Infinities

Factual Questions
81

Oh my. Yeah, well, I was trying to make it jive with the way I presented cardinality. The point still stands, however, that we can avoid the sticky notion of a ‘list,’ of which IMHO, ianzin is reasonably skeptical. This is why we speak of functions, no?

PLUS! Once we know this definition, we can prove 2^k > k for every cardinal k. Hint for those playing at home: it is the same proof that 2^(aleph null) > (aleph null).
Oh, and Indistinguishable? I’ve never heard the usage ‘inhabited set.’ I like it.

82

I’m tempted toward two related answers:

  1. I could take issue with the stipulation that L could be relevantly like English. For if it can be like English in a relevant way, I see no reason we shouldn’t allow it to be English. But English-heterological is definable in English. This leads to paradox, to be sure, but that does not mean the term is indefinable. It just means its definablity leads to paradox. But since English-heterological is definable in English, the argument you gave shouldn’t be interpreted as applying to all languages whatsoever, and in particular, shouldn’t be interpreted as being applicable to “a language like English.” This is to take issue with the stipulation contained in line one that L could be a language like English.

  2. But having given that argument, I might as well just go whole-hog and say that just because saying L-heterological is definable in L leads to paradox, that does not mean L-heterological is not definable in L.

In response to 2, though, I might say (I guess to myself :slight_smile: ) the following:

The paradox we run into with L-heterological is of the general form “Both if A then not-A and if not-A then A.” But from this we can derive a contradiction. And so no sentence of that general form can be true.

And if no sentence of that form can be true, then we can treat our having reached a paradox as a basis for a reductio argument.

This swings me back to the intuition I express in response 1 above: Even though I’ve just seen what is apparently a sound proof that English-heterological is not definable, nevertheless, I can define, in English, the term English-heterological.. (A term is English-heterological if it is not an instance of its own definition.)

I think maybe a response to that point would try to argue that English is not definable in English, so that while “heterological” may be definable in English, English-heterological is not. But I’m now finished stumbling through this area where angels (and people who actually know what they’re talking about) fear to tread.

-FrL-

83

For me, the issue is not whether “indescribable numbers” exist or not. Mathematicians are free to create or construct any entitites they like. And if they do so, the entity in question exists. For me,the more important question is "what hole is filled by indescribable numbers.

For example, the concept of imaginary numbers allows us to give an answer to the question “what is the square root of negative one” and to come up with an interesting and consistent system.

I’m not a mathematician, but I would reject the unspoken assumption that every question has a yes or no answer; that every claim is true or false.

Here’s a simpler paradox:

Fundamentally, how is that any different from the paradox you described? And in any event, what does it matter? Even with indescribable numbers, you still have liar type paradoxes.

Again, the real question (for me) is whether indescribable numbers fill some hole, just as negative numbers and imaginary numbers fill a hole.

84

That could conceivably be a fine path to go down; I have sympathy for that. But it would certainly have ramifications to adopt such a position and working out the details of a corresponding logic which still skirts the paradoxes can be tricky.

It is essentially the same paradox, just not applied to the particular task of constructing “indescribable” entities. It’s good that you see that they are fundamentally the same, though. The existence of indescribable numbers is not meant to prevent liar paradoxes; rather, what I was trying to illustrate is that indescribable entities are an inevitable consequence of certain kinds of reasoning reminiscent of those in liar paradoxes, but tamed so as to fall shy of producing actual contradiction (except, of course, when bundled with the assumption that all entities are describable).

As I (and ultrafilter) said before, they don’t in themselves fill a hole any more than the existence of everywhere non-differentiable functions fills a hole; they’re more just a side effect of assumptions/decisions taken with other goals in mind. (That having been said, speech about functions indefinable in particular systems often fills a kind of hole, in that, fixing the system of definition to particular things such as “That which can be defined by a computer program” or so forth, speech about indefinability becomes speech about concepts such as uncomputable functions, etc., which certainly comes in useful in analyzing the obviously related phenomena)

(I was composing a reply to Frylock’s post as well, but it was getting rambling at about the same rate that I was getting tired, so I think I’ll save it for finishing up tomorrow.)

85

Well, the answer given by complex numbers seems to be nothing more than “well, it’s the square root of negative one (which we’ve just postulated up and perhaps given the name i)”. But, yes, the resulting system is interesting.

Indescribable numbers also allow us to do such things: they let us give an answer to “What do you get when you put all the describable binary sequences in a table, ordered alphabetically by description, and then run through the diagonal and flip all the bits?”. The answer being “Well, clearly, you get some indescribable sequence”. Not terribly intrinsically insightful or informative, but systems which deal with such topics can be very interesting as well.

