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But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the real

But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.

Nevermark 1y ago
Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
- throwaway81523 1y ago
  It's hard to claim that an infinite (or anyway unbounded) collection of integers exists in nature either. If you accept the idea of an infinite collection, why not an infinite sequence? Write down a decimal point, then start flipping a coin, 1's and 0's forever: .011010010111... So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact.
  - Nevermark 1y ago
    > So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact. Well no, you will never have it. You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it. That is what countably infinite means. It is a very practical kind of infinity. And the concept comes directly, and inevitably, from the integers. Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints. The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals. Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
    - threatofrain 1y ago
      > It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.
      - Nevermark 1y ago
        > If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Absolutely. When most people think of reals, they are thinking constructible reals. Which are countably infinite in number. If there is an equation (known or in principle), they are constructible. If there is an algorithm (known or in principle), they are constructible. Limits, continuous fractions, all those are constructible. From a mathematics perspective, I think it's a loss that the distinction between countable/constructible numbers vs. uncountable/un-constructible is completely blurred under one name "reals" early in everyone's math education. Even though the difference is significant when reasoning about information, the relationship between math and physics, math and computation, etc. And about infinites. Most of us famously know that there are infinitely more reals than integers. But how many people know that all well defined reals, constructible reals, calculable reals, and their equations, probably everything they imagine when they think of "real" numbers in practice, remain countably infinite. Exactly like the integers. The set of constructible reals is the same size as the set of integers.
- NoahZuniga 1y ago
  > None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. Basically all of calculus needs all the reals because if you don't you get some really pathological results. If you do calculus over the constructible reals, almost all continuous functions will have infinitely many jumps (and these jumps will be infinitely dense too). This means that if a function is continuous, that doesn't mean the integral exists. The only way to avoid these pathological functions is to apply some restriction on how they would behave over the reals, but then you're back to square one.
  - Nevermark 1y ago
    > Basically all of calculus needs all the reals because if you don't you get some really pathological results. This is how deceptive the name and early introduction is. No, algebraic numbers, calculus, diff eq, topology [ , ... ] only need constructible numbers, structures, relationships and other forms and concepts. Pi, e, infinitesimals, converging series, limits, smooth connected continuity, are all constructible. They all have finite symbolic expressions (with finitely describable algorithms for arbitrarily accurate decimal expansion). There is never a situation in calculus where an un-constructible real does anything useful. There is never a situation in calculus where the concept of un-constructible reals is necessary. Bonus challenge: Name a single un-constructible real constant that you have ever encountered. What is the 0, -1, pi, i or e of un-constructible numbers?
    - NoahZuniga 1y ago
      Lets take some un-constructible real number and call it r. Now think of the function f(x)=0 if x<r and 1 if x>r. Edit: I originally said this is a function over the un-constructible reals. That was wrong! This is a function over the constructible reals. What I mean with that is that it only has a value for constructible real numbers, because r isn't constructible, f doesn't have a value at r. That's why I haven't defined what f(r) is. While this function exists (or something like it does, I can prove this fact if you like), this is not a construction because of course r is un-constructible This function is continuous. Let's prove this with the definition of continuity. We say that a function is continuous at a point x if lim a->x f(x) = f(x). We say that a function is continuous if it is continuous at all points. The only point where this function could hypothetically be discontinuous is at the boundary point r, but actually this boundary point doesn't exist for this function (because f(r) isn't defined), so this function is continuous at every point, and is thus a continuous function. Edit: forgot to add, but based on this concept you can then show that almost all continuous functions over the constructible reals have these jumps at infinitely many places, and that means that you can't hope to define a way to integrate continuous functions. I appreciate that this is a pretty complex topic, so that I probably haven't been that clear (or made a mistake), so I value any and all comments. This argument is somewhat adapted from an argument about why we do calculus with the reals and not the rationals. See also this video: https://www.youtube.com/watch?v=vV7ZuouUSfs https://www.youtube.com/watch?v=vV7ZuouUSfs

But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.

