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I've always called BS on the math community for this. Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I ca

I've always called BS on the math community for this. Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I can (count) enumerate all of them. Simply flip the digits over the decimal point, so 0.14159 become integer 95141. Done. Now you ask where pi sits? It's infinitely far down the list of integers. It's position is as precise as your definition. We can extend this to (I forget the terminology) enumerate the numbers that can be named, by converting the names to integers. Now pi has several location in this mapping, as do a lot of real numbers. Infinite sets are interesting, but I've never been convinced that there are more reals than integers. It seems plausible but I don't think pi not mapping to a specific integer in the above scheme invalidates the mapping.

101008 4y ago
If you sort this set after you apply your transformation, what's the number that comes right after pi?
- dsr_ 4y ago
  You can begin looking for it knowing just two facts: the last digit of pi, and the magnitude of that digit. Easy! [sarcasm]
  - phkahler 4y ago
    This is all related to the ability to name a specific number. If there are reals that can't be named in a finite number of symbols, why can't there be integers like that as well?
    - davrosthedalek 4y ago
      Because a property of the numbers we call integers is that we can name them. You can certainly rename some of the numbers which are indescribable and call that set your integers, but these are different integers than what everyone else talks about.
chriswarbo 4y ago
> Simply flip the digits over the decimal point, so 0.14159 become integer 95141 This sounds similar to an interesting (and often counter-intuitive) way of writing down numbers called 'p-adic numbers' (although that terminology usually requires 'p' to be prime; in your case you'd want the '10-adic' numbers, since you're using base-10) See e.g. https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number (here's a video specifically about 10-adic numbers https://www.youtube.com/watch?v=XXRwlo_MHnI https://www.youtube.com/watch?v=XXRwlo_MHnI )
iroddis 4y ago
“Flip around the decimal”… you haven’t accounted for irrational numbers. The reals are not countably infinite. Even if they were, pi is not between zero and one, so would not appear in your list. The rest of your argument is difficult to parse. If you’re interested in this kind of stuff though, I’d really recommend picking some good books on set theory and maybe number theory. Set theory in particular will cover a lot of what you’re talking about.
deleted 4y ago
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SantalBlush 4y ago
You can't flip irrationals over the decimal the way you described, because their decimals have infinitely many non-repeating values. You would end up with an infinite sequence of numbers, not a real number.
- phkahler 4y ago
  You mean there aren't infinity large integers? Hmm, that seems weird.
  - xvkektjdicou 4y ago
    All integers are by definition finite. Positive integers are defined by S(S(S(...S(0)...))) a finite number of times, where S is the successor function. (And negatives in the obvious way.)
jacobmartin 4y ago
Where is 1⁄3? Is it before or after (pi - 3)?
- phkahler 4y ago
  We need to switch to the scheme that converts name/description to an integer for that. If you mean the repeating decimal, I don't know if it's before or after pi because I can't identify (name) their locations. It's the same argument. If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?
  - jacobmartin 4y ago
    > If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions? Because the integers are defined as the finite successors to the number 0 and their additive opposites. See, for instance, the construction section of: https://en.m.wikipedia.org/wiki/Integer https://en.m.wikipedia.org/wiki/Integer You can start talking about numbers that can't be constructed in this way, but then you've ceased to be talking about the integers.
    - phkahler 4y ago
      Is the size of the set of integers an integer?
      - jacobmartin 4y ago
        No, usually the cardinalities of infinite sets are represented by things called aleph numbers: https://en.m.wikipedia.org/wiki/Aleph_number https://en.m.wikipedia.org/wiki/Aleph_number. But it isn't, in any case, an integer.
        phkahler 4y ago
        Thanks! That to me is the most useful response to my ramblings today!
kjaa 4y ago
I believe you are confusing rational for real numbers. Rational numbers really do map to the integers, but I believe I can "break" any purported isomorphism from reals to ints, by taking two adjacent ints and asking what number the midpoint between the corresponding reals map to.
- phkahler 4y ago
  Please be specific. I never said the mapping has nice properties regarding the ordering. BTW I know rational numbers map to integers, and hence any length tuple can be mapped to integers by pairwise combining them in similar ways. But if any length tuple of integers can map to a single integer, why can't an infinitely long set of integers map to an integer? Like the string of digits in an irrational number represented as a decimal?
  - Warwolt 4y ago
    The problem with mapping the reals to the integers is that a list of real numbers is infinite in "two directions". Each real can be potentially described with an infinite number of digits, and you can have an infinite list of real numbers (i.e. infinite sequences of digits). This is, I believe, the basis for Cantor's diagonal argument, which is a quite beautiful proof of there being more reals than integers. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
SamBam 4y ago
Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of literally any pair of irrational numbers in your system. > It's position is as precise as your definition. This wand-waving is actually transforming irrational numbers into rational numbers, and so means you are not actually working with irrational numbers. You can't just say "this works with all irrational numbers, like Pi, so long as we call Pi '3.14'." Pi is not 3.14, so any mapping system that depends on you cutting off your precision at some finite number of digits is worthless.
- phkahler 4y ago
  >> Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of even a single irrational number in your system. That is my point exactly. Why do we accept that there are real numbers that cannot be defined by a finite string of symbols, but there are no integers of that nature. Getting back to the axiom of choice thing. How do you pick a specific item from one of those ill-defined sets that include numbers that don't have finite descriptions? How do you choose something you can't name? It's absurdity all the way down. ;-)
  - SamBam 4y ago
    That did not seem in any way to be your point. You said > I've never been convinced that there are more reals than integers I'm explaining that your method of mapping reals to integers is flawed, and so does not prove that reals and integers have the same cardinality.
  - WastingMyTime89 4y ago
    > Why do we accept that there are real numbers that cannot be defined by a finite string of symbols We don't accept it. It's easy to prove that it's the case. I think you are unfamiliar with how R is defined. The traditional definition of R is a completion of Q with respect to the metric |x-y| (getting all Cauchy sequences to converge). That's larger than infinite sequence of decimals.
boole1854 4y ago
> Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I can (count) enumerate all of them. Simply flip the digits over the decimal point, so 0.14159 become integer 95141. Every integer has a finite number of numerals. Proof: by definition, each natural number is either 0 or a successor to 0. 0 is finite. And if n is a finite length, then n + 1 is a finite length. So by mathematical induction, all natural numbers have finite length. A similar argument covers the negative numbers, so all integers have finite lengths. Since the decimal expansion of certain rational numbers are infinite (e.g., 1/3 = 0.333...), the strategy of mapping each number between 0 to 1 to an integer by flipping digits over the decimal point does not work for all rationals, let alone all reals.
alecbz 4y ago
Don't want to pile on too much, but my attempt at an explanation: > Simply flip the digits over the decimal point, so 0.14159 become integer 95141 They key here is that both 0.14159 and 95141 contain a _finite_ number of symbols/digits. There are infinitely many numbers like this, but each of those numbers are themselves finite. The decimal expansion of pi, OTOH, is _itself_ infinite. The decimal expansion of any irrational number is infinite. There are infinitely many of these things, each of which are themselves infinite. This is the key thing that causes uncountability. Each of the numbers has infinitely many "parameters" that can be tweaked. If you try to produce an infinite (countable) list that contains all of these infinite numbers, I can always tweak a different parameter for each number in your list and be guaranteed to have produced a number that your list missed.

