Barnaby Sheppard - Collected Quotes

"For group theoretic reasons the most impressive paradoxical decompositions occur in dimension at least three, but there are also interesting decompositions in lower dimensions. For example, one can partition a disc into finitely many subsets and rigidly rearrange these subsets to form a square of the same area as the original disc. Even without the Axiom of Choice it is possible to construct some counterintuitive subsets of the plane, so Choice cannot be held responsible for all that is counterintuitive in geometry. Some more radical alternatives to the Axiom of Choice obstruct such constructions more effectively." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Given any collection of infinite sets the Axiom of Choice tells us that there exists a set which has one element in common with each of the sets in the collection. Choice, which seems to be an intuitively sound principle, is equivalent to the much less plausible statement that every set has a well-ordering. Although many tried to prove Choice, they only seemed to be able to find equivalent statements which were just as difficult to prove." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Intuition is reliable only in the limited environment in which it has evolved. Unable to abandon its prejudices completely, we must constantly question what appears to be obvious, often revealing conceptual problems and hidden paradoxes. One intuitive notion which is ultimately paradoxical is that of arbitrary collections."  (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Objections to the Axiom of Choice, either the strong or the weak version, are typically either philosophical, based on the intuitive temporal implausibility of making an infinite number of choices, or on the non-constructive nature of the axiom, or are based on a peculiar identification of continuum-based models of physics with the physical objects being modelled; properties of the model which are implied by the Axiom of Choice are deemed to be counterintuitive because the physical objects they model don’t have these properties. Motivated by these objections, or just for curiosity, several alternatives to Choice have been explored." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Since the membership relation is well-founded, well-founded relations can be defined on any class, however, the existence of a well-ordering of every set cannot be proved without appealing to the Axiom of Choice. Indeed, the assumption that every set has a well-ordering is equivalent to the Axiom of Choice." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Some intuitive sets seem to have a less tangible existence than others, but it is difficult to draw a clear boundary between these different levels of concreteness. Naively we might expect to be able to include everything, to associate a set with any given property (precisely the set of all things having that property), but by a clever choice of property this leads to a contradiction. One of the tasks of set theory is to exclude these paradox-generating predicates and to describe the stable construction of new sets from old. Mathematics has a long history of creating concrete models of notions which at their birth were difficult abstract ideas."  (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most obvious variations of the Axiom of Choice are those that restrict the cardinality of the sets in question. Other variations impose relational restrictions between the sets. When the early set theorists tried to prove the Axiom of Choice they invariably ended up showing it is equivalent to some other statement that they were unable to prove. This collection of equivalent statements has grown to an enormous size. One of its striking features is that some of the statements seem intuitively obvious while others are either wildly counterintuitive or evade any kind of evaluation." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most well-known equivalent of the Axiom of Choice is the Well-Ordering Theorem, which states that every set has a well-ordering. Choice is also equivalent to the statement that every infinite set is equipollent to its cartesian square (we have already seen some concrete examples of this equipollence, without having to appeal to Choice, in the case of Z and R). One of the reasons that the Axiom of Choice is so widely adopted is that it is so useful, and the contortions one must make to prove a statement without it, if this is possible, are often painful." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The so-called ‘imaginary numbers’ were successfully used long before they were properly defined as elements of a concrete field extension of the real numbers. The axiomatic description of a theory generally appears only in its mature stages, after many of its properties have been informally explored. Perhaps the longest duration between the usage and formalization of a notion is that of the natural numbers." (Barnaby Sheppard, "The Logic of Infinity", 2014)