86

But didn’t you just describe it? I guess not, since the description would require you to loop forever to come up with the number.

I think I see the point now. If you want to limit your system to “describable” numbers, you need a precise definition of “describable” Once that’s done, it’s possible to generate a number that is not describable. So there’s a hole in the system. Not as obvious as the hole occupied by negative numbers, but it’s still there.

87

Yes. Now you see it.

Further points: To say “But didn’t you just describe it?” is quite rightly to point out the unsettling question at the root of the apparent paradox (because I clearly did describe it, and yet it also clearly cannot be describable) to which responses are tricky. A standard response relies on a strict stratification of object language from metalanguage (so that when we speak about “undescribable in language L” we can only do so via some other language M; our description in M of something undescribable in L doesn’t cause any contradiction), but this is quite unsatisfying, as Frylock notes above, since it certainly seems like English is able to speak about English descriptions without requiring us to pass to some other language of meta-English. I’ll… say more later.

Point 2: When you said “I guess not, since the description would require you to loop forever to come up with the number”, that was nicely put; it brings to mind that one good example of application of these concepts is when we precisify our language of definition to be some programming language. What these indefinability results end up showing, in that context, is that there is no computable surjection from strings (representing source code) to the computable maps from strings to {true, false}. However, there is a computable surjection from strings to a larger set, that of computable maps from strings to {true, false, loop forever}. The key thing is that “loop forever” is a fixed point of computable negation; if you have a program that sends true to false and false to true, it sends input which loops forever to output which loops forever. If you legitimize things like “loop forever” as outputs, then you legitimize the possibility of internalizing the description “Run down the diagonal and flip everything”. And this will take the bite out of the diagonalization arguments, so to speak. Or, more positively, it can be seen as turning the diagonalization arguments on their head, converting them into not reductios but fixed-point existence theorems. I… might say more about this later too. Maybe.

(I’ve stayed up far too late and gotten far too little work done)

88

(Tying 1 and 2 together just to make one thing clear: Note that while the description I gave would require you to loop forever if it were itself in the language it was speaking about, it does not automatically require you to loop forever if it is itself in some other language. The indirect self-reference vanishes; there would then be no recursion since the description would no longer be speaking about (a language containing) itself)

89

Yes, it’s a bit more evocative than the dry “non-empty”. Actually, the terminology exists because of an intuitionistic distinction between “sets which don’t have 0 elements” (non-empty) and “sets which have at least one element” (inhabited), though it didn’t matter too much in the classical context where I used it above.

(I really don’t mean to be an intuitionistic shill; it’s just the most well-known and most widely-applied of the various “non-classical” logical systems interesting to speak of, and happens to naturally model many of the phenomena that relate to the discussions in this thread)

90

I’m confused about this: “But didn’t you just describe it? I guess not, since the description would require you to loop forever to come up with the number.”

Can’t Indistinguishable’s number be written as the limit of a convergent series? Does this mean that a limit does not necessarily describe a number?

91

Indeed, like all real numbers, it can be written as the limit of a convergent series of rationals or such things. The trick is, you still have to describe the series whose limit you want to refer to. The number’s indescribability (in language L) will tell us that the series you have in mind is also indescribable (in language L).

92

Not at home right now to salvage the post I was writing yesterday, but a quick response anyway.

As it happens, when I said “like English”, I did intend for actual English to be one of the possibilities; that is, I agree with your initial “I see no reason we shouldn’t allow it to be English” (I basically said “like English” as not to worry about “Which English?” and “Why not German?” and so forth). Though you then go on to a sort of Moorean counter-argument, I suppose (to indefinability in the context of English). Rather than actually point out some logical flaw in the proof, you claim that its conclusion is too implausible to be considered, and thus rejects its premises on precisely and seemingly no more than the grounds that they lead to its conclusion. (Though it seems odd to deny line 1, that L could be a language like English, when I haven’t assumed anything about L yet other than that it’s a language. Surely, if implicit constraints on L arise, they must be pointable to only at later lines in the proof?)

I’m not entirely sure what you’re saying here. I guess you are considering the possibility of simply not always accepting reductio from “X leads to paradox” to “Not X”?

Well, even if you don’t consider yourself to know what you’re talking about, the things you’re saying seem reasonably close to what knowledgeable people could say. Everyone agrees that English can define hetorologicity in the abstract, and that it can define L-heterologicity for many L other than English. What those of Tarskian persuasion would deny is only that it can define English-heterologicity, and specifically, what they will conclude (as you, I think, essentially do by considering “English” undefinable in English) is that the interpretation function, relating English definitions to the objects they define, is one which is undefinable in English. To speak of such semantic features of English, they would say, one must do so from the vantage point of another language (a meta-English). Of course, what other language could we hope for with more expressive power than a natural human language like English? Tarski was understandably led to conclude that there is simply no possibility of giving a full and logically coherent account of semantic notions for natural language; that people could try to speak of such things, but that this everyday language could not be logically formalized.

