Infinity, Sets, and Causality

[Raelynn]

I'm a math guy and not so much a physics guy, so I'll mostly talk about the math side.

Representation vs. Concept: Infinity

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.

Now suppose we have sets of "Unit" ("Unit"s grouped together) - how "big" are the sets?  "Magnitude" is the concept of how big the sets are.  {*} is defined as a set of a certain magnitude, 1 in a lot of representation.  {*, *} has a magnitude of 2 in Decimal, 10 in Binary, 11 in Unary, and Two in English.  Decimal 10, Binary 1010, Unary 1111111111, and English Ten all represent the same set are equivalent - 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".

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 {***}.

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.  There are a lot of rules that you can use with the magnitudes of finite sets - addition, subtraction, multiplication, division, less than, greater that and so on.  We are very familiar with the magnitudes of finite sets -  The positive integers are all magnitudes of finite sets.

This was a fairly good description of sets using easy to understand analogies. :)

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.  So, we use sets of sets - usually, sets of the magnitude of finite sets.  For instance, the positive integers: if we put all the positive integers in a set, how "big" is it?  Let's represent that set as {1, 2, 3, ...}.  If those "numbers" are the magnitudes of finite sets, it's {{*}, {*,*}, {*,*,*},...}, but the first representation is easier to handle.  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.

All of math is based on Peano's Principles, which basically say 1+1=2 so the natural numbers exist. Just about everything in math is based on this principle. If you don't assume that it's true, then everything in math, not just sets, becomes defunct.

Inifite sets can be bounded or unbounded too, which again we don't have direct experience with - if we put 'infinite' stones in a bucket, we'd think that no matter how small the stones were, you'd never get an infinite number of them in a mop bucket.  Howver, the magnitude of the set of all real numbers between 0 and 1 is inifinite, but each member of that set is less than 1 and more than 0.

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.

To make things even more confusing, there are as many real numbers between 0 and 1 as there are all the real numbers :blink:

An infinite set "never ends".  What if we define infinite (represented by B for this) as so many, that you add itself to it and still have B, or

B is infinite.
0 is not infinite.

B = B + B

Then, by algebra

B - B = B

0 = B

Here's the big complaint that I have. Infinity is a concept, not a number. This means that you can't actually do arithmetic operation on it. Because of this, the math, though it seems true, doesn't hold.

B or infinite is not in the same class of representation as, say, 12, and the same rules don't apply to it (I know this is faulty logic and missing some steps).  It's hard to define what infinite is, but it is not too hard to define what is not infinite - all the things that work with the rules we expect such as addition and subtraction.

The question, "What is infinity plus one?" has no meaning - you can't add one and infinity and come up with an answer any more meaningful than to "What is a pile plus one?".

This basically says what I had said above. Since it's a concept instead of a number, methods that we attach to numbers don't work with infinity.

Back to the quote at hand

Taking the ponts above that a) sets are ordered to make them more easily represented and B) the same rules don't apply to infinite magnitudes as to finite ones, lets look back at the quote.
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.

Inifinite as I have defined it (the magnitude of a set of units) is non-directional - it only goes one way ("up" in the counting sense).

I don't believe that your third definition is a legal definition, but I don't know a ton about sets. In terms of numbers, it is commonly accepted that there is a "positive infinity" and a "negative infinity", which arise from going in each direction. Since you talk only of the number of elements, you are actually using a function to turn turn the elements of the set into the elements of a set of positive numbers. This means that there can only be a positive infinity in your definition, because you start at 0 and count upwards. This doesn't mean there can't be a negative one though.

Personally, I believe that the positive and negative infinity are connected an possibly the same. If you allow for that, geometrically all polynomials with power of any number, positive or negative, would be continuous, as opposed to the current definition which only allows for polynomials with positive powers to be continuous. This of course is a discussion all on its own.

As far as time goes, this is where causality comes in.

Causality and Time as a 4th Dimension

I bet we can all imagine the universe as a 4-dimesional thing: you need 4 measurements (distances) to find a certain thing, three spatial coordinates (call them x, y, z) and a time coordinate (t).  Missing one, you won't find the thing - it does no good to look for Duncan Donuts at x, y, z if you don't know when it is there.  Also, it would do no good to go there and expect to find it a x, y, and t if you show up 30m underground (if z is a measure of height relative to some reference).

If we looked at a set of coordinates (x,y,z,t) in 4-dimensional space, how would we order them?  We could order them first by x, so that the set is {...,(-1,y,z,t), (0,y,z,t), (1,y,z,t),...} - the direction of this ordering is abitrary, and is based on the distance from some arbitrary reference coordinate.

We could order it by t, which would also be based on the distance to an arbitrary reference point, but what does distance in t mean?  What makes something more or less "t long"?

Causality gives a direction and meaning to our order in t.  For us to be able to interpret the universe, if the stuff in the universe can be described by coordinates in the set we mentioned, causes must happen before effects, and this gives us an order: effects have a value of t that is more than their cause.  So, the coordinates describing my butt sitting on my chair goes before the series of coordinates of the chair breaking because I sat on it.

An awfully Anthropic view, but very scientific: we observe that causes come before in time from effects.  As soon as we observe an effect happening before a cause, this is out the window, but how would we know that something is a cause when it happens after the effect?  How would we prove it scientifically?  Who knows.

I like this definition of time, but it is based on a one-dimensional time, which may or may not be the case. I personally think time is greater than 1 dimensional, though it may possibly be 0-dimensional (time is an instant, we just interpret it as something else). The two dimensional time also leads to some other theories I have concerning ghosts, gods, and some supernatural phenomenon, but again that's another discussion.

Where does the Universe start in time?  Where does it end?

If we say the Universe is again a set of coordinates ordered by cause-before-effect in t, where does it start or end?  Does it need to?

If time 'starts' somewhere, then that coordinate has no cause in the Universe (or else another coordinate would go before it, and would be the start).  I guess it could have a cause in some other set, but it all breaks down there as far as our rules go.  If it 'ends', then all coordinates, if followed in cause-effect order, eventually causes no effect in the Universe, though it could cause one in something that isn't the Universe.

It could have a start but no end, an end but no start (depressing, eh?), or no start and no end.

There is another mathematical theory call the Incompleteness Theorem which states that there always exists some problems which cannot be solved using only the rules of a certain set. This goes along with what you state here, that there may be an explanation if we expand our set outward to include another set as the cause.

There are two ways it could have no start and no end in time: it could either go on infinitly in both directions (there's always a cause before this, and a cause after this), or it could wrap around (for instance, a->b, b->c, c->d, d->a).

What the beginning, end, or wrap of time really represents and means, is beyond me.

Long and rambling, but not infinite!

Cyclical time would be interesting. It's one of the common theories of time. The other possibility is that there exists something that breaks causality, something that is only a cause, and not an effect. Just because no one has found it, doesn't mean it doesn't exist. :)

These are actually some of the topics I thoroughly enjoy talking about. It's been fun writing out this response.