# “Nontheist” is not the same as set size as “Atheist” due to sentience of members- PROOF by Contradiction

ASSUMPTION: Nontheist := Atheist

Purpose of proof: To show that claiming that the set of “nontheist” is exactly equivalent to set of “atheiest” leads to category error by contradiction. It has been often argue that “atheist v theist” is a “strict dichotomy”, and I prove that by prescribing Google’s definitions as “by definition” leads to an inescapable contradiction in set theory.

Oxford definition of “Atheist”:
“a person who disbelieves or lacks belief in the existence of God or gods.”

Theist: Set of sentient beings who believes in God
Atheist: Set of sentient beings who do not believe in God.
Nontheist (Non-theist) is any element not in the set of theist.

Complementary Set Relationships

1. Set A and A’ are in U: A’=U\A
2. Assume A’ ⊣ A and A ⊣ T
3. A = (U \ T)
4. If x ∈ U then x must be {x ∈ T | x ∈ A}:
5. A = {x ∈ U | x ∉ T}

Let:

U = Universal set of all entities (U)
T = Set of theists (A’)
A = Set of atheists (A)
S = Set of sentient beings capable of holding beliefs (S)

1. Theist: x ∈ T ⇔ (x ∈ S ∧ Believes(x, God exists))
If x is a theist ⇔ x is a sentient being and believes God exists.
2. Atheist: x ∈ A ⇔ (x ∈ S ∧ ¬Believes(x, God exists))
If x is an atheist ⇔ x is a sentient being and does not believe God exists.
3. NonTheist: x ∈ (U \ T)
x is in U, but is not in T

Proof:

1. Assume for contradiction that A = (U \ T)
(Given by initial assumptions of A’=U\A) where “A” is representing A’ and T is representing A)

2. Consider an entity r where r ∉ S (e.g., a rock)
∀x (x ∉ S → x ∉ A) (from definition of Atheist)
r ∉ S
∴ r ∉ A
(A rock is not a sentient being. A rock does not believe God exists. Being an atheist requires sentience, therefore a rock is not an atheist)

3. However, r ∈ (U \ T) because:
r ∈ U (as U is the universal set)
r ∉ T (as r ∉ S, and all theists must be in S)
∴ r ∈ (U \ T)
(A rock is a nontheist because, a rock is in the Universal set, but a rock is not sentient. All theists must be sentient, therefore rocks are not theists)

4. From steps 2 and 3 and assumptions:

r ∈ (U \ T) ∧ (r ∉ A ∧ r ∉ T)
(A rock is a nontheist and rock is also not and atheist nor a theist)

5. This is a contradiction of A = {x ∈ U | x ∉ T}
(If x is in U, it must be either in T or A)

6. Therefore, our assumption must be false.

Conclusion: Nontheist (the set of entities that are not theists) is not the same set size as atheist (the set of sentient beings who do not believe in God).

QED

Short version:
“NonTheist” is not the same as set size as “Atheist” due to sentience of members- PROOF by Contradiction

Assumptions:

1. Nontheist := Atheist
2. Theist: Set of sentient beings who believe in God
3. Atheist: Set of sentient beings who do not believe in God
4. Non-theist (Nontheist) is any element not in the set of theists

Complementary Set Relationships:

1. Set A and A’ are in U”: A’=U\A
2. Assume A’ ⊣ A and A ⊣ T
3. A = (U \ T)

Proof:
1. Assume for contradiction that A = (U \ T)
2. Consider an entity r where r ∉ S (e.g., a rock)
3. Since r ∉ S, ∀x (x ∉ S → x ∉ A) (from definition of Atheist), r ∉ A
4. However, r ∈ (U \ T) because r ∈ U and r ∉ T (from definition of Theist).
5. Therefore we get a contradiction as r ∈ (U \ T) ∧ (r ∉ A ∧ r ∉ T) which contradicts the initial assumptions and complimentary set relationships.
6. (Nontheist := Atheist) = ⊥

QED

Author: Steve McRae