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Stuart_Armstrong comments on The Last Number - Less Wrong
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Comments (57)
I doesn't mean anything. It's a fiction about the breakdown of arithmetic. If arithmetic breaks, then any conclusion is possible, any statement is true. Including such things as:
Last Number + 1 = "the sensation female urangutangs have when scratching your back during unnaturally hot winters on Mars"
The concepts discussed in the fiction are still supposed to mean something. It's like with hypotheticals: they aren't asserted to be probable, or even apply to this our real world, but they weave certain ideas in people's minds, and these ideas lend them meaning.
You may make certain statements about the language, like "all well formed formulas of this particular system are theorems", but you can't cross over into arbitrary real-world statements.
What about the statement of the type: "the reals are a model of peano arithmetic"?
Nice pun.
Pun? Where?
"Arbitrary real-world statements", "the reals are a model of peano arithmetic".
What about it?
Can that statement be proved if arithmetic is inconsistent?
From an inconsistent system (such as ZFC would be if arithmetic were), yes. An inconsistent system has no models.
That would imply that 4.2 is an object of Peano arithmetic; but there is a simpler way of getting this.
The first-order statement: "there exists an x, such that x times 10 is 42" can be phrased in artithmetic. Therefore if arithmetic is inconsistent, it is true. And I define 4.2 to be a shorthand for this x.