Diagonal argument is not an assumption. You can, of course, disagree with any of the assumptions it relies on, and some mathematicians do so.
Some reject the axiom of infinity, and the use of actual infinities in general. Others reject the axiom schema of specification, or rather put some limits on the formulas it can be used with. (There may be other objections I haven’t heard of.) These are not referred to as crackpots. They are choosing a different collection of axioms—i.e. their “sets” are not the same as Cantor’s “sets”—and they can show the exact step where Cantor’s diagonal argument fails to apply in their system.
I think the label crackpot is applied to people who vocally disagree with the argument, but can’t pinpoint the place where it fails; in other words they have a strong opinion on something they don’t understand.
.
I also think it’s fair to say “I disagree with the conclusions of Cantor’s diagonal argument, although I can’t explain where exactly it fails, because these things are too abstract for me—it’s just that the outcome feels wrong according to my intuition, so there is probably a step I would disagree with, even if I can’t tell which one it is”.
Such people might even be happy to learn about Skolem’s paradox, which used to drive “the opposite camp” crazy. “There are countable models of ZF set theory that satisfy the theorem that there are uncountable sets.”—which at first sight seems like absolute nonsense, but it’s a logical consequence of following the axioms.
Nonetheless, modern set theorists are okay with this, too. So the problem is not with disagreeing with the outcomes per se, but with not being able to play the game according to its rules.
Let me explain. I get the following impression. When making basically any other argument in philosophy of mathematics, people are generally highly disinterested, but as soon as the debate touches on issues involving transfinite set theory, like diagonalization, at least people online (on blogs, Wikipedia, Stack Exchange etc, though journals might be similar) are quick to heavily downvote comments and declare people to be “fringe” or “cranks”. Even though they have opinions that were completely mainstream until a hundred years ago, and were defended by a number of well-known mathematicians and respectable philosophers like Wittgenstein.
To give a more specific example: Adding a “criticism” or “controversy” section to certain Wikipedia articles on topics around infinity in set theory seems almost as impossible as writing a Wikipedia article covering certain taboo research involving race and IQ.
The only way to argue in a different direction is to word things very carefully and conservatively to not step outside the narrow Overton window. Optimally one shouldn’t be directly arguing for anything and instead just engage in explaining arguments other respectable people like Weyl or Wittgenstein have hopefully already made. Failing that, one should always introduce statements with hedges like “From the perspective of finitism, (diagonalization / …) is of questionable validity because...” However, just saying directly “(Diagonalization / …) is of questionable validity because...” is seen as beyond the pale, cranky, or otherwise unacceptable.
I think the reason for these hostilities is that in these disagreements there is an implicit battle going on over where to draw the line between mathematics (the land of definitive facts) and philosophy of mathematics (where nothing is certain and diverse positions may be defended). The reason why hedging a statement with “from the perspective of finitism, …” makes it more acceptable, is that this phrase places it in the realm of philosophy, where things may be questioned.
But in philosophy-of-mathematics land, this same thesis is called Hume’s principle. When referred to by this name, disagreement is suddenly very much allowed. Not only that, the principle is generally seen as resting on a somewhat shaky ground, or anyway being in need of further justification. The only reason these two inconsistent perspectives on the same topic can co-exist in Wikipedia is that they simply don’t acknowledge each other. But if someone were to propose to merge the articles, the philosophy perspective would likely lose.
Regarding these points:
Diagonal argument is not an assumption. You can, of course, disagree with any of the assumptions it relies on, and some mathematicians do so.
I’m really not aware of any assumptions one can disagree with here while still being perceived as respectable. The most one can say is something like “Finitists might argue that given any way to construct the infinite table, the rows must approach infinity faster than the columns, in which case the diagonal number is always on the table” or something like that. But even this would be suspect and in any case not constitute a specific assumption one is permitted to disagree with.
Some reject the axiom of infinity, and the use of actual infinities in general.
But these assumptions are only directly related to Cantor’s power set argument (“Cantor’s theorem”), which is different from his diagonal argument. The former involves (power) sets, the latter real numbers. The latter is also not directly related to Skolem’s paradox.
opinions that were completely mainstream until a hundred years ago, and were defended by a number of well-known mathematicians and respectable philosophers like Wittgenstein.
