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Vladimir_Nesov comments on The Last Number - Less Wrong
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Comments (57)
Huh?
Prove to me that this can't happen :-)
And feel free to make use of any axiom of arithmetic that you want...
I like your idea of defining "The Last Number" to be the smallest number which, if it exists, directly contradicts Peano arithmetic when summed to one. It satisfies my inner-child desire to think "but what if..."
Anyone who who hasn't yet, check out Hilbert's second problem.
We really don't know if arithmetic is consistent... perhaps that's no big failure, but what's worse is that we have not even found a simpler logical system that implies its consistency (so then it would rest on the consistency of the simpler system).
EDIT: Actually apparently we have (see Stuart's reply), but I haven't followed up on it yet...
Look up: http://en.wikipedia.org/wiki/Self-verifying_theories
"for instance there are self-verifying systems capable of proving the consistency of Peano arithmetic."
Interesting. Why would one want a theory that can prove its own consistency? This doesn't really tell us anything, because inconsistent theories can prove any statement, including their own consistency.
I don't agree that it doesn't tell us anything ... an inconsistent theory can prove all statements, yes, but not all with proofs shorter than its shortest proof of a contradiction. That is, if Peano arithmetic has a trillion-line proof that 3.2 is an integer, then it can prove anything in about a trillion and two lines... but it can't prove everything as easily as say (1+1+1)(1+1)=(1+1+1+1+1+1). It might be something special when a theory can prove its own consistency elegantly, sort of the way a human can have non-zero credence that s/he is usually rational.
I'm not sure I know what a proof of consistency is, except that I wouldn't want an inconsistent theory to be capable of one.
An inconsistent theory can prove anything - including its own consistency.
I'm familiar with "anything statement can be derived from an inconsistent theory" but I really am confused by how any such derivation could be a proof of consistency. If proofs of consistency are possible for inconsistent theories then how exactly are they proofs of consistency?
It's a "proof" in that it follows the formal rules of the proof system. You can "prove" anything if your rules are sufficiently ridiculous, but that doesn't mean the proof actually means anything.
Thanks.
If I tell the truth, I cannot say: "I lie".
But if I lie, I can say: "I tell the truth".
So, a theory's proving its own consistency is strong Bayesian evidence that it's inconsistent ;).
If that's all you know about the theory, I'd say yes - but not "strong" evidence.
I probably should have given more than just a winkie to indicate that I was joking.
Huh, did not know! Unfortunately they don't imply the results of Peano arithmetic, but that would be asking for too much, heheh.
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?
What about it?
Can that statement be proved if arithmetic is inconsistent?
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.
It is interesting that the relation "on the same successor (adding 1 repeatedly) number line" isn't expressible in first-order predicate calculus (the type of logic that Godel's thm. is talkign about).
It is also interesting that there is at least one model of that first-order logic+Peano axioms that has infinitely many disconnected successor lines - http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem (Lowenheim-Skolem) . That is, starting with 0, adding 1 repeatedly, you can never reach most of the numbers.
But what the story seems to posit is merely a finitely large number on the same number line as 0, whose successor isn't on that line. But the line is defined (in a statement too powerful for 1st order logic, by me) as all the things reachable from 0 by repeatedly adding one, so that's impossible.
Or, looked at another way, you allude to an algorithm for adding 1 to what is merely a very long string of digits. Ripple carry counting, for example, will always take digit strings to digit strings. An algorithm is likely telling you much more detail than Peano+1st-order does.