This recent publication is another source that can be looked into on this topic.

--------

The Logic of Infinity by Barnaby Sheppard

http://www.amazon.com/The-Logic-Infinit ... 1107678668No other question has ever moved so profoundly the spirit of man;

no other idea has so fruitfully stimulated his intellect; yet no other

concept stands in greater need of clarification than that of the infinite.

– David Hilbert

--------

There is great power in the set theoretic description of the mathematical

universe. As we shall see in detail, one of Cantor’s questions was, naively: ‘how

many points are there in a line?’. This had little more than vague philosophical

meaning before the arrival of set theory, but with set theory the question can be

made precise. The punch-line in this case is that the standard theory of sets – a

theory which is powerful enough to sensibly formulate the question in the first

place – is not strong enough to give a definite answer, it only gives an infinite

number of possible answers, any one of which can be adopted without causing

contradiction. The solution remains undecidable; the theory does not know the

size of the beast it has created! This flexibility in the solution is arguably far

more intriguing than any definite answer would have been – it transfers our focus

from a dull fixed single universe of truths to a rich multiverse, one consistent

universe for each alternative answer.

--------

Although the ideas discussed in this book are now well-established in mainstream

mathematics, in each generation since Cantor there have always been a

few individuals who have argued, for various philosophical reasons, that mathematics

should confine itself to finite sets. These we shall call Finitists. Despite

my occasional sympathies with some of their arguments, I cannot claim to be

a Finitist. I’m certainly a ‘

Physical Finitist’ to some extent, by which I mean

that I am doubtful of the existence of infinite physical aggregates, however I

do not extend this material limitation to the Platonic constructions of mathematics

(the set of natural numbers, the set of real numbers, the class of all

groups, etc.), which seem perfectly sound as ideas and which are well-captured

in the formalized language of set theory. However, I cannot pretend to be a

strong Platonist either (although my feelings often waver), so my objection is

not based on a quasi-mystical belief that there is a mathematical infinity ‘out

there’ somewhere. In practice I find it most convenient to gravitate towards

a type of Formalism, and insofar as this paints a picture of mathematics as a

kind of elaborate game played with a finite collection of symbols, this makes

me a Finitist in a certain Hilbertian sense – but I have no difficulty in imagining

all manner of infinite sets, and it is from such imaginings that a lot of

interesting and very useful mathematics springs, not from staring at or manipulating

strings of logical symbols. I am not alone in being so fickle as to

my philosophical leanings; ask a logician and a physicist to comment on the

quote attributed to Bertrand Russell ‘any universe of objects...’ which opens

Section 11.1 of Chapter 11 and witness the slow dynamics of shifting opinions

and perspectives as the argument unfolds.

--------

Formal mereological systems make use of the relation of parthood and the axioms

generally reflect the properties one would expect if a part of a whole is

interpreted as a subset. An atom is an object with no proper parts. A natural

antisymmetry condition on parthood ensures that finite mereological systems

will always have atoms, but an infinite system need not, unless a special axiom

is introduced to force it to have one. A classical example of an atomless fusion

is given by the line, regarded as a fusion of line segments: any line segment can

be divided into subsegments indefinitely. The notion of a line as a set of points

is a relatively modern one which emerged hand in hand with the development

of set theory. Classically one could always specify a point on a line, but since

there always exists a new point between any given pair of points it was perhaps

thought absurd that one could saturate the entire line by points. We will soon

see how such a saturation can be realized.

------

In trying to define finiteness we seek a set theoretic property that distinguishes

between intuitively finite sets and intuitively infinite sets. One such property is

the existence of a proper self-injection. A set which does not have this property

is Dedekind finite. A further assumption, the Axiom of Choice, is required to

ensure that this definition coincides with various other proposed definitions of

finiteness. The existence of a proper self-injection is our first glimpse of the

peculiar behaviour of infinite sets. At the other extreme of the cardinal spectrum

is a similarly misunderstood set, the empty set.

------

1.1.5 FinitenessI have freely used the terms finite and infinite expecting the reader to have an

intuitive grasp of these notions. Clearly we wish infinite to mean ‘not finite’,

but then how are we to define ‘finite’? A natural approach, and one that is

encoded within the theory of sets we are about to describe, is to say that a set

A is finite either if it has no elements at all or if there exists a natural number

n such that

A is equipollent to {0, 1, 2, . . . , n}. It will be convenient to call this

property ‘ordinary finiteness’. This is a fair definition, but to be comfortable

with it we need to know precisely what we mean by ‘natural number’, and that

is to come (we assume an intuitive familiarity with the set of natural numbers

below, for the purposes of an example).

There are alternative definitions of finiteness which make no mention of

natural numbers. The equivalence of these ‘number free’ definitions follows if

we are permitted to assume the truth of a certain far-reaching principle called

the Axiom of Choice. We will have much to say about Choice later; briefly

it asserts that for any collection C of sets there exists a set with exactly one

element in common with each set in C.

If a set

A is ordinary finite then there does not exist a bijection from A onto

a proper subset of

A – a property known as Dedekind finiteness. The Axiom

of Choice is needed to prove that every Dedekind finite set is ordinary finite.

Contraposing the Choice-free implication we see that a set

A is ordinary infinite

if there exists a bijection from

A onto a proper subset of itself. This property

of proper self-injection may seem absurd to someone who has not encountered

the phenomenon before, but any such surprise is only symptomatic of the fact

that

we live in a finite world subject to the relatively sober properties of finite

sets, infinite collections being inevitably unfamiliar.

On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''