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.
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.