Epistemology of infinite sets

Question

I'm trying to understand the difference between sets from an epistemological point of view.

Let S be a finite set of cardinality = n, I find intuitive that I can hold epistemically the set S without any special effort. For sure the greater n is the harder it's to hold it, but I don't see any logical bound.

Now let P a numerable infinite set (like N), it becomes harder to think that I can hold simultaneously all the elements of N, It seems that I know the set of natural numbers less than one of its subsets.

Let's now increase in complexity, let R be a non numerable set of cardinality Aleph1, my epistemic grasp of it seems fading away more and more.

We can extend our reasoning to complexities far greater than the real numbers like the manifold of cardinals.

In what sense our epistemic grasp of sets dicrease? and why? Which feature of sets bounds our ability to know them?

Answer 1

I agree with the above answer, but I'll attack the issue from another point ov view.

Let S be a finite set of cardinality = n;

clearly, with n "little, we have no problem to "hold epistemically" (to imagine ? to visualize ?) it.

I'm not sure about a set S with n = 3567 elements.

For sure, a set S with n = 6665734529976967438675338965633321266643790584532111111 elements is quite far from beeing "easy to hold".

Consider nowe the set N wich include an "initial" element, call it 0, and for each element n, it includes also its "successor", call it S(n).

We have "specified" in a easy and understandable way the set N of natural numbers, which has infinite (exactly : countable many) elements.

The basic concept of an unlimited possibility of iteration, which is the "core" of the potential infinite is much more easy to grasp than a "very very big" finite namber.

What about aleph-one ?

Assuming that it is equal to c, here we have to start from the so-called real line; what is the role of a "continuous" magnitude in our thinking ? We need it "only" inside mathematics ?

Answer 2

For low numbers we rely on our physical understanding. I can see three cups and I can throw 5 cricket-balls, For high numbers this is not possible and other capabilities take over. Consider:

678

1256745

The second number is much larger than the first. But I have no physical appreciation of this fact. (It is possible to increase this by training in quantitative physics); but here its much simpler as we can rely on the same low number paradigm as before. I don't actually see the number physically, but I do see quite quickly that the second has more digits than the first. Now lets consider the following:

3567

6665734529976967438675338965633321266643790584532111111

The second is much larger than the first, now reading the number of digits is something that can't be done at a glance, but looking at it geometrically, one can judge that the second is perhaps ten times as long.

Practically, one has to consider that large numbers are fairly pointless. There are after all only about a googles worth of particles in the universe. One can write this number down - 1 with a hundred zeroes after it. Most of the time we actually deal with quite small numbers - 6, 30 maybe 500,000. Hence, for most people, for most of the time, a fairly primitive understanding of quantity is all thats required: nothing, one, two, a little and a lot.

The picture becoes clearer when we begin to look at cardinal numbers. Here there is no hope of usingphysical imagination, and what one uses is how they function within the body of mathematics, that is by the statements that use it,and are understood by it. In Saussearean terminology we understand it structurally. That is by an element in a whole, and its relations. One could develop from here , I suppose, a structural epistemology of quantity.

Answer 3

In what sense [does] our epistemic grasp of sets d[e]crease? and why? Which feature of sets bounds our ability to know them?

I think with some training in mathematics (undergraduate level), dealing with infinite sets in different ways becomes pretty natural. It is obvious that any human mind can only comprehend finite things, so instead of trying to understand an infinite thing you reason about a finite description of how to construct it. You then use this finite construction to reason about the set.

For example: instead of trying to keep in your head what the real numbers are, you instead just consider the reals as the completion of the rational numbers. So although the real numbers have a much greater cardinality, understanding that cardinality doesn't have to be that much harder.

The best example of something that seems to escape human reasoning about sets is the Continuum Hypothesis: whether there is a set whose size is greater than that of the integers but less than that of the reals.

Answer 4

Now let P a numerable infinite set (like N), it becomes harder to think that I can hold simultaneously all the elements of N, It seems that I know the set of natural numbers less than one of its subsets.

But that's true about every object, physical OR abstract. Suppose I hold in my hand a rock. I have a direct physical experience of the rock ... its size, weight, texture, and visual appearance.

But I have no knowledge or experience whatsoever of the atoms, quarks, gluons, strings, etc. that make up the rock. There are parts of the rock that physicists haven't even discovered yet.

Is this an epistemological problem? I don't think so. To know a thing, I need not know all of its parts. I know about China, even though I don't know each and every citizen of China by name.

You are correct that we have an intuition of N. But we do not have a direct experience of each and every one of the number 1, 2, 3, ..., let alone each element of the uncountable power set of N.

But as I say, this is no different than anything else I know. I know the sun will rise tomorrow morning; but I do not have a direct experience of each and every one of the photons that it will send out in my direction.