Morris Kline, *Mathematics: The Loss of Certainty*, Oxford 1980, p. 200:

. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.

Here may be one answer to the question that got me going on this series of posts. The question was whether one could prove the existence of actually infinite sets. Note, however, that Kline's talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I have already explained more than once. Pleonasm, however, is but a peccadillo. But let me explain it once more. A potentially infinite set would be a set whose membership is finite but subject to increase. But by the Axiom of Extensionality, a set is determined by its membership: two sets are the same iff their members are the same. It follows that a set cannot gain or lose members. Since no set can increase its membership, while a potentially infinite totality can, it follows that that there are no potentially infinite sets. Kline therefore blunders when he writes,

However, most mathematicians -- Galileo, Leibniz, Cauchy, Gauss, and others -- were clear about the the distinction between a potentially infinite set and an actually infinite one and rejected consideration of the latter. (p. 220)

Kline is being sloppy in his use of 'set.' Now to the main point. Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.4142136. . . . Despite the nonterminating decimal expansion, the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.

If someone can put the argument rigorously, have at it.

" Suppose you have a right triangle. If two of the sides are one unit in length..."

I smell a begged question. I can't quite pin it down yet but "two of the sides are one *unit* in length" seems suspicious.

Posted by: Alfred Centauri | Sunday, August 15, 2010 at 08:42 PM

I've been on holiday (Turkey again - where the local authories are now painting orange lines in the middle of the road to encourage driving on the side - seems to have worked).

I see there have been some fascinating posts on the subject of 'actual infinity'. Sorry to have missed these, hope to catch up later.

Will

Posted by: William | Monday, August 16, 2010 at 01:50 PM

>>If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.

This is, as you say, not an argument.

Suppose we represent each number in the decimal expansion 1.414214... as a line segment. I.e. the first segment is one unit, the second, 0.4 units, the third 0.01 units and so on. Clearly no finite expansion will produce a line long enough to complete the hypotenuse. It will always fall a little short. So, 'every segment' must be there. But why does this justify an 'infinite set'? We agree that there are infinitely many segments, and that each is a part of the hypotenuse. But why does that imply the existence of a 'set'?

Posted by: William | Monday, August 16, 2010 at 01:58 PM

Alfred,

Huh?

William,

I was wondering where you were. Turkish drivers are the worst I have ever seen anywhere. Orange lines won't help them; nothing will. I have seen Turks drive on sidewalks. And I once witnessed a Turk backing up in the face of oncoming traffic on a busy thoroughfare because he missed a right-hand turn! Just unbelievable.

Posted by: Bill Vallicella | Monday, August 16, 2010 at 04:45 PM

William,

There have been two questions we have been discussing. One is whether there are sets at all (and what arguments one could give for them). The other more recent question is whether, given that there are finite sets, there are also infinite sets. The above argument which is from Cantor is supposed to show that there are infinite sets, not that there are sets. You are reverting to the first question.

Supposing you were to grant me that there are sets, would you not have to also grant that there are infinite sets given your admission that there are infinitely many line segments in the hypotenuse?

Posted by: Bill Vallicella | Monday, August 16, 2010 at 04:59 PM

William wrote: "the first segment is one unit, the second..."

Again with the "unit". What is this alleged unit and how is it established?

Isn't it the case that this unit length side Bill mentions *must* also be the hypotenuse of some isosceles right triangle the sides of which another might reasonably claim to be *unit* length?

Then your argument, on the basis of your unit, would be "but the length of the sides is perfectly definite, perfectly determinable and yet be a non-terminating decimal...".

But the argument from the perspective of the other *unit* would be "but the length of the hypotenuse is perfectly definite, perfectly determinable and yet be a non-terminating decimal..."

This all seems quite circular to me.

And what exactly does perfectly definite and perfectly determinate mean in this context?

If you hand me the number 1 then I will agree with you that it is definitely a number one, that it is whole, and that it is rational.

If you hand me a unit length, what am I to make of it?

Posted by: Alfred Centauri | Monday, August 16, 2010 at 08:08 PM

>>The above argument which is from Cantor is supposed to show that there are infinite sets, not that there are sets. You are reverting to the first question.

Surely a proof of the existence of infinite sets is a proof of the existence of sets? The argument from 'completion' (irrational numbers and so on) is traditionally the starting point for the existence of the 'actual infinite', and seems to me the strongest argument for the existence of sets.

>>Supposing you were to grant me that there are sets, would you not have to also grant that there are infinite sets given your admission that there are infinitely many line segments in the hypotenuse?

Yes. However as I said, it is rather the other way round. The main reason for positing sets is that any decimal approximation is incomplete. Cantor writes to Hilbert "I have translated the word "set" [Menge] (when it is finite or transfinite) into French with "ensemble" and into Italian with "insieme". And so too, in the first article of the work, "Contributions to the founding of transfinite set theory" I define a "set" (meaning thereby only the finite or transfinite) at the very beginning as an "assembling together" [Zusammenfassung]. But an "assembling together" is only possible if an existing together [Zusammenseins] is possible."

