One infinity greater than another infinity

Question

Somebody told me this piece of logic:

"If we consider in our thoughts something which is actually infinite, and we take a part from it, the remainder will undoubtedly be less than what it was before. And if the remainder is also infinite, then one infinite will be greater than another infinite, which is impossible."

I believe its from Aristotle. Can anyone explain the logic behind it? in mathematics the size of an infinite quantity does not decrease when you take a subset of the infinite quantity.

Answer 1

How you deal with infinity depends on what your priorities are.

If you care only about sheer cardinality — as Frege did, as he was considering set theory — you can quite easily have an infinite set, with a proper subset having the same size. But to have this, you must ignore most if not all structure in the infinite set, and define "size" by considering bijections in a very flexible way.

It's quite possible to consider a notion of "size" for subsets, where you consider not whether you can describe a bijection between the subset and its superset, but just whether the set-difference has any non-zero elements. But how then to compare two sets for which neither is a subset of the other? It depends then on which functions you consider to be "size"-preserving.

In measure theory, we consider sets not by cardinality, but by how we may describe it as a (limit of a) union of disjoint intervals; and the mappings which preserve "size" are just translations by positive or negative shifts. Removing individual elements may be seen as infinitesimal decreases in size. But in any case, this requires a commitment to certain priorities in how to describe infinite sets; so that an uncountable set such as the Cantor Set has the same measure as a finite set, i.e. zero.

There are multiple formalised ways to describe and consider the infinite. None is obviously "truer" than the others; they are all merely tools which are better or worse for considering different questions. So what is most important is to make sure that you are asking the right question about infinity, and then to identify the right tool to solve your problem.

Answer 2

We should be careful not to disregard Aristotle for his ignorance of set theory. There are a lot of people, old and not as old, who have had good insights but were not informed by the technical definitions of set theory.

Aristotle's notion of infinity is not the same as that defined in set theory, and is in fact the more intuitive notion of infinite: Having some property, extent, or quantity that is not limited. He also had an intuitive definition of "more" and "less": Suppose you have some two groups or extents, X and Y. If X contains all of the objects that Y contains, or its extent contains all of that which Y contains, but Y contains some object or extent that X does not, then Y is greater in number or magnitude.

These ideas together, if accepted, show that nothing can be infinite. Suppose X is infinite and is, say, a group of objects. Then consider the group Y which is the same as X but with the first object omitted. Then both are still infinite and, being not limited in their quantity, for each object in X it has a "corresponding" object in Y and so the two groups are equinumerous. But they share all the same objects except that X contains one which Y doesn't, and so X is greater than Y, a contradiction.

Of course, set theory doesn't employ Aristotle's definition of "more" or "greater" and has a consistent definition of infinite sets that employs only the notion of a one-to-one onto correspondence (or bijection). The natural numbers are infinite by this definition because they are bijective with the natural numbers starting at 2, and since the ordering relation is defined such that, for any natural number n, n < n+1 and further satisfies irreflexivity, asymmetry, and transitivity, and if we extend this definition to transfinite numbers then, for any finite n, the transfinite aleph_0 is greater than n and all sets bijective with the natural numbers have equal size. By these definitions no contradiction is reached.

This doesn't mean that infinity is real, or that Aristotle's definitions are wrong; nor does it mean that Aristotle was right. To my mind, the conclusion to take from this is: If we use Aristotle's definition, then nothing can be infinite. If we use set theoretic definitions, then things can be infinite but it is possible to pose the question whether there is anything that actually is infinite in reality. By analogy two people might disagree about whether something is large, and it merely depends on what your definition of "large" is. If you think all things smaller than Russia are small, then the average American is small. If you think that all things larger than 100 lbs are large, the average American is massive.

Answer 3

To imagine infinity, ask yourself how many coats of blue paint are required to paint a wall red. The answer is an infinite number, and this is a flavour of what infinity is... because no matter how many you add, it will always be insufficient as blue paint will never paint a wall red.

That is why, if you were to require two fewer coats of paint, it would not reduce the number required one iota since the application of paint would continue infinitely and never satisfy the requirement.

Answer 4

"...remainder will undoubtedly be less than what it was before"...

Why on earth remainder should be necessarily less than it was before? Unless this assumption is justified, any inference drawn from it is questionable. I won't be go with the case where we can assume it should be really less than what it was before (and the follow up claims about infinity). Consider this:

How do you decide one entity (with discrete parts) is less or more than a similar one? A good way is to try to establish a part-by-part correspondance. If you consume all the parts in one entity and yet have non-paired parts in the other, you would say that the entity with unpaired parts are "larger" than the other. If no parts left in either entity that have a pair in the other, than those entities cannot be said larger or smaller with respect to the other.