Some respected scientists of the past have defended elan vital or aether. And yet, if someone did the same today, they wouldn’t get any respect for that. One could say that elan vital and aether are now outside the Overton window, but there are good reasons for that.
one should always introduce statements with hedges like “From the perspective of finitism, (diagonalization / …) is of questionable validity because...” However, just saying directly “(Diagonalization / …) is of questionable validity because...” is seen as beyond the pale, cranky, or otherwise unacceptable.
Well yes, if you use different axioms, you get different results. Using axioms X, Y, Z, statement S is true; using axioms A, B, C, statement S is false. So if there is an article explaining how XYZ implies S, saying “however, from the perspective ABC, S is false” makes sense, but saying merely “however, S is false” is confusing. Are you saying that XYZ does not imply S? That would be wrong. Or are you saying that you disagree with XYZ and we should use ABC instead? Then say it clearly.
If you want to argue that two sets that have a 1:1 correspondence do not have the same size, start by saying what you actually mean by “size”? (Do you see the repeating pattern here? It is the unclear argumentation that is often a problem. Not merely disagreeing with some definition, but using words in an unusual way without bothering to explain that. Perhaps without even being aware that different definitions can lead to different outcomes.) You can successfully argue that a 2-inch line does not have the same size as a 1-inch line, but you have to make it clear that you are talking about length rather than about cardinality.
But these assumptions [axiom of infinity] are only directly related to Cantor’s power set argument (“Cantor’s theorem”), which is different from his diagonal argument.
The diagonal argument assumes that you write down the real numbers. But how could you write down the decimal representation of 1⁄3, if you are not allowed to use infinitely many 3′s? More importantly, what are you going to do about uncomputable numbers, if you can neither provide the algorithm that returns the digits on demand, nor assume that the infinite sequence of their digits simply exists?
So now you are making a list of computable numbers (or something like that) rather than the list of real numbers. And Cantor’s diagonal argument still finds a number that was not included in the original list.
There is a “Controversy over Cantor’s theory” article in Wikipedia.
Well, just read the introduction. It says nothing about the controversy and instead presupposes that Cantor is correct:
Cantor’s theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers.
That is, the introduction asserts that Cantor’s power set argument is correct and doesn’t mention any criticisms, despite the title of the article. Even later it mentions Skolem nowhere, nor any criticism of the diagonal argument. Absurdly enough, the article neutrality header even claims that the article is biased in the anti-Cantor direction, when clearly the opposite is the case.
Plus articles on finitism, etc.
Yes, as I said previously: Under the term “finitism” these positions are allowed, because this places them merely in the philosophy of mathematics bin, but not in mathematics articles in general.
Some respected scientists of the past have defended elan vital or aether. And yet, if someone did the same today, they wouldn’t get any respect for that. One could say that elan vital and aether are now outside the Overton window, but there are good reasons for that.
This analogy is inappropriate. Finitism is not like élan vital. In fact, finitism doesn’t say anything specific (like élan vital or aether) exists, but that something doesn’t exist (or isn’t required to assume to exist), namely the actual infinite. And it is hardly the case that infinitism is a scientific fact now while finitism is disproved. So finitism is nothing like the examples you gave; finitists today are not cranks like defenders of élan vital would be. Or do you think otherwise?
Well yes, if you use different axioms, you get different results. Using axioms X, Y, Z, statement S is true; using axioms A, B, C, statement S is false. So if there is an article explaining how XYZ implies S, saying “however, from the perspective ABC, S is false” makes sense, but saying merely “however, S is false” is confusing. Are you saying that XYZ does not imply S? That would be wrong. Or are you saying that you disagree with XYZ and we should use ABC instead? Then say it clearly.
If only it were that simple. It is not clear what axioms Cantor’s diagonal argument uses. It is largely informal. Which axioms do you think it relies on?
If you want to argue that two sets that have a 1:1 correspondence do not have the same size, start by saying what you actually mean by “size”?
