Why is axiom of choice controversial?

The axiom of choice was controversial because it proved things that were obviously false, in most people’s intuition, namely the well-ordering theorem and the existence of non-measurable sets.

What is axiom of choice in set theory?

axiom of choice, sometimes called Zermelo’s axiom of choice, statement in the language of set theory that makes it possible to form sets by choosing an element simultaneously from each member of an infinite collection of sets even when no algorithm exists for the selection.

Is axiom of choice independent of ZFC?

The axiom of choice is not the only significant statement which is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC.

What is the axiom of equality?

“The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for …

Why is the axiom of choice unique among the axioms?

The axiom of choice states that arbitrary products of nonempty sets are nonempty. Clearly, we only need the axiom of choice to show the non-emptiness of the product if there are infinitely many choice functions.

Can every set be ordered?

In mathematics, the well-ordering theorem, also known as Zermelo’s theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.

Is the axiom of choice accepted?

It states that for any collection of sets, one can construct a new set containing an element from each set in the original collection. In other words, one can choose an element from each set in the collection. Nevertheless, most mathematicians generally accept the axiom of choice as true and use it when necessary.

What does it mean to be independent of ZFC?

A statement is independent of ZFC (sometimes phrased “undecidable in ZFC”) if it can neither be proven nor disproven from the axioms of ZFC.