition of, mathematical infinity if we introduce an extremely useful structural term, 'equivalence'. Two processes, arrays., between which it is possible to set up, by some law of transformation, a
one-to-one correspondence are said to be equivalent. A process, array., which is equivalent to a part of itself, is said to be infinite. In other words, a process, array., which can be put into a one-to-one correspondence with a part of itself is said to be infinite. We can define a finite process, array., (class, aggregate.,) as one which is not infinite. The following is valid exclusively because of the use of the 'etc'A few examples will make this definition clearer. If we take the series of positive integers, 1, 2, 3, 4, ...
etc., we can always double every number of this row provided we retain the process-character, but not otherwise. Let us write the corresponding row of their doubles under the row of positive integers, thus:1, 2, 3, 4, S, . . .
.etc.2, 4, 6, 8, 10, . . .
e t c. or we can treble them, or «-ble them, thus:1, 2, 3, 4, 5, ... .
etc.3, 6,9, 12, 15, . .
etc.there are obviously as many numbers in each row below as in the row above,
provided we retain the 'etc.', so the numbers of numbers in the two rows compared must be equal. All numbers which appear in each bottom row also occur in the corresponding upper row, although they only represent a part of the top row, again provided that we retain the 'etc'The above examples show another characteristic of infinite processes, arrays,. In the first example, we have a
one-to-one correspondence between the natural numbers and the even numbers, which are equal in number at each stage. Yet, the second row results from the first row by14 |
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