That’s not the relevant question. The question is what we (humans in general) commonly mean with something being infinite. Finitists say potential infinity (or unboundedness) is the answer, infinitists say we mean something stronger. There are various arguments for and against, and that’s precisely the disagreement between finitism and infinitism. In the case of potential infinity, size comparisons are done via comparisons of rates of growth of sequences, not via mappings between unordered sets. Two quantities have the same size if they “approach infinity” at the same rate. This notion of size is used all the time in calculus. (Which has the distinction of having a lot of real world applications, unlike transfinite set theory.) And it doesn’t introduce new exotic properties, like that a proper subset can have the same “size” as the original set, which never happens for finite quantities.
You can successfully argue that a 2-inch line does not have the same size as a 1-inch line, but you have to make it clear that you are talking about length rather than about cardinality.
This doesn’t answer the question of whether it makes more sense to call a line with “infinite” length “unboundedly long” or “of actual infinite (countable? uncountable?) length”. I would argue the former makes sense, the latter doesn’t, because it isn’t clear what “countably infinite length”, or the like, would mean.
But these assumptions [axiom of infinity] are only directly related to Cantor’s power set argument (“Cantor’s theorem”), which is different from his diagonal argument.
The diagonal argument assumes that you write down the real numbers. But how could you write down the decimal representation of 1⁄3, if you are not allowed to use infinitely many 3′s? More importantly, what are you going to do about uncomputable numbers, if you can neither provide the algorithm that returns the digits on demand, nor assume that the infinite sequence of their digits simply exists?
The axiom of infinity is only about actual infinity. It is not required for potential infinity. Even without the axiom of infinity, there is no largest set with some finite size. Finitism, despite its (misleading) name, doesn’t say there are no infinities, it just disagrees with infinitism about the meaning of “infinite”, as I emphasized above. So infinite decimal expansions are not ruled out. (The actual rejection of any infinite quantities is called ultrafinitism.)
So now you are making a list of computable numbers (or something like that) rather than the list of real numbers. And Cantor’s diagonal argument still finds a number that was not included in the original list.
Why? Cantor assumes we have an infinite table of decimal digits. Under finitism, this means both the number of columns and rows are “infinite”, that is, they are unbounded, they grow without bound. The diagonal number would only not be on the table if the number of columns grows at least as fast as the number of rows. Which wouldn’t be the case for a lexicographic ordering. E.g. for base 10, if the number of columns/digits grows with n, the number of rows grows with 10n. So for any rectangular n×10n table, the diagonal number is on the table, for any n. The table is only square for the unary number system, but with those the diagonal argument doesn’t work in the first place, as no digits can be swapped.
I’ll try to rephrase it as I understood it (the objection against the diagonal argument):
.
Actual infinities do not exist. We only have sequences that can be extended without limits.
If you see the diagonal not as a finished object, but as a sequence that keeps growing, then for any (finite) length of that diagonal (i.e. what an infinitist would call a “prefix” of the diagonal), you can find a row, usually much lower in the table, that actually contains the negation of that (unfinished) sequence. (“Negation” = when you change every digit to something different.)
That is, yes you can have a(n unfinished) table that contains the negation of its own (every unfinished part of) diagonal.
.
My issue with this answer is that you seem to say “this table contains the negation of its own diagonal”, but then you keep changing your mind about where exactly it is. Like, at the moment we only get five columns, you can point at a row 678 and say “this row contains the negation of the diagonal (so far)”. But when we later get to 678 columns, and obviously the digit at [678, 678] cannot be a negation of itself, you say “actually, it is the row 123456789 that contains the negation of the (first 678 digits of) the diagonal”. And of course, once we get to 123456789 columns, you will have to change your mind again. And again.
So it’s like… the negation of the diagonal supposedly is there, but… not at any specific place?
.
Also, if we say that actual infinities are not real, then what are we actually trying to prove by the diagonal argument? That real numbers between 0 and N grow faster than integers between 0 and N? That the number of all subsets of N items grows faster than N? Those statements still seem true to me.
So it’s like… the negation of the diagonal supposedly is there, but… not at any specific place?
Why should this be a problem? On this view, there is no “the diagonal”; there are only diagonals of particular tables for particular values of n, which each have their own negations.
I guess one could similar argue that there is no “the table”; there are only finite tables that are gradually expanded rightwards and downwards… and whether they contain their anti-diagonals, that kinda depends on whether they expand downwards much faster than they expand rightwards.
