Sentence examples for cardinal theory from inspiring English sources

On another front there are logicians that are fond of making the point that some of the further reaches of large cardinal theory, routinely disregarded and even disparaged by many mathematicians nevertheless have implications for a kind of low-level mathematics whose core significance is not in dispute.

For example, the replacement of "cardinal" utility theory by "ordinal" utility theory (see below Section 5.1) in the 1930's, which is generally regarded as a major step forward, involved the replacement of one theory by another that was strictly weaker and which had no additional empirical content.

(For a detailed discussion of cardinal utility theory see Section 3.5 of the entry on interpretations of probability as well as the entry on the St . Petersburg Paradox.

The background motivation is this: First, we know that in the presence of large cardinal axioms the theory of second-order arithmetic and even the entire theory of L is invariant under set forcing.

As the investigations of Kit Fine (1998: 515; 2002) have revealed, any attempt to combine such an abstractive account of transfinite cardinals with set theory must resort to treating the abstracted cardinals as Urelemente rather than as sets.

From studies on infinitary logics, William Hanf, an American logician, was able to define certain cardinals, some of which have been studied in connection with the large cardinals in set theory.

Ultrapowers are an essential tool for handling large cardinals in set theory (see the entry on set theory).

For more on the case of singular cardinals and pcf theory see Abraham & Magidor (2010) and Holz, Steffens & Weitz (1999).

He adhered to Hilbert's "original rationalistic conception" in mathematics (as he called it);[1] and he was prophetic in anticipating and emphasizing the importance of large cardinals in set theory before their importance became clear.

Lex Talionis (section 3.4) offers a theory of cardinal proportionality.

However, what is clear is the absolute centrality of large cardinal axioms in set theory.