6 ms·

Yeah, it's important to think of these axioms as choosing the rules of the game, rather than what intuitively makes sense. The real question is if playing the g

Yeah, it's important to think of these axioms as choosing the rules of the game, rather than what intuitively makes sense. The real question is if playing the game produces useful results.

btilly 1y ago
Spoken like a true formalist. It doesn't really have to mean anything when we say that the reals are a larger set than the natural numbers - that's just the conclusion of the game that we are playing. What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
- skissane 1y ago
  > What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with? The claim that there are more reals than naturals holds given classical ZF(C) set theory. But there are alternative set theories in which the reals are countable, e.g. NFU+AxCount. These alternative set theories ensure all reals are countable by rendering Cantor’s diagonalisation argument invalid, since their axioms are too weak to validate it. But, they contain all the same reals as the high school mathematics concept of “reals”. So, there are many reals, and that some of them are countable and others are not are indeed “eternal truths” (it is an eternal truth that whatever axioms have the consequences they do), but the everyday (non-expert) concept of reals isn’t any of them in particular - and it is unclear if the dominance of classical notions in mainstream professional mathematics was historically inevitable or a historical accident - maybe, on the other side of the galaxy, there exists some alien civilisation, in which different foundations of mathematics are mainstream, because their mathematics took a different evolutionary course from ours - maybe for them, reals are classically countable, and uncountability is an exotic notion belonging to alternative foundations of mathematics
  - btilly 1y ago
    As I pointed out at https://news.ycombinator.com/item?id=44271589 https://news.ycombinator.com/item?id=44271589, there are systems that can accept Cantor's argument, without concluding that there are more reals than rational numbers. As you point out, there are many mathematical systems that contain all of the numbers in the high school mathematics concept of "reals". Since those with a high school understanding of reals do not know which of those systems they would agree with, they should not be asked to accept as true, any results that hold in only some of those systems. And that is why I don't like mathematicians telling lay audiences that there are more reals than rationals.
    - zozbot234 1y ago
      "Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument. Whether these systems truly accept "Cantor's argument" is perhaps only a matter of perception and intuition, that people may perhaps disagree about.
      - btilly 1y ago
        It does not matter what your best understanding of Cantor's diagonalization argument is. In some mathematical systems it means, "there are more reals than natural numbers", and in others it means, "the reals encode self-reference in a more direct way than the natural numbers do". The result is that it is possible for the acceptance of the argument to lead to very different consequences about what we then conclude.
      - skissane 1y ago
        > "Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument. Okay, but can you prove Lawvere’s theorem in NFU+AxCount? And even if you can, since NFU+AxCount proves that the reals are countable, if NFU+AxCount proves Lawvere, then (to echo what btilly says in a sibling comment) NFU+AxCount+Lawvere couldn’t entail the countability of the reals, since that would render NFU+AxCount trivially inconsistent, and we know it is isn’t trivially inconsistent (as with any formal system, consistency is ultimately unprovable, but if a system is taken seriously as an object of mathematical research, then any inconsistency must be highly non-trivial.)
- deleted 1y ago
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- karmakurtisaani 1y ago
  > Spoken like a true formalist. Doesn't seem to be a bad thing. There are some famous cranks who reject the concept of infinity, since I suppose they have problems wrapping their head around it. > What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with? People misunderstand mathematics all the time. It's ok, it's part of the journey.
woopsn 1y ago
Axioms are also introduced in practical terms just to make proofs and results "better". Usually we talk in terms of what propositions are provable, saying that indicates the strength/power of these assumptions, but beyond this there are issues of proof length and complexity. For example in arithmetic without induction, roughly, theorems remain the same (those which can still be expressed) but may have exponentially longer proofs because of the loss of those `∀n P(n)`-type propositions. In this sense it does sometimes come back to intuition. If for all n we can prove P(n), then `∀n P(n)` should be an acceptable proposition and doesn't really change "the game" we are trying to play. It just makes it more intuitive and playable.
- karmakurtisaani 1y ago
  Good point. I would argue, however, that having nicer proofs is a "useful" result of the game.
- SabrinaJewson 1y ago
  I’m not sure what you mean by “theorems remain the same”. If you take away induction from Peano arithmetic, you get Robinson arithmetic, which has many more models, including (from https://math.stackexchange.com/a/4076545 https://math.stackexchange.com/a/4076545): - ℕ ∪ {∞} - Cardinal arithmetic - ℤ[x]⁺ Obviously, not all theorems that are true for the natural numbers are true for cardinals, so it seems misleading to say that theorems remain the same. I also believe that the addition of induction increases the consistency strength of the theory, so it’s not “just” a matter of expressing the theorems in a different way. I would agree more for axioms that don’t affect consistency strength, like foundation or choice (over the rest of the ZF axioms).
  - deleted 1y ago
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  - woopsn 1y ago
    If I had to write again I might say "same theorems about natural numbers" and capitalize ROUGHLY. It is a conversation, what exactly I am weaseling around (not just nonstandard model theoretic issues), and I take your caveat about consistency strength - with that said would you still call it misleading? Why is it that eg x+y=y+x for x y given takes exponential length proof in Robinson compared to PA? For the reason stated, which is true in a very broad sense.