There are probably other strange consequences, such as inability to say whether two real numbers are equal or not; all you can say is “so far, the digits I have checked do match”, but by that logic π actually equals 22⁄7, at least at the beginning; and 3×(1/3) could or count not be 1, no one knows.
And the question of whether there are more real numbers than integers reduces to “there is only a finite amount of each already written, so it depends on whether the guy who writes down the real numbers is faster than the guy who writes down the integers”. Does this resemble the math as we know it?
If someone bites the bullet and says “yeah, the math where you can’t figure out how much is 3×(1/3) is philosophically preferable to math with infinities (because infinities do not really exist in nature, and frankly neither does perfect equality)”, I can respect that. But I would like people to say all of that up front, rather than hide behind “but there are problems with Cantor’s definition”. There are problems with every definition, you just have to pick your poison.
What are real numbers then? On the standard account, real numbers are equivalence classes of sequences of rationals, the finite diagonals being one such sequence. I mean, “Real numbers don’t exist” is one way to avoid the diagonal argument, but I don’t thinks that’s what cubefox is going for.
Diagonal argument is not an assumption. You can, of course, disagree with any of the assumptions it relies on, and some mathematicians do so.
Some reject the axiom of infinity, and the use of actual infinities in general. Others reject the axiom schema of specification, or rather put some limits on the formulas it can be used with. (There may be other objections I haven’t heard of.) These are not referred to as crackpots. They are choosing a different collection of axioms—i.e. their “sets” are not the same as Cantor’s “sets”—and they can show the exact step where Cantor’s diagonal argument fails to apply in their system.
I think the label crackpot is applied to people who vocally disagree with the argument, but can’t pinpoint the place where it fails; in other words they have a strong opinion on something they don’t understand.
.
I also think it’s fair to say “I disagree with the conclusions of Cantor’s diagonal argument, although I can’t explain where exactly it fails, because these things are too abstract for me—it’s just that the outcome feels wrong according to my intuition, so there is probably a step I would disagree with, even if I can’t tell which one it is”.
Such people might even be happy to learn about Skolem’s paradox, which used to drive “the opposite camp” crazy. “There are countable models of ZF set theory that satisfy the theorem that there are uncountable sets.”—which at first sight seems like absolute nonsense, but it’s a logical consequence of following the axioms.
Nonetheless, modern set theorists are okay with this, too. So the problem is not with disagreeing with the outcomes per se, but with not being able to play the game according to its rules.
Let me explain. I get the following impression. When making basically any other argument in philosophy of mathematics, people are generally highly disinterested, but as soon as the debate touches on issues involving transfinite set theory, like diagonalization, at least people online (on blogs, Wikipedia, Stack Exchange etc, though journals might be similar) are quick to heavily downvote comments and declare people to be “fringe” or “cranks”. Even though they have opinions that were completely mainstream until a hundred years ago, and were defended by a number of well-known mathematicians and respectable philosophers like Wittgenstein.
To give a more specific example: Adding a “criticism” or “controversy” section to certain Wikipedia articles on topics around infinity in set theory seems almost as impossible as writing a Wikipedia article covering certain taboo research involving race and IQ.
The only way to argue in a different direction is to word things very carefully and conservatively to not step outside the narrow Overton window. Optimally one shouldn’t be directly arguing for anything and instead just engage in explaining arguments other respectable people like Weyl or Wittgenstein have hopefully already made. Failing that, one should always introduce statements with hedges like “From the perspective of finitism, (diagonalization / …) is of questionable validity because...” However, just saying directly “(Diagonalization / …) is of questionable validity because...” is seen as beyond the pale, cranky, or otherwise unacceptable.
I think the reason for these hostilities is that in these disagreements there is an implicit battle going on over where to draw the line between mathematics (the land of definitive facts) and philosophy of mathematics (where nothing is certain and diverse positions may be defended). The reason why hedging a statement with “from the perspective of finitism, …” makes it more acceptable, is that this phrase places it in the realm of philosophy, where things may be questioned.
For example, in mathematics land, two sets have the same size iff there is a definable one-to-one correspondence. Full stop. Anyone who disagrees is a flat-Earther.
