The nature of number

[Categories+Sheaves]

I am dyslexic, but that is only covers part of it, I am by nature ill defined, I could come up with an alternative way of making the statement, but in all truth it probably would not be any more clear. And anyway I think you understand the gist of it.

Okay, but... it's really general... Which end of this goes up?
If you have some asymmetrical thing that comes out of a singular point, you've got some lack of symmetry in the stuff that happened along the way... but what are you getting at with "adversely affect"?

[jonb]

Getting there, I think I know what I want to ask just have to formulate it, meanwhile.......... http://www.youtube.com/watch?v=3wHKBavY_h8

[Categories+Sheaves]

That sketch goes perfectly with the ABC conjecture thread.

[jonb]

I think I may have distilled my difficulty.
Is a series of all the numbers between two points say 6 to 7 thought of as the same as say an infinity of (to use Cantor) multiples of three? Are these thought of as different categories of series?
I don't think they are the same sort of series.

[Categories+Sheaves]

I think I may have distilled my difficulty.
Is a series of all the numbers between two points say 6 to 7 thought of as the same as say an infinity of (to use Cantor) multiples of three? Are these thought of as different categories of series?
I don't think they are the same sort of series.

The sense in which you're using series is still not entirely clear to me. But are these two things equivalent as sets? If we're talking about all the real numbers between 6 and 7 or all the relevant points in such-and-such region of euclidean space, then no, the first set is strictly bigger than the second one.

[jonb]

Bigger?
If it could be engineered so the sets were of equal size, I think we maybe looking at different types of sets. But that would depend on what is meant by bigger could you explain?

[Categories+Sheaves]

HAY ADMINS, HOW DO I HTML TABLE? PLZ HALP OR FIX

Bigger?
If it could be engineered so the sets were of equal size, I think we maybe looking at different types of sets. But that would depend on what is meant by bigger could you explain?

"Bigger" = Greater cardinality. In Cantor's diagonalization proof, we see that we can have two infinite sets A and B such than the first can be mapped onto the second (covering everything) but the second cannot be mapped onto the first: the first set is "bigger" in a sense similar to how we can draw from a container containing 8 balls to replace a missing container containing 6 balls, but this doesn't work if we reverse the roles. If we assume the axiom of choice, then by that assumption, given any random pair of infinite sets A and B, there exists an onto function from A to B or from B to A (both may exist, but at least one does). And once we can have this, it takes just a bit more work to show (I can run through it if you like, but I need to finish up this post) that you can make a strict order relation (either |A| <, =, or > |B|; no matter what, their cardinalities can be measured against one another).
And in the default approach to sets--it's just a boring old set, a box with prima facie bland, featureless objects in it. As far as the semantics go, it's a set before we start talking about the shapes or structures the objects form. Let's take two sets that have the same cardinality and show how we can pretend one is the other.
How to pretend the integers are the same thing as the rationals:
Are you familiar with the bijection between the rationals and the integers? Well, we're mapping 0 to 0, and we're going to map the positive integers onto the positive rationals (and use this to define our map between both flavors of negative number).
All rational numbers have some unique 'fully reduced' form m/n, where m and n are coprime (no common factors) integers. So we make ourselves a grid. The entry in the nth row and mth column is "the nth natural number coprime to m", divided by m. Here's the first 49 squares:
Pardon my formatting, I don't know how to make a nice, regularly spaced table in bbcode.

Code:

<table border="0" cellspacing="2" cellpadding="2"><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>...</td></tr>
<tr><td>1/2</td><td>3/2</td><td>5/2</td><td>7/2</td><td>9/2</td><td>11/2</td><td>13/2</td><td>...</td></tr>
<tr><td>1/3</td><td>2/3</td><td>4/3</td><td>5/3</td><td>7/3</td><td>8/3</td><td>10/3</td><td>...</td></tr>
<tr><td>1/4</td><td>3/4</td><td>5/4</td><td>7/4</td><td>9/4</td><td>11/4</td><td>13/4</td><td>...</td></tr>
<tr><td>1/5</td><td>2/5</td><td>3/5</td><td>4/5</td><td>6/5</td><td>7/5</td><td>8/5</td><td>...</td></tr>
<tr><td>1/6</td><td>5/6</td><td>7/6</td><td>11/6</td><td>13/6</td><td>17/6</td><td>19/6</td><td>...</td></tr>
<tr><td>1/7</td><td>2/7</td><td>3/7</td><td>4/7</td><td>5/7</td><td>6/7</td><td>8/7</td><td>...</td></tr></table>

And we throw the natural numbers at them in a zigzag pattern like so... (the natural number occupying the nth row and mth column of this is mapped to the rational number occupying the same position in the table above)

1 2 6 7 15 16 28 ...
3 5 8 14 17 27 30 ...
4 9 13 18 26 31 43 ...
10 12 19 25 32 42 49 ...
11 20 24 33 41 50 62 ...
21 23 34 40 51 61 72 ...
22 35 39 52 60 73 85 ...

