Concept
Transfinite number
Overview
number larger than all finite numbers
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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What constitutes the importance of this memoir is that it has given the first example of those antinomies met in the study of transfinite numbers and making since some years the despair of mathematicians. The aim, says Burali-Forti, of this note is to show there may be two transfinite numbers (ordinals), _a_ and _b_, such that _a_ is neither equal to, greater than, nor less than _b_. To reassure the reader, to…
Henri Poincaré, The Foundations of Science
Names Transfinite number several times.
Now, instead of the relation of father and son, consider the relation of a finite number to its immediate successor, that is, the relation which holds between 0 and 1, between 1 and 2, between 2 and 3, and so on. If a property of numbers is hereditary with respect to this relation, then if it belongs to (say) 100, it must belong also to all finite numbers greater than 100; for, being hereditary, it belongs to 101…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Transfinite number is.
It is a theorem readily proved by means of a syllogism of transposed quantity, but not otherwise. Of finite collections it is true, of infinite collections false. Thus, a part of the whole numbers are even numbers. Yet the even numbers are no fewer than all the numbers; an evident proposition since if every number in the whole series of whole numbers be doubled, the result will be the series of even numbers.
Charles S. Peirce, Chance, Love, and Logic
Close in meaning to Transfinite number; the term itself is not named.
Philosophers who discussed Transfinite number
Related ideas
- Finite set
- Georg Cantor
- Zermelo–Fraenkel set theory
- Mathematics
- Infinity
- Real number
- Natural number
- Bulletin of the American Mathematical Society
- Continuum hypothesis
- Cardinal number
- Cardinality
- Axiom of choice
Place in the tradition
Part of Zeno of Elea.