But in philosophy-of-mathematics land, this same thesis is called Hume’s principle. When referred to by this name, disagreement is suddenly very much allowed. Not only that, the principle is generally seen as resting on a somewhat shaky ground, or anyway being in need of further justification. The only reason these two inconsistent perspectives on the same topic can co-exist in Wikipedia is that they simply don’t acknowledge each other. But if someone were to propose to merge the articles, the philosophy perspective would likely lose.
Regarding these points:
I’m really not aware of any assumptions one can disagree with here while still being perceived as respectable. The most one can say is something like “Finitists might argue that given any way to construct the infinite table, the rows must approach infinity faster than the columns, in which case the diagonal number is always on the table” or something like that. But even this would be suspect and in any case not constitute a specific assumption one is permitted to disagree with.
But these assumptions are only directly related to Cantor’s power set argument (“Cantor’s theorem”), which is different from his diagonal argument. The former involves (power) sets, the latter real numbers. The latter is also not directly related to Skolem’s paradox.
There is a “Controversy over Cantor’s theory” article in Wikipedia. Plus articles on finitism, etc.
Some respected scientists of the past have defended elan vital or aether. And yet, if someone did the same today, they wouldn’t get any respect for that. One could say that elan vital and aether are now outside the Overton window, but there are good reasons for that.
Well yes, if you use different axioms, you get different results. Using axioms X, Y, Z, statement S is true; using axioms A, B, C, statement S is false. So if there is an article explaining how XYZ implies S, saying “however, from the perspective ABC, S is false” makes sense, but saying merely “however, S is false” is confusing. Are you saying that XYZ does not imply S? That would be wrong. Or are you saying that you disagree with XYZ and we should use ABC instead? Then say it clearly.
If you want to argue that two sets that have a 1:1 correspondence do not have the same size, start by saying what you actually mean by “size”? (Do you see the repeating pattern here? It is the unclear argumentation that is often a problem. Not merely disagreeing with some definition, but using words in an unusual way without bothering to explain that. Perhaps without even being aware that different definitions can lead to different outcomes.) You can successfully argue that a 2-inch line does not have the same size as a 1-inch line, but you have to make it clear that you are talking about length rather than about cardinality.
The diagonal argument assumes that you write down the real numbers. But how could you write down the decimal representation of 1⁄3, if you are not allowed to use infinitely many 3′s? More importantly, what are you going to do about uncomputable numbers, if you can neither provide the algorithm that returns the digits on demand, nor assume that the infinite sequence of their digits simply exists?
So now you are making a list of computable numbers (or something like that) rather than the list of real numbers. And Cantor’s diagonal argument still finds a number that was not included in the original list.
Well, just read the introduction. It says nothing about the controversy and instead presupposes that Cantor is correct:
That is, the introduction asserts that Cantor’s power set argument is correct and doesn’t mention any criticisms, despite the title of the article. Even later it mentions Skolem nowhere, nor any criticism of the diagonal argument. Absurdly enough, the article neutrality header even claims that the article is biased in the anti-Cantor direction, when clearly the opposite is the case.
Yes, as I said previously: Under the term “finitism” these positions are allowed, because this places them merely in the philosophy of mathematics bin, but not in mathematics articles in general.
This analogy is inappropriate. Finitism is not like élan vital. In fact, finitism doesn’t say anything specific (like élan vital or aether) exists, but that something doesn’t exist (or isn’t required to assume to exist), namely the actual infinite. And it is hardly the case that infinitism is a scientific fact now while finitism is disproved. So finitism is nothing like the examples you gave; finitists today are not cranks like defenders of élan vital would be. Or do you think otherwise?
If only it were that simple. It is not clear what axioms Cantor’s diagonal argument uses. It is largely informal. Which axioms do you think it relies on?
That’s not the relevant question. The question is what we (humans in general) commonly mean with something being infinite. Finitists say potential infinity (or unboundedness) is the answer, infinitists say we mean something stronger. There are various arguments for and against, and that’s precisely the disagreement between finitism and infinitism. In the case of potential infinity, size comparisons are done via comparisons of rates of growth of sequences, not via mappings between unordered sets. Two quantities have the same size if they “approach infinity” at the same rate. This notion of size is used all the time in calculus. (Which has the distinction of having a lot of real world applications, unlike transfinite set theory.) And it doesn’t introduce new exotic properties, like that a proper subset can have the same “size” as the original set, which never happens for finite quantities.
