Sleight of Mind by Matt Cook

Author:Matt Cook
Language: eng
Format: epub
Tags: Paradox; mathematics; logic; philosophy; social sciences; physics; Zeno; infinity; supertask; probability; social choice; game theory; self-reference; induction; geometry; special relativity; quantum mechanics
Publisher: MIT Press

This says that for a number x to be prime, it has to be greater than 1. Additionally, x is not divisible by any numbers less than or equal to itself, other than by 1 and itself.

The prime number function relied on the definition of divisibility. To reach the second rung on the conceptual ladder, the first step had to be reached. So the rungs continue, each building from previous rungs, each attaining a new level of complexity—but each able to be encoded and arithmetized, and also expressed (albeit in ways that are hard to follow) in terms of very simple building blocks. Among the other rungs are the factorial function; a function used to identify the nth prime number by order of magnitude; functions that identify objects as constants, variables, terms, and formulas; and even rungs that identify logical implication. The ladder climbs very high indeed.

The penultimate rung, rung forty-five, formulates a particularly important concept. It introduces a proof(x, y) relation, which says in plain English: “x is the Gödel number of a proof of a formula whose Gödel number is y.” The previous sentence is unwieldy, but it’s critical to understanding what follows. Take a few moments to read and reread it. And remember that because each rung of the ladder is perfectly arithmetized, supported at the bottom and every step of the way by basic primitives, the statement proof(x, y) has a Gödel number itself for Gödel numbers x and y.

The highest rung of the ladder, rung forty-six, is the provability function, provable(x), which says “x is (the Gödel number of) a provable formula.” It is constructed from its conceptual predecessor like this:

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