Apply this to entities with infinite parts and see what happens! Lets say the one entity is the set of all natural numbers( {0, 1, 2, ....N, N+1, N+2, N+3,... } and the other entity be the just even natural numbers. Note that the latter is a subset of the former. Now, if for each part (arbitrary natural number) in the firs entity can be tied to a part in the second entity (just even numbers), and similarly, if each part in the second entity can be tied to a part in the first one, we can say that neither entity is larger than the other. Can we find such a way? Yes: From the first entity to second one direction, simply multiply by 2. We always get a uniqe even number for each selection. This means, no unpaired parts (arbitrary natural number) in the first entity are left by this pairing. For the other direction simply divide the even number by two. You get a unique natural number for any selection in the second set. So, we can make a one-to-one pairing between the parts of those entities. So neither is larger (or smaller) than the other.

What can we infer from this? Parts are not necessarily smaller than the whole at list with entities with infinite parts. However, if you deny (insist / beleive) that no infinite things exists, than you can claim parts should be less than the whole. You certanly allowed to make this assumption. However, what you can't do is use this result back to prove your assumption. It is like this.

You want to prove proposion Q is true (no infinities exist). You PRE-assume R is true ("parts are less/smaller than the whole"). You discover that, if R is true than Q is true. Unfortunately, if R is true only if Q is true, than you cannot use R to prove Q. Above, we saw an example with natural and even numbers that parts are not necessarily less or smaller than the whole if part and the whole are both infinite.

To prove infinities do not exist or infinities cannot be compared, you cannot presume any proposion (like "parts are less than the whole") which holds only if infinities do not exist or they don't compare to prove infinities cannot exist or they cannot be compared.

Answer 5

Infinity is hard to grasp (esp. the second half). It is difficult to define the infinite from the finite. One interesting definition, attributed to Dedekind, is (see here for a math version):

Set S is infinite if and only if there exists a proper subset P (proper means that the subset is not S itself) of S and a bijection f that maps S onto P.

In mudane words, P has at least one element less than S (to differ and be proper), but still be in bijection, so any element of S uniquely corresponds to one element of P. For instance, you can take the set of even integers 2p, in bijection to the set of integers, because for each 2p you can uniquely associate p. But the set of even integers seems to have half the quantity. Which is wrong. Hence, the asumption:

take something which is actually infinite, and we take a part from it, the remainder will undoubtedly be less than what it was before

is not valid for infinite sets. It is just a projection that is valid on finite sets, and that our intuition projects (wrongly) over infinite quantities.

But there are different kinds of infinities, on which one can design an ordering, some infinites are bigger that others, because there are no bijections between them.

Answer 6

Infinity is not a number. It's not like it is somewhere on the number line. When you start walking now, you will walk 1 mile, 2 miles, 3 miles, and so further, but you'll never reach the point that you've actually walked infinity miles.

You cannot think of infinity as the amount of a set of items; you cannot have infinity apples - in reality, that is. Therefore, you cannot think of decreasing and increasing that amount.

The only place in the physical world where we might find infinity is, I believe, the nothing: space. Space can be infinite as it isn't really anything, merely something that can't really be but yet holds the possibility of being used by something that is.

Your quote, ...

If we consider in our thoughts something which is actually infinite, and we take a part from it, the remainder will undoubtedly be less than what it was before. And if the remainder is also infinite, then one infinite will be greater than another infinite, which is impossible.

... cannot be applied on a set of items. You cannot reasonably consider an infinite amount of apples. When you apply the quote on space, it makes sense: take a part from nothing, and it still is nothing, as big as it was.

Answer 7

Without further context, the statement appears to, quite simply, point out an incompatibility between concepts of infinity and those of mereology or, indeed, measurement of any sort.

A "part" can only be defined in relation to some definable "whole." To "define" is, of course, to render the object of definition "finite" in some sense. It is defined only between some specified limits or "from the outside," so to speak. The old problem of whether a point on a line is a "part" of the line, thus partaking of its two-dimensionality, or a purely mathematical, dimensionless "partition" of the line.

So if we grant an "actual" world in which things are in some sense measurable and have "parts" we cannot also have infinity... it doesn't "fit in," you might say. We do reduce things to parts. So an "actual" infinity is impossible, is not commensurate with an actuality of dimensions, wholes, and parts.

At least, this would seem to be the negative demonstration that the author, Aristotle or whomever, is driving at. Perhaps a clue to the deeper antinomy here is the whole implication of considering "in our thoughts" some "actual infinity..." Kant might rebut that we can "think" such things but can "know" nothing about them, nor fill them with "actual" content. This "infinity" that has "parts" is, at the very least, not actual.

Which is perhaps why Kronecker regarded young Cantor's sets as his generation's corrupting equivalent of LSD, the unleashing into physics of pure intoxicating, useless phantasms. Perhaps he actually had a... point.