The nature of number

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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...)