It seems we can only explain the existence of a length corresponding to an infinite expansion by this concept of 'togetherness'. And if 'togetherness' is the principle of a set, the 'completion argument' is an argument for the existence of an infinite set.

1. A set is an assemblage, a Zusammenfassung.

2. The length of the hypotenuse cannot exist without an assemblage of its infinitely many segments

3. The length of the hypotenuse exists.

4. Therefore an infinite assemblage exists

5. Therefore an infinite set exists.

Does that clarify?

Posted by: William | Tuesday, August 17, 2010 at 12:58 AM

>>Turkish drivers are the worst I have ever seen anywhere.

As I pointed out to you last year, you have not visited Egypt! This time we made the acquaintance of the Turkish bus. English buses remain standing while the driver collects and counts the money, works out the number and destinations of the fares, works out the change, hands to passenger and only then drives off. Turkish driver immediately drives off and performs said operations while negotiating difficult road conditions, and also conducting heated conversation with another passenger at the back.

Posted by: William | Tuesday, August 17, 2010 at 12:59 AM

" If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite?"

There are a few difficulties with this question. First, the idea that a line or a line segment is composed of points presupposes that there are infinitely many such points. Second, the idea that the length of any given line segment is determinate presupposes that there is a function, called a metric, on the set of line segments taking values in the set (again presupposed) of real numbers. This function is a surjection, which means that there are at least as many line segments as there are real numbers. The set of real numbers is of course infinte by construction, which depends upon the assumption of infinitely many natural numbers. Third, it is not clear to me how we might formulate the idea of a determinate length without such assumptions. Consider: we are given two line segments and we wish to compare their lengths. We do not know the length of either line segment, and we do not know anything about the limits of rational sequences, either. One segment happens to be exactly "square root of 2" units long, while the other happens to be 1.1414214 units long. How can we decide that the two segments are not congruent? Surely not by measuring them with an instrument.

We would of course like to be able to say that there are two distinct numbers corresponding to two distinct lengths, but in order to do so, we must make use of a theory of numbers which assumes the existence of infinte sets.

Posted by: Phoenix Kyle MacGregor | Tuesday, August 17, 2010 at 07:12 AM

As I was pondering all this in the shower this morning, it occurred to me that length is not a number and thus cannot be characterized as rational or irrational.

That length exists is self-evident but to associate a number with the length of a line segment requires another line segment the length of which is an arbitrary standard. The number we associate with a length is than actually, for lack of a better word, a ratio.

Imagine taking a line segment and "flipping" it end over end, marking out some whole number multiple of its length.

Imagine doing the same with the standard line segment. It might be that after P flips of our line segment and Q flips of our standard line segment, the marked off lengths are equal.

We can then associate the ratio P/Q with the length of our line segment with our standard line segment as a basis.

It is reasonable to ask if there might be lengths for which there are no P and Q for which the marked off lengths are equal.

Evidently, if the isosceles right triangle exists then lengths with such a relation do exist.

But, I don't see how any of this is evidence for the existence of infinite sets.

Posted by: Alfred Centauri | Tuesday, August 17, 2010 at 09:57 AM

Errrr... that should have been "associate the ratio Q/P with the length...".

Or is it (Q-1)/(P-1)? Well, you get the drift...

Posted by: Alfred Centauri | Tuesday, August 17, 2010 at 10:04 AM

William,

Thanks for the quotation from Cantor. Yes indeed, for him

eine Zusammenfassungpresupposesein Zusammenseins. Your reconstruction:1. A set is an assemblage, a Zusammenfassung.

2. The length of the hypotenuse cannot exist without an assemblage of its infinitely many segments

3. The length of the hypotenuse exists.

4. Therefore an infinite assemblage exists

5. Therefore an infinite set exists.

That is good as a reconstruction of what Cantor is saying, but I don't see that it is an absolutely compelling argument. (2) presupposes that there is an actual infinity of line segments in a line -- and this begs the question. Besides, it is not obvious. An Aristotelian could say that a line is not composed of points or of smaller line segments, but is a primmary phenomenon that we divide up into segments. The infinity of segments and points is thus potential only. I.e., lines are not actually divided but merely divisible.

Posted by: Bill Vallicella | Tuesday, August 17, 2010 at 11:51 AM

Alfred,

A unit is any abitrary unit of measure, an inch, a cubit, a millimeter, etc.

Posted by: Bill Vallicella | Tuesday, August 17, 2010 at 12:00 PM

Phoenix writes:

>>Consider: we are given two line segments and we wish to compare their lengths. We do not know the length of either line segment, and we do not know anything about the limits of rational sequences, either. One segment happens to be exactly "square root of 2" units long, while the other happens to be 1.414213 units long. How can we decide that the two segments are not congruent? Surely not by measuring them with an instrument.<<

That is a good point. We cannot determine whether the line segments are congruent or not. For all practical purposes there is no difference between the hypotenuse's being sqrt of 2 units long and 1.414213 units long. Either way, the length will be quite definite. It now seems to me that this consideration refutes Cantor's argument.

