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You in fact did write down "Graham's number" and "googleplex". Most real numbers cannot even be written down as a finite sequence of symbols in any way. However

You in fact did write down "Graham's number" and "googleplex". Most real numbers cannot even be written down as a finite sequence of symbols in any way. However for algebraic numbers ("Plenty of irrational numbers"), every single one has a finite description. There are uncountably many reals and only a countable number of "names" we can give them. This is a very real problem that makes people say that the real numbers are not real.

dragontamer 4y ago
> finite description Oh jeez, its like the space of rational numbers is well described by Arabic Numerals or something. And that anything that's a non-rational number cannot be described by Arabic Numerals + decimal points. :-p Don't confuse our current way of writing down numbers with alternative writing methods. That's my point. "Rational numbers" are simply one-and-the-same as "The space described by finite-length Arabic Numerals + decimal points", no more, no less. EDIT: To put it another way: you're describing a limitation of Arabic Numerals. Not a limitation to math in general. Much like how Roman Numerals are a terrible representation of modern math, we need other representations (ie: Algebraic notation) to represent other concepts in modern math.
- thethirdone 4y ago
  I did not mention Arabic numerals at all in my comment; I think you have missed my point entirely. I was actually thinking of sequences of symbols like "lim(inf, (1 + 1/x)^x)" being a name for e. And all algebraic numbers can be expressed by a polynomial equation and a description of which root you are referring to. Given that I literally wasn't even thinking about Arabic numerals, I was not describing a limitation of them. The limitation is with math. When talking about numbers you need a way to write down a specific number (Needs a finite representation). If it can be put into a computer, that finite representation comes from some countable set. There are uncountably many real numbers. That means we cannot given a computer understandable name to every real number. You can give a name to every algebraic number or rational number. That is a key difference that is not tied in any way to Arabic numerals.
  - dragontamer 4y ago
    > When talking about numbers you need a way to write down a specific number (Needs a finite representation) There are plenty of rational numbers that cannot be written down. Indeed, an infinite number of them. All the numbers larger than say... a googleplex, or the numbers approaching Grahams' number (or larger than that number) are unwritable. We don't even need to stray from the integer numbers if we want to get to unwritable numbers. Plenty of numbers too large to be ever written down even if we use every single atom in our universe (or any such combination of atomic-states with all the atoms in our universe). "X cannot be written" seems to be a rather arbitrary limit.
kortex 4y ago
Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or a one-to-one correspondence of f(x). This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)". But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness. https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox
- thethirdone 4y ago
  If unicorns exist, but humans can never observe them do you really care whether they exist or not? I generally am reducing questions that are only relevant to philosophy such as "Do real numbers exist?" to questions which you can actually answer. To make something clear, rationals have a finite description. If you allow infinite descriptions, all real numbers can be described by Cauchy sequences. When I talk about countably many names, I am talking about the set of finite names humans use for numbers. You cannot meaningfully have infinitely many sets of finite names that humans agree on so you cannot cover the real numbers with names. The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.
  - dragontamer 4y ago
    When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary. Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely. Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.

dragontamer 4y ago

> finite description Oh jeez, its like the space of rational numbers is well described by Arabic Numerals or something. And that anything that's a non-rational number cannot be described by Arabic Numerals + decimal points. :-p Don't confuse our current way of writing down numbers with alternative writing methods. That's my point. "Rational numbers" are simply one-and-the-same as "The space described by finite-length Arabic Numerals + decimal points", no more, no less. EDIT: To put it another way: you're describing a limitation of Arabic Numerals. Not a limitation to math in general. Much like how Roman Numerals are a terrible representation of modern math, we need other representations (ie: Algebraic notation) to represent other concepts in modern math.

thethirdone 4y ago
I did not mention Arabic numerals at all in my comment; I think you have missed my point entirely. I was actually thinking of sequences of symbols like "lim(inf, (1 + 1/x)^x)" being a name for e. And all algebraic numbers can be expressed by a polynomial equation and a description of which root you are referring to. Given that I literally wasn't even thinking about Arabic numerals, I was not describing a limitation of them. The limitation is with math. When talking about numbers you need a way to write down a specific number (Needs a finite representation). If it can be put into a computer, that finite representation comes from some countable set. There are uncountably many real numbers. That means we cannot given a computer understandable name to every real number. You can give a name to every algebraic number or rational number. That is a key difference that is not tied in any way to Arabic numerals.
- dragontamer 4y ago
  > When talking about numbers you need a way to write down a specific number (Needs a finite representation) There are plenty of rational numbers that cannot be written down. Indeed, an infinite number of them. All the numbers larger than say... a googleplex, or the numbers approaching Grahams' number (or larger than that number) are unwritable. We don't even need to stray from the integer numbers if we want to get to unwritable numbers. Plenty of numbers too large to be ever written down even if we use every single atom in our universe (or any such combination of atomic-states with all the atoms in our universe). "X cannot be written" seems to be a rather arbitrary limit.

kortex 4y ago

Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or a one-to-one correspondence of f(x). This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)". But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness. https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox

thethirdone 4y ago
If unicorns exist, but humans can never observe them do you really care whether they exist or not? I generally am reducing questions that are only relevant to philosophy such as "Do real numbers exist?" to questions which you can actually answer. To make something clear, rationals have a finite description. If you allow infinite descriptions, all real numbers can be described by Cauchy sequences. When I talk about countably many names, I am talking about the set of finite names humans use for numbers. You cannot meaningfully have infinitely many sets of finite names that humans agree on so you cannot cover the real numbers with names. The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.
- dragontamer 4y ago
  When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary. Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely. Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.