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Daily Dose of Definition: Tarski's Undefinability Theorem

The set of true sentences of arithmetic is not definable within arithmetic itself.

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Daily Dose of Definition: Recursive Enumerable (r.e.) Set

A set A ⊂ ℕ is recursively enumerable if there is a Turing machine that halts on precisely the elements of A. Not necessarily decidable.

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Godelian jokes

A mathematician and a contrarian walk into a bar

Bartender: "So what'll you girls be having?" Mathematician: "The contrarian won't order anything." Contrarian: "I'll have a drink." Mathematician: "It's now obvious that she's having a drink."

The mathematician has contradicted herself. She is inconsistent.

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A mathematician and a contrarian walk into a bar

Bartender: "So what'll you girls be having?" Mathematician: "The contrarian will have a drink." Contrarian: *leaves* Mathematician: "She'll be back."

The contrarian never comes back. The mathematician is consistent, but unsound.

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A mathematician and a contrarian walk into a bar

Bartender: "So what'll you girls be having?" Contrarian: "I'll have the opposite of whatever the mathematician thinks I'll have". Bartender: "So... uh..." Mathematician: "If I'm consistent, then I won't speak for her and she won't order anything." Bartender: "Are you? Consistent, I mean?" Mathematician: "Let me have a drink while I think about it."

The mathematician never decides. She is consistent and sound, but doesn't know it. The contrarian doesn't have a drink.

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Several hours later...

Bartender: "I'm pretty sure she's consistent. My shift is over, I'm going home."

The bartender has a higher consistency strength than the mathematician.

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An infinite amount of time passes. An oracle walks into the bar.

Oracle: "The mathematician never decided, the contrarian never had a drink, and the Bartender was consistent too."

She speaks in an ancient and impossible tongue which nobody can understand. Everyone else is dead. The oracle is uncomputable.

A contrarian oracle walks into the bar....

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Daily Dose of Definition: Well-Founded Relation

A binary relation R on a set X is well-founded if every non-empty subset of X has an R-minimal element.

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Daily Dose of Definition: Henkin Construction

A method of proving completeness for first-order logic by extending a consistent set of sentences to a maximally consistent set and constructing a term model from it.

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Daily Dose of Definition: Cut Rule

The cut rule in sequent calculus allows one to derive Γ, Γ’ ⊢ Δ, Δ’ from Γ ⊢ A, Δ and Γ’, A ⊢ Δ’. The cut-elimination theorem states that this rule is admissible: any proof using it can be transformed into one that does not.

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Daily Dose of Definition: Löwenheim–Skolem Theorem

If a first-order theory has an infinite model, then it has models of every infinite cardinality greater than or equal to the cardinality of its language.

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Daily Dosage of Definition: Compactness Theorem

A set Σ of first-order sentences is satisfiable if and only if every finite subset of Σ is satisfiable.

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Forget away brother.

Interesting math fact of the day #156:

Either p is a square mod q and q is a square mod p or neither is true.

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Daily Dosage of Definition: Admissability

A rule is admissible in a formal system if whenever the premises are derivable, so is the conclusion, but the rule is not necessarily a part of the system’s axioms or inference rules.

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It is no treason, second root of the sum of my square and the square of 5 and the square of 1, is the very one you defend.

Take the difference between my square and his own, and coupled with his own square you shall see that it forms a contradiction to his very words.

It is the duty of us, as his friends, to, in true brotherly fashion, never let this become forgotten.

You do not betray him. This is our duty.

Interesting math fact of the day #156:

Either p is a square mod q and q is a square mod p or neither is true.

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Daily Dosage of Definition:

Sequent

A sequent is a formal expression of the form

Γ⊢Δ

where Γ and Δ are (possibly empty) sets* of formulas. It expresses that the conjunction of formulas in Γ entails at least one formula in Δ.

*(most definitions here add in that the set must be finite, but some forms of logic do not have this requirement.)

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Are we not your other brothers?

Interesting math fact of the day #156:

Either p is a square mod q and q is a square mod p or neither is true.

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25 is a square mod 73, but 73 is not a square mod 25.

(I'm here!)

Interesting math fact of the day #156:

Either p is a square mod q and q is a square mod p or neither is true.

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dear fbi agent. me suddenly laughing maniacally and having disney villains manneurisms is not out of insanity, i just figured out the math problem

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I'm going to hold a survey, please reblog for sample size :]

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Luckily 0 still won.

Some people think the natural numbers should start at zero, and get very worked up about this somehow being the "true", most logical, practical, etc. definition.

Some people think the natural numbers should start at one, and get equally worked up about this being the "true", most logical, most practical, etc. definition.

Unfortunately, to date -- and despite my vigorous letter-writing campaign -- the hidebound reactionaries that comprise the mathematical elite have rejected out of hand my elegant compromise position.

Apparently it "misses the point completely" and "wouldn't make any sense" and "1/2 isn't even an integer".