Posted by: Bill Vallicella | Tuesday, August 17, 2010 at 12:19 PM

Bill, that a unit is an arbitrary standard of measure isn't what I was aiming for with my question

Both you and William seem to be singling out the length of the hypotenuse as "special" in some sense and somehow connected to the existence of infinite sets.

But, as far as I can tell, the length of the hypotenuse is no more special than the length of any other line segment. What if the length of the hypotenuse was taken to be the "unit" instead? Would the length of the sides then become "special" with the non-terminating decimal expansion 0.70710678...?

In my next to last post, I attempt to argue that it is not the length that has any numeric "specialness" but rather the relationship between two lengths that does. In that context, the "ratio" of the hypotenuse length to side length for an isosceles right triangle can only be approximated and exists only as a limit.

Posted by: Alfred Centauri | Tuesday, August 17, 2010 at 12:50 PM

There seem to be two questions here. First, if we interpret "set" as "instance of the kind of objects studied by axiomatic set theory", it depends on the axioms whether finite sets exist. The class of all finite sets obeys the Zermelo-Fraenkel axioms excluding the axiom of infinity, which is why the existence of an infinite set needs to be postulated separately (using the axiom of infinity).

This does not answer the vaguer (to me at least) question whether, and in what sense, an infinity of objects can be proved to exist. The class of all finite sets

appearsto be infinite, but without the axiom of infinity one cannot construct this class as a set. This leaves open four mutually exclusive possibilities, none of which seems to be obviously wrong:(1) there is no such thing as "the class of all finite sets";

(2) the class of all finite sets exists, and is potentially but not actually infinite;

(3) the class of all finite sets exists, and is actually infinite, but not a set;

(4) the class of all finite sets exists, and is an infinite set.

Posted by: Peter Bruin | Tuesday, August 17, 2010 at 12:53 PM

Bill, in reply to Phoenix you say

If the unit is a lightyear then the difference is about 168 metres. If the unit is one metre then the difference is about eight times the wavelength of the D line in the spectrum of sodium. Would you allow that there is a possible world in which the difference does make a practical difference?Posted by: David Brightly | Wednesday, August 18, 2010 at 02:43 AM

Correction: for eight times please read 0.9 times. By implication, detectable by interferometry.

Posted by: David Brightly | Wednesday, August 18, 2010 at 02:58 AM

David is absolutely right, of course. But the point stands: since we do in fact have real analysis, we know that we can approximate an irrational number with arbitrary precision. This means that we can always find lengths which are theoretically distinct, but practically indistinguishable. This was not meant as a refutation of Cantor's argument, but as a caveat: without a theory of limits and all of its infinite baggage, it may be that measurements cannot in fact be determinate.

What is intended as a refutation of Cantor's argument I perhaps stated unclearly. The problem with 2 and 3 of the reconstructed argument above is the notion of "length". In the classical analysis of which I am aware, this is defined with reference to sets which are infinite more or less by fiat. This was implicit at least as far back as Descartes' work, so I suppose it must have been accessible to Cantor.

Posted by: Phoenix Kyle MacGregor | Wednesday, August 18, 2010 at 08:33 AM

David,

Your point is well-taken. Much depends on what the unit of measure is. I suppose what I am trying to say is that the decimal expansion need not be actually infinite for the length of the hypotenuse of the isoceles triangle to have a definite length when the sides are say 1" in length.

Posted by: Bill Vallicella | Wednesday, August 18, 2010 at 07:24 PM

Bill, I having trouble understanding your point in your reply to David.

It seems to me that the hypotenuse and the sides have a definite length independent of any unit.

I'm currently consulting chapter 2 of "Gravitation" (Misner, Wheeler, and Thorne) and in particular, the section on geometric objects. Consider the following:

"More fundamental than the components of a vector is the vector itself. It is a geometric object with a meaning independent of all coordinates....

Coordinates enter the picture when analysis on a computer is required (rejects vectors; accepts numbers)."The point is that the geometric object

vectorhas direction and length even in the absence of a basis for associating numbers to those quantities.More interesting (to me, at least) is the fundamental geometric operation of the

contractionof a one-form (co-vector, dual vector...) and a vector to produce a number.Particularly, consider the case of the vector addition of two orthogonal vectors with the same length.

The resultant vector, I assume, is indeed a vector with a perfectly definite length.

Consider the contraction of the one-form dual to either of the orthogonal vectors with the resultant vector which, as it turns out, is the number 2.

In this particular case, this implies that the length of the resultant vector is sqrt(2) times the length of either of the orthogonal vectors.

The point being that no unit has been introduced here, i.e., the value of the contraction is independent of any unit of measure.

But, and again, it isn't the length that has the infinite decimal expansion.

Posted by: Alfred Centauri | Thursday, August 19, 2010 at 08:16 AM

(sigh - I'm too eager to hit the "Post" button...)

"Consider the contraction of the one-form dual to either of the orthogonal vectors with the resultant vector which, as it turns out, is the number

1"Posted by: Alfred Centauri | Thursday, August 19, 2010 at 08:37 AM