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.
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.