101008 4y ago

If you sort this set after you apply your transformation, what's the number that comes right after pi?

dsr_ 4y ago
You can begin looking for it knowing just two facts: the last digit of pi, and the magnitude of that digit. Easy! [sarcasm]
- phkahler 4y ago
  This is all related to the ability to name a specific number. If there are reals that can't be named in a finite number of symbols, why can't there be integers like that as well?
  - davrosthedalek 4y ago
    Because a property of the numbers we call integers is that we can name them. You can certainly rename some of the numbers which are indescribable and call that set your integers, but these are different integers than what everyone else talks about.

chriswarbo 4y ago

> Simply flip the digits over the decimal point, so 0.14159 become integer 95141 This sounds similar to an interesting (and often counter-intuitive) way of writing down numbers called 'p-adic numbers' (although that terminology usually requires 'p' to be prime; in your case you'd want the '10-adic' numbers, since you're using base-10) See e.g. https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number (here's a video specifically about 10-adic numbers https://www.youtube.com/watch?v=XXRwlo_MHnI https://www.youtube.com/watch?v=XXRwlo_MHnI )

iroddis 4y ago

“Flip around the decimal”… you haven’t accounted for irrational numbers. The reals are not countably infinite. Even if they were, pi is not between zero and one, so would not appear in your list. The rest of your argument is difficult to parse. If you’re interested in this kind of stuff though, I’d really recommend picking some good books on set theory and maybe number theory. Set theory in particular will cover a lot of what you’re talking about.

deleted 4y ago

[deleted]

SantalBlush 4y ago

You can't flip irrationals over the decimal the way you described, because their decimals have infinitely many non-repeating values. You would end up with an infinite sequence of numbers, not a real number.

phkahler 4y ago
You mean there aren't infinity large integers? Hmm, that seems weird.
- xvkektjdicou 4y ago
  All integers are by definition finite. Positive integers are defined by S(S(S(...S(0)...))) a finite number of times, where S is the successor function. (And negatives in the obvious way.)

jacobmartin 4y ago

Where is 1⁄3? Is it before or after (pi - 3)?