Nevermark 1y ago

Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)

throwaway81523 1y ago
It's hard to claim that an infinite (or anyway unbounded) collection of integers exists in nature either. If you accept the idea of an infinite collection, why not an infinite sequence? Write down a decimal point, then start flipping a coin, 1's and 0's forever: .011010010111... So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact.
- Nevermark 1y ago
  > So now you've got a binary fraction that most would say specifies a real number. An almost surely non-constructible one in fact. Well no, you will never have it. You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it. That is what countably infinite means. It is a very practical kind of infinity. And the concept comes directly, and inevitably, from the integers. Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints. The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals. Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
  - threatofrain 1y ago
    > It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.
    - Nevermark 1y ago
      > If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Absolutely. When most people think of reals, they are thinking constructible reals. Which are countably infinite in number. If there is an equation (known or in principle), they are constructible. If there is an algorithm (known or in principle), they are constructible. Limits, continuous fractions, all those are constructible. From a mathematics perspective, I think it's a loss that the distinction between countable/constructible numbers vs. uncountable/un-constructible is completely blurred under one name "reals" early in everyone's math education. Even though the difference is significant when reasoning about information, the relationship between math and physics, math and computation, etc. And about infinites. Most of us famously know that there are infinitely more reals than integers. But how many people know that all well defined reals, constructible reals, calculable reals, and their equations, probably everything they imagine when they think of "real" numbers in practice, remain countably infinite. Exactly like the integers. The set of constructible reals is the same size as the set of integers.
NoahZuniga 1y ago
> None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. Basically all of calculus needs all the reals because if you don't you get some really pathological results. If you do calculus over the constructible reals, almost all continuous functions will have infinitely many jumps (and these jumps will be infinitely dense too). This means that if a function is continuous, that doesn't mean the integral exists. The only way to avoid these pathological functions is to apply some restriction on how they would behave over the reals, but then you're back to square one.
- Nevermark 1y ago
  > Basically all of calculus needs all the reals because if you don't you get some really pathological results. This is how deceptive the name and early introduction is. No, algebraic numbers, calculus, diff eq, topology [ , ... ] only need constructible numbers, structures, relationships and other forms and concepts. Pi, e, infinitesimals, converging series, limits, smooth connected continuity, are all constructible. They all have finite symbolic expressions (with finitely describable algorithms for arbitrarily accurate decimal expansion). There is never a situation in calculus where an un-constructible real does anything useful. There is never a situation in calculus where the concept of un-constructible reals is necessary. Bonus challenge: Name a single un-constructible real constant that you have ever encountered. What is the 0, -1, pi, i or e of un-constructible numbers?
  - NoahZuniga 1y ago
    Lets take some un-constructible real number and call it r. Now think of the function f(x)=0 if x<r and 1 if x>r. Edit: I originally said this is a function over the un-constructible reals. That was wrong! This is a function over the constructible reals. What I mean with that is that it only has a value for constructible real numbers, because r isn't constructible, f doesn't have a value at r. That's why I haven't defined what f(r) is. While this function exists (or something like it does, I can prove this fact if you like), this is not a construction because of course r is un-constructible This function is continuous. Let's prove this with the definition of continuity. We say that a function is continuous at a point x if lim a->x f(x) = f(x). We say that a function is continuous if it is continuous at all points. The only point where this function could hypothetically be discontinuous is at the boundary point r, but actually this boundary point doesn't exist for this function (because f(r) isn't defined), so this function is continuous at every point, and is thus a continuous function. Edit: forgot to add, but based on this concept you can then show that almost all continuous functions over the constructible reals have these jumps at infinitely many places, and that means that you can't hope to define a way to integrate continuous functions. I appreciate that this is a pretty complex topic, so that I probably haven't been that clear (or made a mistake), so I value any and all comments. This argument is somewhat adapted from an argument about why we do calculus with the reals and not the rationals. See also this video: https://www.youtube.com/watch?v=vV7ZuouUSfs https://www.youtube.com/watch?v=vV7ZuouUSfs