I’m not really happy with
this,
but I’ve been working on it on and off for about a year and a half now.
I’ve decided to solve the worst problems by breaking it into three parts,
which will hopefully make it less a failure than it seems,
and in any case easier to improve later.

In If God is infinite, how can anything else exist?

Where would there even be room for anything else?

The answer, of course, is that God is immaterial, and an infinite immaterial thing
does not take up “space”.
We can illustrate this via numbers, which are themselves immaterial.
This argument does not require you to believe that numbers exist, by the way;✗
I myself don’t believe that numbers exist Where would there even be room for anything else?

What are the different kinds of numbers?

Mathematicians work with sets of numbers.
The definition of a “set” is a little hard to get right
(one reason that multiple set theories exist)
but you can think of it as a way of identifying numbers.
We’re interested in two particular sets of numbers:
- The
**“natural“ numbers**, denoted ℕ, consist of 0, 1, 2, … . Some people call them “counting” numbers, perhaps because they are good at modeling the ability to “count in nature”: you can count 5, 10, or more than a billion people; and there was a time where God would have counted zero of us.✗Some mathematicians exclude 0 from the natural numbers, illustrating an important point about mathematics:**definitions matter**. A difference in definition can lead to a difference in results. Sometimes, the differences in definition matter enough that different schools will converge on a consensus definition, which makes communication easier. An example would be the definition of a**prime number**, which is any natural number with exactly two natural divisors. Thus,- 2 is prime because it has exactly two natural divisors: 1 and 2;
- 4 is not prime because it has three natural divisors: 1, 2, and 4; and
- 1 is not prime (this is important!) because it has only one natural divisor: itself.

- A
**negative**number is one that, when added to a particular natural number, gives you zero. For example, -2 + 2 = 0. Negatives are good at modeling things like debt or spending: you may earn $20, but if you already have to spend $20, your balance is effectively $0; the debt is a “negative” quantity. - The
**integers**, denoted ℤ, consist of 0, ±1, ±2, … . In other words, you get the integers by including negative numbers with the natural numbers. (When mathematicians write ±1 we mean 1 and −1.) - The
**“real” numbers**, denoted ℝ, consist of all values we can in principle measure along a line.*a*is a “real” number because we can measure it along a line. Precisely*which*real number it is, we’ll return to in a moment. Real numbers need not be restricted to a quantity of distance; one can also include direction. Think of “two steps forward, one step back.”

- The
**“rational” numbers**, denoted ℚ, consist of- 0,
- ±1/1, ±1/2, ±1/3, …
- ±2/1, ±2/2, ±2/3, …
- ±3/1, ±3/2, ±3/3, …
- …

**ratio**nal numbers by taking**ratio**s of integers, with the exception that*0 cannot appear in the denominator*. Some rational numbers are equivalent; for example, 1/1 = 2/2 = 3/3 = … - The
**“irrational” numbers**are precisely those real numbers that are not “rational”.✗You may be wondering why these numbers have the name “irrational”, which means "not reasonable". We can blame the Pythagoreans. They first thought rational numbers sufficient to describe reality, until one of their school discovered that the real number*a*apparent in the triangle above, also known as √2, is not a ratio of two integers. Thus, if real numbers exist, or if the phenomena they model exist, then irrational numbers must also exist, or rather the phenomena they model. But in a truly bizarre intellectual twist, the Pythagoreans didn’t like irrational numbers, which is why they have the name “unreasonable”.

- Every natural number is also an integer, by definition of an integer.
- However, not every integer is also a natural number: -1 is not natural.
- Every natural number is real, because it measures a distance: 1 foot, for example.
- Every integer can be viewed as real when we consider direction in addition to distance: walking five feet forward is 5, and walking five steps backward is -5.
- Every integer is equivalentI’m using
“equivalent” instead of “equal”
for an important reason; they don’t actually mean the same thing.
However, for all intents and purposes, the lay reader can consider “equivalent”
to be interchangeable with “equal”. to a
rational number:
1 = 1/1 .In fact, every integer is equivalent to
*multiple*rational numbers:1 = 1/1 = 2/2 = 3/3 = … -
However, “most” rational numbers are
*not*equivalent to integers. Rational numbers like 1/2 lie “between” two integers. - Every rational number
*a*/*b*is real, inasmuch as you can obtain it by measuring out a segment, calling that length 1/*b*, and placing*a*copies of that segment next to each other. - However, “most“ real numbers are
*not*rational. The diagram above shows that √2 is real, but a slightly technical proof shows that it is not rational.✝In brief,- Suppose that √2 =
*a/b*, where*a*and*b*have no common factor. - Rewrite the equation as
*b*√2 =*a*. (Clear denominators.) - Rewrite again as
*b*^{2}×2 =*a*^{2}. (Square both sides.) - Notice that Equation 3 tells us that
*a*^{2}is, by definition, even. Since odd numbers don’t square to give us even numbers — it is easy to convince yourself of this, and not hard to prove✞- Suppose that
*a*is odd. By definition,*a*= 2*c*+ 1 for some integer*c*. - By substitution and a little algebra, we rewrite
*a*^{2}as (2*c*+ 1)^{2}= 4*c*^{2}+ 4*c*+ 1. - Rewrite again as
2(2
*c*^{2}+ 2*c*) + 1. - Let
*d*= 2*c*^{2}+ 2*c*. - By substitution,
*a*^{2}= 2*d*+ 1.

*a*^{2}is odd. —*a*itself must be even. - Suppose that
- Rewrite
*a*= 2*c*. - By substitution, equation 3 becomes
*b*^{2}×2 = (2*c*)^{2}. - Rewrite the equation as
*b*^{2}×2 = 4*c*^{2}. (Simplify the square.) - Rewrite again as
*b*^{2}= 2*c*^{2}. - Notice that the equation 8 tells us that
*b*^{2}is, by definition, even. Just as*a*^{2}being even forced*a*to be even,*b*^{2}being even forces*b*to be even. - We have found that
*a*and*b*are both even. Pretty cool, right? In fact, no: Statement 1 assumed thatWe have a contradiction!*a*and*b*have no common factor!

**Statement 1 is false.**In other words, √2 ≠*a/b*, where*a*and*b*have no common factor. Any fraction can be reduced to a ratio of integers with no common factor, so**√2 is irrational**. - Suppose that √2 =

ℕ ⊊ ℤ ⊊ ℚ ⊊ ℝ.

That is shorthand for saying, “ℕ is ‘contained’ in ℤ, but not equal to it;
ℤ is ‘contained’ in ℚ, but not equal to it; and
ℚ is ‘contained’ in ℝ, but not equal to it.”
If God is infinite, how can anything else exist?

Where would there even be room for anything else?

We can answer in the following way:
Where would there even be room for anything else?

At least two, mutually exclusive sets of infinite numbers exist:
the rational numbers and the irrational numbers.
Hence, infinite immaterial things exist
without preventing the existence of other things.
In the same way, the immaterial God’s infinity
does nothing to prevent the existence of material creatures…
*even if they were infinite*.