Some bits of the zigzag go outside the 7x7 snapshot I just dumped here--you can see the full course of the pattern up to 28 though--but I'm sure you can see what's going on.
So I have an infinite grid of natural numbers and an infinite grid of positive rationals. Every number of each type is given one and only one place in their respective grid, so I can freely exchange all my natural numbers for rational numbers by swapping them out for the # that takes their place in the other grid (note that this means the '2' in the natural numbers isn't the same as the '2' in the rational numbers; and in fact, all natural numbers besides -1, 0, 1 are going to be mapped to something completely different). So now we're ready to take some rational numbers and pretend they're natural numbers...

(It's getting late, I need sleep, and I'm working all day tomorrow. Will finish later...)

[Categories+Sheaves]

Ok. To avoid confusion, I'm going to use italics to distinguish expressions occurring in the rationals from expressions occurring in the integers. Now lets's make our rationals pretend they're integers.

So now when I write some (italicized) expression in the rationals, interpreting it requires you to:
0. Look at what I wrote
1. Replace the rationals with integers, following the rule above
2. Evaluate this integer expression in the normal way.
3. Translate the result back into a rational number, reversing the earlier rule.
So, an example
0. 1/2 + 2
1. 2 + 3
2. 5
3. 3/2
Or,
0. 2/3 + 4
1. 9 + 10
2. 19
3. 5/4
Italicized rationals then behave in exactly the same way as regular old integers--they just go by very different names that bear no apparent relation to their "real" value. This weird redefinition of addition, multiplication, etc. on italicized rationals was defined using our normal, intuitive operations on the integers--but surely we can say these "italicized addition" and "italicized multiplication" exist as binary operations on the rationals in their own right. So while rational numbers and integers exist as distinct sets, the existence of a bijection between them (by the axiom of choice, this definitely exists between any two sets with the same cardinality) we can take any structure on one, and made the other set mimic that structure perfectly (but all the names will be wildly different). So which is the "original" and which is the "duplicate"? (they're functionally equivalent, so this distinction is useless)

So uh--in terms of the "different types of sets" thing... Yes, the relations we put on sets is the stuff that's actually interesting, but the semantics of modern math insists that those relations come after the existence of the sets (and hence, within that framework, these structures can't say very much about the nature of sets themselves). The rationals look very different from the perspectives of regular and italicized additions, but it's the same set (the same collection of symbols, objects, whatever) at the end of the day, right?

Afterthought: damn, that was a long rant.

[jonb]

So uh--in terms of the "different types of sets" thing... Yes, the relations we put on sets is the stuff that's actually interesting, but the semantics of modern math insists that those relations come after the existence of the sets (and hence, within that framework, these structures can't say very much about the nature of sets themselves). The rationals look very different from the perspectives of regular and italicized additions, but it's the same set (the same collection of symbols, objects, whatever) at the end of the day, right?

Afterthought: damn, that was a long rant.

Long but good

but the semantics of modern math insists that those relations come [i]after the existence of the sets [/i]

Ah This is where I am having my problem. It seems to me one form of set has those relations inherently built into it where as the other form those relations are not originally expressed.

[CliveStaples]

I think I may have distilled my difficulty.
Is a series of all the numbers between two points say 6 to 7 thought of as the same as say an infinity of (to use Cantor) multiples of three? Are these thought of as different categories of series?
I don't think they are the same sort of series.

The sense in which you're using series is still not entirely clear to me. But are these two things equivalent as sets? If we're talking about all the real numbers between 6 and 7 or all the relevant points in such-and-such region of euclidean space, then no, the first set is strictly bigger than the second one.

Erm, what? There are as many points in R as there are in R x R x ... x R, and there are just as many points in [6,7] as there are in R. So the sets have equal cardinality.

Unless I've missed something!