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.