Yeah, it's important to think of these axioms as choosing the rules of the game, rather than what intuitively makes sense. The real question is if playing the game produces useful results.

btilly 1y ago

Spoken like a true formalist. It doesn't really have to mean anything when we say that the reals are a larger set than the natural numbers - that's just the conclusion of the game that we are playing. What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?

skissane 1y ago
> What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with? The claim that there are more reals than naturals holds given classical ZF(C) set theory. But there are alternative set theories in which the reals are countable, e.g. NFU+AxCount. These alternative set theories ensure all reals are countable by rendering Cantor’s diagonalisation argument invalid, since their axioms are too weak to validate it. But, they contain all the same reals as the high school mathematics concept of “reals”. So, there are many reals, and that some of them are countable and others are not are indeed “eternal truths” (it is an eternal truth that whatever axioms have the consequences they do), but the everyday (non-expert) concept of reals isn’t any of them in particular - and it is unclear if the dominance of classical notions in mainstream professional mathematics was historically inevitable or a historical accident - maybe, on the other side of the galaxy, there exists some alien civilisation, in which different foundations of mathematics are mainstream, because their mathematics took a different evolutionary course from ours - maybe for them, reals are classically countable, and uncountability is an exotic notion belonging to alternative foundations of mathematics
- btilly 1y ago
  As I pointed out at https://news.ycombinator.com/item?id=44271589 https://news.ycombinator.com/item?id=44271589, there are systems that can accept Cantor's argument, without concluding that there are more reals than rational numbers. As you point out, there are many mathematical systems that contain all of the numbers in the high school mathematics concept of "reals". Since those with a high school understanding of reals do not know which of those systems they would agree with, they should not be asked to accept as true, any results that hold in only some of those systems. And that is why I don't like mathematicians telling lay audiences that there are more reals than rationals.
  - zozbot234 1y ago
    "Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument. Whether these systems truly accept "Cantor's argument" is perhaps only a matter of perception and intuition, that people may perhaps disagree about.
    - btilly 1y ago
      It does not matter what your best understanding of Cantor's diagonalization argument is. In some mathematical systems it means, "there are more reals than natural numbers", and in others it means, "the reals encode self-reference in a more direct way than the natural numbers do". The result is that it is possible for the acceptance of the argument to lead to very different consequences about what we then conclude.
    - skissane 1y ago
      > "Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument. Okay, but can you prove Lawvere’s theorem in NFU+AxCount? And even if you can, since NFU+AxCount proves that the reals are countable, if NFU+AxCount proves Lawvere, then (to echo what btilly says in a sibling comment) NFU+AxCount+Lawvere couldn’t entail the countability of the reals, since that would render NFU+AxCount trivially inconsistent, and we know it is isn’t trivially inconsistent (as with any formal system, consistency is ultimately unprovable, but if a system is taken seriously as an object of mathematical research, then any inconsistency must be highly non-trivial.)
deleted 1y ago
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karmakurtisaani 1y ago
> Spoken like a true formalist. Doesn't seem to be a bad thing. There are some famous cranks who reject the concept of infinity, since I suppose they have problems wrapping their head around it. > What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with? People misunderstand mathematics all the time. It's ok, it's part of the journey.

woopsn 1y ago

Axioms are also introduced in practical terms just to make proofs and results "better". Usually we talk in terms of what propositions are provable, saying that indicates the strength/power of these assumptions, but beyond this there are issues of proof length and complexity. For example in arithmetic without induction, roughly, theorems remain the same (those which can still be expressed) but may have exponentially longer proofs because of the loss of those `∀n P(n)`-type propositions. In this sense it does sometimes come back to intuition. If for all n we can prove P(n), then `∀n P(n)` should be an acceptable proposition and doesn't really change "the game" we are trying to play. It just makes it more intuitive and playable.

karmakurtisaani 1y ago
Good point. I would argue, however, that having nicer proofs is a "useful" result of the game.
SabrinaJewson 1y ago
I’m not sure what you mean by “theorems remain the same”. If you take away induction from Peano arithmetic, you get Robinson arithmetic, which has many more models, including (from https://math.stackexchange.com/a/4076545 https://math.stackexchange.com/a/4076545): - ℕ ∪ {∞} - Cardinal arithmetic - ℤ[x]⁺ Obviously, not all theorems that are true for the natural numbers are true for cardinals, so it seems misleading to say that theorems remain the same. I also believe that the addition of induction increases the consistency strength of the theory, so it’s not “just” a matter of expressing the theorems in a different way. I would agree more for axioms that don’t affect consistency strength, like foundation or choice (over the rest of the ZF axioms).
- deleted 1y ago
  [deleted]
- woopsn 1y ago
  If I had to write again I might say "same theorems about natural numbers" and capitalize ROUGHLY. It is a conversation, what exactly I am weaseling around (not just nonstandard model theoretic issues), and I take your caveat about consistency strength - with that said would you still call it misleading? Why is it that eg x+y=y+x for x y given takes exponential length proof in Robinson compared to PA? For the reason stated, which is true in a very broad sense.