Now, to me, it seems clear that English does have the expressive ability to speak of its own semantic concepts (don’t think, but look!). It’s only through detached philosophizing that we could train ourselves blind to the fact that English speakers speak of such things readily. But how can one give a logically coherent account of this, in light of the apparent paradoxes? Well, it seems to me the thing to do is to try to discover the logical calculi that do accurately model the structure of this everyday language, rather than try to shoehorn it into a preconceived system of logic it fails to fit. What sort of system would properly describe the role of “This sentence is not true”/“Is ‘heterological’ heterological?”/etc. in English? I don’t know; I’m not sure exactly what the English semantics of those should or could be taken as, other than that they clearly do manage not to cause explosive paradox (English can express those things, and English semantics don’t resultingly trivialize). But though I don’t know all the details, it’s clear there is some interesting, nontrivial structure here, gleamable from the actual practice of English.

93

I think we have to do some kind of “formalized pragmatics” to capture this kind of thing.

A few weeks ago I was toying with the idea of analyzing the move towards heierarchalization of language in response to (what I was thinking of specifically at the time) Berry’s paradox as of a piece with the move towards forbidding use of pejoratives when referring to people of other social classes than one’s own. I forget how it went exactly (and in my judgment it didn’t actually end up working) but the idea was that in both cases we avoid a particular use of a term based not on semantic considerations but rather on pragmatic considerations. In the case of pejoratives, arguably it’s true that, say, Hitler is a Kraut, it’s just that one shouldn’t use that term. The reasons not to use it aren’t semantic, they’re pragmatic. And the pragmatic rules involved can probably be given a fairly clear articulation. In the Berry’s Paradox case, the truth or falsehood of the various paradoxical statements involved is nowhere near as clear, but it’s fairly clear that the reasons for avoiding usages or reinterpreting them heierarchically (which is really another way to avoid certain usages in favor of others) are pragmatic, and I am tempted to argue, are not semantic. (The formulation of the pragmatic principles upon which the moves are made may include semantic concepts–for example "We should avoid that usage because it leads, by the rules of our semantics, to an infinite regress, or an undecideable sentence, or whatever–but the force of the principle is pragmatic–“we shouldn’t do X because it leads to undesirable result Y”–rather than semantic–“this usage applies because it means such-and-such and the thing we’re talking about is such-and-such.”

But I think people generally think that pragmatics can not be formalized. But I wonder if some linguists could be construed as pursuing just such a project?

[/ramble]

-FrL-

94

There are actually several definitions of infinity.

  1. The concept of the largest thing ever conceivable. You can’t multiply this concept by 2, you can’t do anything. It is on the level of god. This is sometimes referred to as “potential infinity.”

  2. Something that isn’t a concept, but something that can be thought of as a value, a value that is bigger than any real-world, finite thing. This was referred to as “actual infinity.”
    Unfortunately, across a span of decades sometime a century ago, some mathematicians came along and didn’t grock the philosophical distinction. In fact, they must have been pretty frustrated by it all, being aesthetically drawn to the concept of the actual, the math-like in #2 but caught in the intuitive, layman idea of #1. Despite their confusion, despite their outright defiance and claim that there were never two concepts to start with, they were able to prove theorems, to even paint a seemingly self-consistent picture (despite some startling claims).

Which brings us to the modern day in when people talk about different ‘sizes’ of infinities. You were spot-on to see the absurdity in their words, because concept #1 does indeed absolutely preclude any “infinity” being larger than any other “infinity.” It is impossible to say such a thing. Yet concept #2 perfectly allows it. There can be any number of actual numbers bigger than anything in our world, and they can be bigger than eachother or what-not.

Finally, there is a third concept that is similar in spirit to #1: “the largest finite number.” This thing, although not an actual number, is smaller than #2.

Notice that all these things look a lot the same, and from the point of view of our mortal selves we may not even be able to tell the difference. Yet the difference exists. For example, when Cantor first talked about cardinality, he, by a slight-of-hand, mixed #3 and #2 with each other. He talked of the “quantity of natural numbers,” which is definition #3. This thing isn’t any actual number – since with a number you can always say “well, add 1 to that” – just a concept. Yet Cantor did stumble upon an actual number (#2) that can be used to count up the points in a line. Cantor proceeded to find sets that his #2 couldn’t count up, but found other #2’s which could. Those are his Alephs. A while later, Godel came along and found certain procedures which have #3 (or #2) number of steps. But, thinking they were #1, deemed them “impossible.” Thus, Godel, like fn Xeno two millenia before him, successfully proved that certain things which are possible, are in fact impossible.