phkahler 4y ago
We need to switch to the scheme that converts name/description to an integer for that. If you mean the repeating decimal, I don't know if it's before or after pi because I can't identify (name) their locations. It's the same argument. If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?
- jacobmartin 4y ago
  > If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions? Because the integers are defined as the finite successors to the number 0 and their additive opposites. See, for instance, the construction section of: https://en.m.wikipedia.org/wiki/Integer https://en.m.wikipedia.org/wiki/Integer You can start talking about numbers that can't be constructed in this way, but then you've ceased to be talking about the integers.
  - phkahler 4y ago
    Is the size of the set of integers an integer?
    - jacobmartin 4y ago
      No, usually the cardinalities of infinite sets are represented by things called aleph numbers: https://en.m.wikipedia.org/wiki/Aleph_number https://en.m.wikipedia.org/wiki/Aleph_number. But it isn't, in any case, an integer.
      - phkahler 4y ago
        Thanks! That to me is the most useful response to my ramblings today!

kjaa 4y ago

I believe you are confusing rational for real numbers. Rational numbers really do map to the integers, but I believe I can "break" any purported isomorphism from reals to ints, by taking two adjacent ints and asking what number the midpoint between the corresponding reals map to.

phkahler 4y ago
Please be specific. I never said the mapping has nice properties regarding the ordering. BTW I know rational numbers map to integers, and hence any length tuple can be mapped to integers by pairwise combining them in similar ways. But if any length tuple of integers can map to a single integer, why can't an infinitely long set of integers map to an integer? Like the string of digits in an irrational number represented as a decimal?
- Warwolt 4y ago
  The problem with mapping the reals to the integers is that a list of real numbers is infinite in "two directions". Each real can be potentially described with an infinite number of digits, and you can have an infinite list of real numbers (i.e. infinite sequences of digits). This is, I believe, the basis for Cantor's diagonal argument, which is a quite beautiful proof of there being more reals than integers. https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

SamBam 4y ago

Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of literally any pair of irrational numbers in your system. > It's position is as precise as your definition. This wand-waving is actually transforming irrational numbers into rational numbers, and so means you are not actually working with irrational numbers. You can't just say "this works with all irrational numbers, like Pi, so long as we call Pi '3.14'." Pi is not 3.14, so any mapping system that depends on you cutting off your precision at some finite number of digits is worthless.

phkahler 4y ago
>> Your method is not a 1:1 mapping of reals to integers, because you can't even define the position of a single irrational number with your method. You don't know the final digit, so you don't know the leading digit in your system. You have absolutely no idea the ordering of even a single irrational number in your system. That is my point exactly. Why do we accept that there are real numbers that cannot be defined by a finite string of symbols, but there are no integers of that nature. Getting back to the axiom of choice thing. How do you pick a specific item from one of those ill-defined sets that include numbers that don't have finite descriptions? How do you choose something you can't name? It's absurdity all the way down. ;-)
- SamBam 4y ago
  That did not seem in any way to be your point. You said > I've never been convinced that there are more reals than integers I'm explaining that your method of mapping reals to integers is flawed, and so does not prove that reals and integers have the same cardinality.
- WastingMyTime89 4y ago
  > Why do we accept that there are real numbers that cannot be defined by a finite string of symbols We don't accept it. It's easy to prove that it's the case. I think you are unfamiliar with how R is defined. The traditional definition of R is a completion of Q with respect to the metric |x-y| (getting all Cauchy sequences to converge). That's larger than infinite sequence of decimals.

boole1854 4y ago

> Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I can (count) enumerate all of them. Simply flip the digits over the decimal point, so 0.14159 become integer 95141. Every integer has a finite number of numerals. Proof: by definition, each natural number is either 0 or a successor to 0. 0 is finite. And if n is a finite length, then n + 1 is a finite length. So by mathematical induction, all natural numbers have finite length. A similar argument covers the negative numbers, so all integers have finite lengths. Since the decimal expansion of certain rational numbers are infinite (e.g., 1/3 = 0.333...), the strategy of mapping each number between 0 to 1 to an integer by flipping digits over the decimal point does not work for all rationals, let alone all reals.

alecbz 4y ago

Don't want to pile on too much, but my attempt at an explanation: > Simply flip the digits over the decimal point, so 0.14159 become integer 95141 They key here is that both 0.14159 and 95141 contain a _finite_ number of symbols/digits. There are infinitely many numbers like this, but each of those numbers are themselves finite. The decimal expansion of pi, OTOH, is _itself_ infinite. The decimal expansion of any irrational number is infinite. There are infinitely many of these things, each of which are themselves infinite. This is the key thing that causes uncountability. Each of the numbers has infinitely many "parameters" that can be tweaked. If you try to produce an infinite (countable) list that contains all of these infinite numbers, I can always tweak a different parameter for each number in your list and be guaranteed to have produced a number that your list missed.