ezio said...

http://www.humanities.mcmaster.ca/~rarthur/papers/LeibCant.pdf

As he wrote to Foucher in 1692:

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said; I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.1

1 Letter to Foucher, Journal de Sçavans, March 16, 1693, G I 416.

In one of these, “On the Secrets of the Sublime, or on the Supreme Being”, written 11 February 1676, he offers the following proof that there is no such thing as infinite number:

It seems that all that needs to be proved is that the number of finite numbers cannot be infinite. If numbers can be assumed as continually exceeding each other by one, the number of such finite numbers cannot be infinite, since in that case the number of numbers is equal to the greatest number, which is supposed to be finite. (A VI iii

477)

Cantor distinguished between two kinds of multiplicities:

[O]n the one hand a multiplicity can be such that the assumption that all of its elements ‘are together’ leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as ‘one finished thing’. Such multiplicities I call absolutely infinite or inconsistent multiplicities.

http://en.wikipedia.org/wiki/Actual_infinite

Actual infinity is the notion that all (natural, real etc.) numbers can be enumerated in any sense sufficiently definite for them to form a set together. Hence, in the philosophy of mathematics, the abstraction of actual infinity is the acceptance of infinite entities, such as the set of all natural numbers or an arbitrary sequence of rational numbers, as given objects.

The mathematical meaning of the term actual in actual infinity is synonymous with definite, not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist actually in nature.

To reject the abstraction of actual infinity, as in intuitionism (see also intuitionistic logic), requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively. One mathematician to hold this view was Georg Cantor, who decided that it is possible for natural and real numbers to be definite sets, and that if we reject the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then we are not involved in any contradiction. This rejection was also adopted by philosophers and mathematicians in the 20th century, such as Michael Dummett.

The mathematical problem of actual infinities is whether this rejection is justified or not.

http://www.brainyquote.com/quotes/quotes/g/georgcanto201207.html

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

Georg Cantor

http://www.mlahanas.de/Greeks/Infinite.htm

If we have to travel a distance then we do not travel over a actual infinite set which is impossible but we travel over a potential infinite set in that this set could be divided actually for ever in smaller and smaller pieces. It was actually considered that the for real objects the actual infinite is not possible, that an object could not have something which exceeds all limits. An actual infinity is regarded as a completed totality. A potential infinity is more like a finite but indefinitely long, unending series of events. According to Aristotle, actual infinities cannot exist, but potential infinities exist in nature and are manifested to us in various ways, for instance the indefinite cycle of the seasons or the indefinite divisibility of a piece of matter.

…

What are the problems according to Aristotle for things such as a infinite Universe as Archytas believes with his “Thought Experiment”?

1. A body is defined as that which is bounded by a surface, therefore there cannot be an infinite body.

2. A number, by definition, is countable, so there is no number called ‘infinity’.

3. Perceptible bodies exist somewhere, they have a place, so there cannot be an infinite body.

But we also cannot say that the infinite does not exist according to Aristotle here are some reasons:

1. If no infinite, magnitudes will not be divisible into magnitudes, but magnitudes can be divisible into magnitudes (potentially infinitely), therefore an infinite in some sense exists.

2. If no infinite, number would not be infinite, but number is infinite (potentially), therefore infinity does exist in some sense.

…

It was Cantor who considered the problems of infinite sets. Against common logic that infinite is something unlimited and larger than anything he discovered that there are sets larger than the infinite sets of natural numbers. He also showed that no infinite set could have as many elements as all possible subsets.

In this way he showed that there are sets infinite larger than the sets of the infinite natural numbers. He arranged all the subsets and the elements of the sets in the form of a table and showed using the diagonal of the table that such a mapping is not possible. In this way we have a similarity with the Pythagorean discovery that the diagonal of a square cannot be described as a ratio of natural numbers as there is no such ratio for the square root of the number 2. In this way actually the ancient Greeks discovered that there is no mapping of the natural to the real numbers that are more infinite than the infinite natural numbers 1,2,3,... etc.

