Axiomatic Set Theory
In 1931, Kurt Gödel proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC.
Moreover the following statements are independent of ZFC (shown by Paul Cohen and Kurt Gödel):
- The axiom of constructibility (V = L);
- The generalized continuum hypothesis (GCH);
- The continuum hypothesis (CH);
- The diamond principle (◊);
- Martin's axiom (MA);
- MA + ¬CH.
Note that we have the following chains of implication:
- V = L → ◊
- V = L → GCH → CH.
Assuming that ZFC is consistent, the existence of large cardinal numbers, such as inaccessible cardinals, Mahlo cardinals etc., cannot be proven in ZFC. On the other hand, few working set theorists expect their existence to be disproved.
Read more about this topic: List Of Statements Undecidable In ZFC
Famous quotes containing the words axiomatic, set and/or theory:
“It is ... axiomatic that we should all think of ourselves as being more sensitive than other people because, when we are insensitive in our dealings with others, we cannot be aware of it at the time: conscious insensitivity is a self-contradiction.”
—W.H. (Wystan Hugh)
“Unfortunately, many things have been omitted which should have been recorded in our journal; for though we made it a rule to set down all our experiences therein, yet such a resolution is very hard to keep, for the important experience rarely allows us to remember such obligations, and so indifferent things get recorded, while that is frequently neglected. It is not easy to write in a journal what interests us at any time, because to write it is not what interests us.”
—Henry David Thoreau (1817–1862)
“The whole theory of modern education is radically unsound. Fortunately in England, at any rate, education produces no effect whatsoever. If it did, it would prove a serious danger to the upper classes, and probably lead to acts of violence in Grosvenor Square.”
—Oscar Wilde (1854–1900)