The confusion about infinities has a very long history. In ancient Greece, in fact, there was a thinker named Xeno who, in another charming anachronism, had a thought experiment. His experiment pitted a turtle vs Achilles in a foot-race, with the turtle given a head-start. Once the race is started, Achilles quickly enough runs to the spot the turtle was. However, by now the turtle, slow thought it is, was a few inches ahead. A moment later Achilles traverses those few inches but finds the turtle to be leading by 10 thous (a hundredth of an inch). Achilles again catches up, but the turtle is ahead again. To sum a long story short, Xeno proved that Achilles couldn’t win. What he proved, in fact, was that to get to anywhere you have to pass through an infinity of points. But which infinity? And is it really impossible to pass through it? Well, the number of points in a line is actually #3. #1 can never be passed through, but #3 can be. (Err… it’s a bit more confusing than that, but this gets the point across).

95

You get a similar name in type theory - inhabited types.

96

These are not definitions of infinity. They are misunderstandings of infinity.

Infinity is not a value or a number or a thing. It merely means unending or unbounded. That’s a wholly different concept from anything you write.

In fact, the cause of most of the problems people have had with infinity is that they tried to use definitions like yours. But these are wholly wrong in every way, and so lead nowhere or to complete confusion. They must be abandoned before any progress can be made.

Mathematicians abandoned them many thousands of years ago. Even the Greeks understood that no such concept as the largest finite number was possible.

97

Yeah. And since most type theories have an underlying intuitionistic logic (via the Curry-Howard correspondence), we could view the non-empty vs. inhabited distinction in that context as well. A type T is inhabited if there is a closed term of type T, while we could say it is (internally) non-empty if there is a term of type (T -> |) -> |, where | stands for the type with no elements (what languages more sensible than C sometimes call “void”, though C bastardized that term to essentially refer to the one-element type). For those who like to think of maps from T to | as continuations accepting arguments of type T, it should be clear how the collapse of the distinction in classical logic corresponds to the ability to move back and forth between the two types in languages containing such features as call-with-current-continuation (indeed, type theories containing call/cc have underlying classical logics).

Also, I’ve noticed that, for some reason, any Internet discussion of “Zeno” will involve a large number of people who think it’s spelled with an ‘X’. Conflation with the warrior princess?

98

What I would be remiss in not pointing out are certain well-known systems still within traditional mathematically-oriented semantics which allow for languages which are semantically closed in precisely the way the indefinability results would suggest they can’t be. Most famously, for example, there is Kripke’s Theory of Truth*. This approach essentially uses a non-traditional, multi-valued logic in which True and False are augmented by a new truth value of Ungrounded (though Kripke himself was quite adamantly against this perspective). Using his techniques, “This sentence is not true”, “This sentence is true”, and so forth can all be seen as being ungrounded, basically precisely because naively determining their status would involve “looping forever.” The work is, ironically, heavily based on the fixed-point theorems of order theory which descended from the Knaster-Tarski theorem. There are some quirks of this system (it requires all logical connectives to be monotonic with respect to certain orderings (usually with U as a least element, and T and F as maximal elements), the result of which is that negation has certain properties one might not naively expect; the statement “x is not equal to T” fails to be true when x = U, being rather merely ungrounded)), but it is quite aesthetically appealing in many ways.

*: These things all get rather misleadingly labelled and conflated as being essentially about Truth, that impressive head-liner of a word, when the matter at hand is really something more broad; the issue is definability of interpretation functions in general (essentially the surjections examined by Cantor’s Theorem), and we needn’t narrow our focus to simply the particular ones handling descriptions of propositions/truth values [e.g., all the essential issues of liar-type paradoxes are motivated just as well with"3 plus the integer referred to by this phrase", which doesn’t really have anything to do with truth].

99

Isn’t it the convention (God knows English doesn’t have spelling rules) in English to spell words to start with the letter “x” instead of “z”? E.g. xylophone, xenophobia (fear of proofs regarding the infinite?), xanax (the treatment?) etc. Zeno is a Greek name, and I think we spell it with a Z only because they spelled with a Z. Anyway, sorry for the mistake, but it did NOT have anything to do with xena!

100

xylophone and xenophobia both come from Greek words that start with their equivalent of the letter x. We pronounce them with a z sound because the x sound is not used to start words in English.