…

From what we have said, then, it is clear that the weight of the infinite body cannot be finite. It must then be infinite. We have therefore only to show this to be impossible in order to prove an infinite body impossible. But the impossibility of infinite weight can be shown in the following way. A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights. For instance, if one weight is twice another, it will take half as long over a given movement. Further, a finite weight traverses any finite distance in a finite time. It necessarily follows from this that infinite weight, if there is such a thing, being, on the one hand, as great and more than as great as the finite, will move accordingly, but being, on the other hand, compelled to move in a time inversely proportionate to its greatness, cannot move at all. The time should be less in proportion as the weight is greater. But there is no proportion between the infinite and the finite: proportion can only hold between a less and a greater finite time. And though you may say that the time of the movement can be continually diminished, yet there is no minimum. Nor, if there were, would it help us. For some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite; so that an infinite and a finite weight must have traversed an equal distance in equal time. But that is impossible. Again, whatever the time, so long as it is finite, in which the infinite performs the motion, a finite weight must necessarily move a certain finite distance in that same time. Infinite weight is therefore impossible, and the same reasoning applies also to infinite lightness. Bodies then of infinite weight and of infinite lightness are equally impossible.

Aristotle, De Caelo I Part 6, 273b 27 - 274a 19

http://ii.best.vwh.net/math/ch/

1.1 What is the Continuum Hypothesis?

In 1874 Georg Cantor discovered that there is more than one level of infinity. The lowest level is called "countable infinity" and higher levels are called "uncountable infinities." The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set. In 1877 Cantor hypothesized that the number of real numbers is the next level of infinity above countable infinity. Since the real numbers are used to represent a linear continuum, this hypothesis is called "the Continuum Hypothesis" or CH.

Let c be the cardinality of (i.e., number of points in) a continuum, aleph0 be the cardinality of any countably infinite set, and aleph1 be the next level of infinity above aleph0. Here are six ways to state CH:

Sets-of-Reals Versions of CH

Any set of real numbers is either finite, countably infinite, or has the same cardinality as the entire set of real numbers.

Any infinite set of real numbers is either countably infinite or has the same cardinality as the entire set of real numbers.

Any uncountable set of real numbers has the same cardinality as the entire set of reals numbers.

Cardinal-Number Versions of CH

There is no cardinal number between aleph0 and c.

c = aleph1

2aleph0 = aleph1 (explained in section 3.1.1)

The Continuum Hypothesis has been, and continues to be, one of the most hotly pursued problems in mathematics. It was the first problem in Hilbert's list of 23 important unsolved problems, ten of which he presented to the Second International Congress of Mathematicians at Paris in 1900. Pursuit of the Continuum Hypothesis has motivated a lot of useful and interesting mathematics in real analysis, topology, set theory, and logic.

aleph0 < c = card(R) = card((0,1)) = card(P(N)) = 2aleph0

…

Thom represents the realist view:

Much emphasis has been placed during the past fifty years on the reconstruction of the geometric continuum from the natural integers, using the theory of Dedekind cuts or the completion of the field of rational numbers. Under the influence of axiomatic and bookish traditions, man perceived in discontinuity the first mathematical Being: "God created the integers and the rest is the work of man." This maxim spoken by the algebraist Kronecker reveals more about his past as a banker who grew rich through monetary speculation than about his philosophical insight. There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity. If one has any consciousness at all, it is consciousness of time and space; geometric continuity is in some way inseparably bound to conscious thought.

…

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now aleph1 is the set of countable ordinals and this is merely a special and the simplest way of generating a highter cardinal. The set C is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach C. Thus C is greater than alephn, alephw, alepha, where a= alephw, etc. This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

- Paul Cohen on p. 151 of Set Theory and the Continuum Hypothesis

…

For the advancing army of physics, battling for many a decade with heat and sound, fields and particles, gravitation and spacetime geometry, the cavalry of mathematics, galloping out ahead, provided what it thought to be the rationale for the real number system. Encounter with the quantum has taught us, however, that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the concept of the continuum has been and will continue to be as indispensable for physics as it is for mathematics. In either field of endeavor, in any given enterprise, we can adopt the continuum and give up absolute logical rigor, or adopt rigor and give up the continuum, but we can't pursue both approaches at the same time in the same application.

- John Archibald Wheeler on p. xii of Weyl's The Continuum

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