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Vladimir_Nesov comments on The Last Number - Less Wrong
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Comments (57)
What can't happen? Before arguing with a statement, one shall do well understanding what meaning is intended (if any).
That there is no integer that, when added to one, produces 4.2... (or alternatively, that arithmetic is consistent).
Again: What does that mean? You are not offering explanations, only words, curiosity-stopping ruses. "4.2"? What kind of object is that? Is it even in the language?
4.2 is a number such that when multiplied by 5 yields 21.
So one interpretation is that a Turing machine implementing successive additions of 1 in Peano arithmetic, starting at 1, storing the results, and multiplying each result by 5=1+1+1+1+1, might eventually output the result 21=(1+1+..+1), which is easily shown to be a contradiction.
If you're not happy with what is meant by "contradiction", then lets just say it would be extremely surprising if that happened, and a lot of people would be very upset ;)
It is meaningful to pose the possibility that our map has a certain very surprising property. In particular, we can consider the possibility that one of our cartography tools, which we thought was very reliable, doesn't behave the way that we thought it did. The story gives one partially-conceived manner in which this could happen.
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.
4.2 - 1 = 3.2. Simples.
And redefine 3.2 to be an integer. Even more simples!
This was my first reaction. But one way of showing that arithmetic is inconsistent would be to show that under it's axioms some very very large number (edit: I mean integer, thanks Stuart) was equal to 3.2.
Some very large integer.
Huh, integer. I don't know how that got past me when I wrote that.