This doesn’t answer the question of whether it makes more sense to call a line with “infinite” length “unboundedly long” or “of actual infinite (countable? uncountable?) length”. I would argue the former makes sense, the latter doesn’t, because it isn’t clear what “countably infinite length”, or the like, would mean.
The axiom of infinity is only about actual infinity. It is not required for potential infinity. Even without the axiom of infinity, there is no largest set with some finite size. Finitism, despite its (misleading) name, doesn’t say there are no infinities, it just disagrees with infinitism about the meaning of “infinite”, as I emphasized above. So infinite decimal expansions are not ruled out. (The actual rejection of any infinite quantities is called ultrafinitism.)
Why? Cantor assumes we have an infinite table of decimal digits. Under finitism, this means both the number of columns and rows are “infinite”, that is, they are unbounded, they grow without bound. The diagonal number would only not be on the table if the number of columns grows at least as fast as the number of rows. Which wouldn’t be the case for a lexicographic ordering. E.g. for base 10, if the number of columns/digits grows with n, the number of rows grows with 10n. So for any rectangular n×10n table, the diagonal number is on the table, for any n. The table is only square for the unary number system, but with those the diagonal argument doesn’t work in the first place, as no digits can be swapped.
I’ll try to rephrase it as I understood it (the objection against the diagonal argument):
.
Actual infinities do not exist. We only have sequences that can be extended without limits.
If you see the diagonal not as a finished object, but as a sequence that keeps growing, then for any (finite) length of that diagonal (i.e. what an infinitist would call a “prefix” of the diagonal), you can find a row, usually much lower in the table, that actually contains the negation of that (unfinished) sequence. (“Negation” = when you change every digit to something different.)
That is, yes you can have a(n unfinished) table that contains the negation of its own (every unfinished part of) diagonal.
.
My issue with this answer is that you seem to say “this table contains the negation of its own diagonal”, but then you keep changing your mind about where exactly it is. Like, at the moment we only get five columns, you can point at a row 678 and say “this row contains the negation of the diagonal (so far)”. But when we later get to 678 columns, and obviously the digit at [678, 678] cannot be a negation of itself, you say “actually, it is the row 123456789 that contains the negation of the (first 678 digits of) the diagonal”. And of course, once we get to 123456789 columns, you will have to change your mind again. And again.
So it’s like… the negation of the diagonal supposedly is there, but… not at any specific place?
.
Also, if we say that actual infinities are not real, then what are we actually trying to prove by the diagonal argument? That real numbers between 0 and N grow faster than integers between 0 and N? That the number of all subsets of N items grows faster than N? Those statements still seem true to me.
Why should this be a problem? On this view, there is no “the diagonal”; there are only diagonals of particular tables for particular values of n, which each have their own negations.
I guess one could similar argue that there is no “the table”; there are only finite tables that are gradually expanded rightwards and downwards… and whether they contain their anti-diagonals, that kinda depends on whether they expand downwards much faster than they expand rightwards.
There are probably other strange consequences, such as inability to say whether two real numbers are equal or not; all you can say is “so far, the digits I have checked do match”, but by that logic π actually equals 22⁄7, at least at the beginning; and 3×(1/3) could or count not be 1, no one knows.
And the question of whether there are more real numbers than integers reduces to “there is only a finite amount of each already written, so it depends on whether the guy who writes down the real numbers is faster than the guy who writes down the integers”. Does this resemble the math as we know it?
If someone bites the bullet and says “yeah, the math where you can’t figure out how much is 3×(1/3) is philosophically preferable to math with infinities (because infinities do not really exist in nature, and frankly neither does perfect equality)”, I can respect that. But I would like people to say all of that up front, rather than hide behind “but there are problems with Cantor’s definition”. There are problems with every definition, you just have to pick your poison.
The bullet-biting here is just “‘real numbers’ are fake”. That makes most of the questions you cite moot.
What are real numbers then? On the standard account, real numbers are equivalence classes of sequences of rationals, the finite diagonals being one such sequence. I mean, “Real numbers don’t exist” is one way to avoid the diagonal argument, but I don’t thinks that’s what cubefox is going for.
“Real numbers don’t exist” seems like a good solution to me.
Would it be fair to describe finitism as “a belief in at most one infinity”?