display all the ideas for this combination of texts
7 ideas
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
A reaction: [He credits Sierpinski (1918) with this discovery] |
10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |