The largest number is about 45,000,000,000, although mathematicians suspect that there may be even larger numbers.
[45,000,000,001?]
Prove to me that this can't happen :-)
And feel free to make use of any axiom of arithmetic that you want...
Prove to me that this can't happen :-)
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 ;)
Again: What does that mean?
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"
I doesn't mean anything. It's a fiction about the breakdown of arithmetic.
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.
If arithmetic breaks, then any conclusion is possible, any statement is true. Including such things as:
Last Number + 1 = "the sensation female orangutans have when scratching your back during unnaturally hot winters on Mars"
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.
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"?
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.
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.
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.
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?
So, a theory's proving its own consistency is strong Bayesian evidence that it's inconsistent ;).
Actually, it is a serious point. If you choose thories at random, according to some universal prior, then a lot of them are going to be inconsistent. And most of the theories that can quickly prove their own consistency are the inconsistent ones. So this does provide some information (depending on how the consistency proof was arrived at, of course).
That was pretty much what I was getting at. But since I'm not in a position to quantify how strong the evidence is, I took the cheap route of making it a joke :).
Huh, did not know! Unfortunately they don't imply the results of Peano arithmetic, but that would be asking for too much, heheh.
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.
Greg Egan also wrote two amazing stories about inconsistent arithmetic: "Luminous" and "Dark Integers".
This story was a bit weird, main characters seemed to take it a bit too strongly that arithmetic failed.
I mean, there is some sense in which 1 does not equal 2. Even if arithmetic fails to define a framework where they're separate, that would simply mean that we need to redefine stuff, figure out a better way to describe our intuitive way of understanding how they're different, and thus changing the framework towards that intuitive understanding.
It would of course be a bit embarrassing, when much of the work had to be revisited and all that sort of stuff. But the characters seemed to be plain overreacting.
Even if arithmetic fails to define a framework where they're separate, that would simply mean that we need to redefine stuff, figure out a better way to describe our intuitive way of understanding how they're different, and thus changing the framework towards that intuitive understanding.
Man that would be hard, though!
I imagine that if this happened, math would become, at least for a while, an empirical science. People would study the derivations that led to contradictions. Proscriptions of such derivations would be taken as provisional, "this-is-how-our-universe-seems-to-work" axioms.
(Of course I don't actually expect arithmetic to ever be shown to be inconsistent.)
This story was a bit weird, main characters seemed to take it a bit too strongly that arithmetic failed
ROT13: Uhu? Bar znva punenpgre gbbx vg uneq. Rirelobql ryfr jrag "uhu, gung'f jrveq", naq pbagvahrq ba. Guvf znva punenpgre jnf n zngurzngvpvna jub unq onfrq ure frafr bs gehgu naq pbeerpgarff naq zrnavat ba zngu -- zngu juvpu unq qvfcebirq vgfrys.
The story is supposed to be the icing, not the cake.
If I read an elaborate story that describes something that is apparently mathematically absurd (big integer + 1 = a small decimal) then I expect the post to go on to illustrate or explain some counter-intuitive truth or at least a speculative mathematical theory.
Problem is, everything collapses with one contradiction. So rigorously, there is nothing more to tell.
Now you can conceive of some sort of world in which the truths of mathematics are empirical, contingent and changeable, for instance, so that one contradiction is not much of a biggie. That would be quite fun, but, alas, I don't have time for much fiction nowadays. Maybe someone else could try?
There are systems (relevant logic, for example) which do not collapse under one contradiction - to some extent, the fragility of classical logic is due to very strong assumptions that were built into it about how powerful math will turn out to be before Godel's incompleteness and other undecidability results were discovered (and, at least in pop math like you and I are familiar with, they're still not really fully digested).
Charlie Stross has also written post-arithmetic-consistency sf: "Dark Integers". I'm a little surprised that Ted Chiang's story didn't contain any attempts to build devices to exploit the inconsistency.
Charlie Stross
It's a Greg Egan story actually.
"But a few hours ago, a cluster of propositions on our side started obeying your axioms.”
Double "huh?" -- I am intrigued, but puzzled. Is this supposed to be an allegory about how doing too much math isn't good?
No, it's just trying to envisage a world in which arithmetic really was inconsistent, and show how non-sensical it seems.
May I be enlightened as to what this mysterious "consitency" mentioned in your tags section is?
"...90116633393348054920083..."
He paused for a moment, and licked his recently reconstructed lips. He was nearly there. After seventeen thousand subjective years of effort, he was, finally, just seconds away from the end. He slowed down as the finish line drew into sight, savouring and lengthening the moment where he stood there, just on the edge of enlightenment.
"...4...7...7...0...9...3..."
Those years had been long; longer, perhaps, in the effects they had upon him, than they could ever be in any objective or subjective reality. He had been human; he had been frozen, uploaded, simulated, gifted with robotic bodies inside three different levels of realities, been a conscript god, been split into seven pieces (six of which were subsequently reunited). He had been briefly a battle droid for the army of Orion, and had chanted his numbers even as he sent C-beams to glitter in the dark to scorch Formic worlds.
He had started his quest at the foot of a true Enlightened One, who had guided him and countless other disciples on the first step of the path. Quasi-enlightened ones had guided him further, as the other disciples fell to the wayside all around him, unable to keep their purpose focused. And now, he was on the edge of total Enlightenment. Apparently, there were some who went this far, and deliberately chose not to take the last step. But these were always friends of a friend of an acquaintance of a rumour. He hadn't believed they existed. And now that he had come this far, he knew these folk didn't exist. No-one could come this far, this long, and not finish it.
"...2"
There, he had done it. He had fully pronounced, defined and made his own, the last and greatest of all integers. The Last Number was far too large for standard decimal notation, of course; the first thousand years of effort, while there were still many other disciples around, filling the air with their cries and their joys, had been dedicated entirely to learning the mathematical notions and notations that were needed to correctly define it. But it seemed that for the last ten trillion digits of the Last Number, there was no shorter way of stating them than by listing them all. Entire books had been written about this fact, all untrue or uninteresting (but never both).
He willed a pair of lungs into existence, took a deep shuddering breath, and went on:
"... + 1 ..."
Had he been foolish enough to just list the Last Number, then he would have had to spend another seventeen thousand years calculating that sum - or most likely, given up, and contented himself with being semi-enlightened, one who has seen the Last Number, but not the Final Sum. However, he had been building up the mathematics of this addition as he went along, setting up way-stations with caches of buried theorems and lemmas, and carrying the propositions on his back. It would take but a moment to do the Final Sum.
"... = 4.2"
It was finished. Gödel had been more correct than that old Austrian mathematician could ever have imagined. Two integers, summed according to all the laws of arithmetic, and their sum was not an integer. Arithmetic was inconsistent.
And so, content, he went out into the world as an Enlightened One, an object of admiration and pity, a source of wisdom and terror. One whose mind has fully seen the inconsistency of arithmetic, and hence the failure of all logic and of all human endeavours.