*What is a contradiction?* We define a contradiction as a statement that leads to the principle of explosion. The simplest contradictions take the form of “X is Y and X is not Y”. Yet there are many other rules that must be considered before we can formally do anything here:

Tortoise: But we must be careful in combining sentences. For instance you’d grant that “Politicians lie” is true, wouldn’t you?

Achilles: Who could deny it?

Tortoise: Good. Likewise, “Cast-iron sinks” is a valid utterance, isn’t it?

Achilles: Indubitably.

Tortoise: Then, putting them together, we get “Politicians lie in cast iron sinks”. Now that’s not the case, is it? - Godel, Escher, Bach

*What is a consistent logical system?* A consistent logical system is one that does not lead to a contradiction. This is a definition by negation, the term is more properly “not-inconsistent”.

*What is a logical system?* A logical system is made up of axioms and rules of inference. It defines a subset of grammatical statements which are included in the logical system.

The incompleteness theorems state no logical system is complete; that is there are statements where neither the statement nor its negation are part of the logical system.

What is Gödel numbering? Gödel numbering is a reversible encoding (in fancier words, a bijective function) between logical statements and numbers. There are many ways to do this; the simplest is to consider the binary Unicode representation as a number. An explicit enumeration of grammatical statements (as would be found in a diagonalization proof) or a prime-power-encoding based system are often more useful. However, the details are completely irrelevant to our purposes.

## Set Building 101

We consider a set S, with the following axioms.

(∃ X ∈ S) ∧ (∃ X ∉ S)

(X ∈ S ⇒ X ∈ ℂ)

X, Y ∈ S ⇒ X + Y ∈ S

X, Y ∈ S ⇒ X × Y ∈ S

X ∈ S ⇒ X⁻¹ ∈ S

It is fairly straightforward to prove that the number 1 is in S. By repeated addition, we can then prove that all ℕ are in S, and eventually all positive rational numbers.

It is also straightforward to prove that the number 0 is not in S. There is no multiplicative inverse of zero in ℂ. You could make a change to the super-set to be {ℂ ∪ ℵ₀} and say 1 ÷ 0 = ℵ₀, and possibly should do that. For today, it is more interesting to pretend otherwise.

*Is the square root of 2 in S?* It is indeterminate. It certainly could be in S, yet we have no way to deduce it without adding another axiom. We could add an axiom that √(2) ∉ S. Going the other way, there are multiple axioms we could add:

We could simply add “the square root of 2 is in S” as an axiom. This would add 1+√(2) and 2 - √(2) and other (positive) combinations to S.

We could add an axiom X ∈ S ⇒ √(X) ∈ S and get all the square roots in one fell swoop.

We could use Cauchy sequences or Dedekind cuts to add all positive real numbers to S.

*Is negative one in S?* No. If it were, then -1 + 1 = 0 would be in S.

*What about negative square root of two? *I’m fairly certain this is just as indeterminate as the positive square root of two. It will take university-level algebra to prove this, though. Feel free to post your proof in the comments. You may want to start with the term “ideal” if you are looking things up.

## The positive number “about negative 1”

*Could “just about negative one” be in S?* That is, -1 ± 𝛆. We arbitrarily take 𝛆 to be one-billionth1 of the reverse-factorial number, a binary number where the Nth bit is 1 iff N is a factorial number. As that definition is likely to induce off-by-one errors, we explicitly note a value of b0.110001000… ≅ 49/64 + 1/(16777216) ≅ .765250596 . This is certainly in ℂ.

To a certain extent, you wouldn’t even notice whether we choose to use +𝛆 or -𝛆.

*What is the only sport where a tie is different from a draw?* *crickets*

The term “billion” is generally considered vague and dis-preferred. In America, “billion” unambiguously refers to a thousand millions. In other countries, the English language term billion refers to a million millions. Also, we are likely to want a “long” billion, relying on base 1024 rather than base 1000.