This recent publication is another source that can be looked into on this topic.
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The Logic of Infinity
by Barnaby Sheppard
http://www.amazon.com/The-Logic-Infinit ... 1107678668
No 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
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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.
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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.
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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.
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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.
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1.1.5 Finiteness
I 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.''