Infinity, Sets, and Causality

[LemmingofGlory]

First, let's differentiate a concept, and the way we represent that concept.  "Unit" is a concept based on something being.  I'm not sure that "Unit" is based on anything.  "Unit" can be represented in many ways; for now, let's represent "Unit" as *.  Many times this is represented as 1, but this makes it hard to seperate "Unit" from the next concept.

The things in a set are called "elements." And using * to represent an arbitrary element makes me wince.

I can ask for Decimal 3 frogs or Binary 11 frogs and I will end up with the same set of frogs, if frog is our "Unit", {frog, frog, frog}.  These are all examples of positive integers, a certain type of finite magnitudes, as I have no easy way to represent part of a "Unit".

Repetition in a set does not change the set, so { frog, frog, frog } = { frog }. Since you say you have 3 frogs, we assume the frogs are distinct so let's indicate that: { frog1, frog2, frog3 }.

You can also have direction, such as positive or negative numbers.  We call negative direction taking things away, and positive direction adding things to, but if I say -3, it's still the set {***}.

In a set, you don't have direction. You need something more sophisticated than a set if you want to talk about direction. Look up posets.

Sets of finite magnitude are most easily understood since we have direct experience with them - the number of stones in a pail, the average number of socks lost per person in the wash last year (which is something abstract and not physical); the number of grains of sand on a beach at a certain moment in time is a very large set, but finite.

Your descriptions are not what you think you're describing.

The stones in the pail are a set, where each stone is an element. The cardinality of the set is the number of stones, say #. "The number of stones in a pail", as a set, is { # }. Similarly, "The average number of socks lost per person in the wash last year", as a set, is the one-element set containing that number. And "the number of grains of sand on a beach", as a set, is a one-element set containing that number.

Sets of infinite magnitude are harder to understand since we don't have any direct experience with them - they are impractical.  It's hard to define precisely a set of infinite "Unit" and be able to do anything with it; {..., *, *, *, ...} doesn't tell us much.

You're over thinking this; don't make something sound difficult when it isn't. Everyone has that moment when they're learning to count where they say "Oh, I could go on and never run out of numbers, couldn't I?" That's direct experience with an infinite set.

  Let's represent that set as {1, 2, 3, ...}.  ...  However, the only reason this is the set of positive integers is that I said so - you assume that the ... means that the next one will be 4, and then 5, but what if it is 0 or -100?  Well, because 4 comes next!  We impose order on the set only because it makes it easier to represent.

Firstly, one who talks about a set having a "next" element does not understand what a set is. A sequence, on the other hand, does have a "next" element.

Secondly, I disagree with your point. You claim that we assume a reader would infer 4 and 5 would also be elements of {1, 2, 3, ...} because of an ordering. The reader would infer 4 and 5 are also elements not because of ordering, but because of association. They identify the set based on representative elements. For example:

{ The Hobbit, The Lord of the Rings, ... }

There are many possible ways for a reader to label this set (which is one reason why listing elements is not a terribly good way to represent a set), but if a reader said "Books by Tolkien" they would do so based on the representative elements of the set.

What differentiates infinite from finite then?  Any easy way to tell is to try to apply the rules we are familiar with for finite magnitudes and see what happens.

Bad criteria.

B is infinite. 0 is not infinite.

Then, by algebra
B + B = B
B - B = B
0 = B

I know you're trying for a proof by contradiction, but your second step is still bull#$%&. oo - oo is undefined. Who told you the additive inverse of oo is -oo? Nobody, because it's not.

Better Criteria:
(1) Is it possible to count the elements? If not, the set has an infinite number of elements. (e.g. the real numbers)
(2) If it is possible to count the elements, will counting ever cease? If not, the set has an infinite number of elements. (e.g. the natural numbers)

To be sure, these are two different notions of infinity.

Let's take {. . ., -2, -1, 0, 1, 2, 3, . . .} as the set of all integers.  The question is, where does it start, and where does it end?  Well, represented like this, and with the elypsis (sp?) taken to mean "and so on forever", it doesn't start or end!  YOu might say, well, it starts at negative infinity and ends at positive infinity, but we've already decided that the same rules don't apply to 0, 1, 2, and 3 magnitudes as to inifinite magnitudes.  The same set can be represented as {0, 1, -1, 2, -2, ...}, which has a start, but no end.  The same set as {0, -1, 2, -2,...,1} is not a good representation, but does indeed have a start and an end - but it could start and end anywhere really, depending on how you ordered it.

Whoever told you the ideal way to represent a set was to list elements needs to have their crack taken away. And that aside, seeing the integers thrown around like a $4 whore is making me nauseous.

If you really want to bake your brain, play with this set for a little while: { z | z is Complex and | z | <= 1 }. Frankly, I think it's a better example for the sort of set you're looking for ("no beginning and no end") because you don't have to eliminate fifty different ways someone might list the elements.

-Lemmy