|Ordinal numbers: ordinals indicate the position of elements within a sequence (expressed by "first", "second", ...). In contrast, cardinal numbers (expressed by "one," two ", ...) indicate the size (cardinality) of sets. See also numbers, sets, order, well-ordering._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
John von Neumann on Ordinal Numbers - Dictionary of Arguments
Thiel I 205
Ordinal numbers/Neumann/Thiel: Today, ordinal numbers are not only introduced differently than in Cantor and Dedekind, but are also defined differently.
John v. Neumann: Axiomatic construction of the set theory. In the foundation of logic certain formulas are recognized as "excellent formulas".
The rules allow us to form unreservedly new sentential connective-logical propositional schemas, in which we can recognize excellent ones and not a. But this does not provide us with a real overview of the sentences of the sentential connectives logic, nor a systematic insight into their connections.
We must distinguish between the logical framework and the sentences themselves in an axiomatic structure.
Axiomatization allows a potentially infinite set of sentences by representing them as a conclusion set from finitely many sentences._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
J. v. Neumann
The Computer and the Brain New Haven 2012
Philosophie und Mathematik Darmstadt 1995