12 ms·

One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless r

by JJMcJ 1y ago

One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".

reaperman 1y ago
Yes, it's more of a convention where we assume language like "...ignoring the trivial case of 1 being an obvious factor of every integer." It's not interesting or meaningful, so we ignore it for most cases.
- gerdesj 1y ago
  I'm no expert but: "...ignoring the trivial case of 1 being an obvious factor of every integer." I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8) Then you have to define what factor means ...
  - Maxatar 1y ago
    Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.
    - stouset 1y ago
      Do you have a link to where I could learn more about this?
      - gerdesj 1y ago
        You might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach That's the GEB I mentioned above.
        pinkmuffinere 1y ago
        I know this is a great book, it’s been on my to-read list for about 5 years. But I never get to it. Is there not another (shorter) discussion I could read on this? Even an academic paper would be acceptable.
        gerdesj 1y ago
        As I said above, I'm not an expert. However, I read GEB on a whim when bored at school and I think it still informs my thinking 35 years later. Move GEB up the reading list right now! The edition I initially read was hard bound and was quite worn. I bought and read it again about 20 years ago and found more treasures. It is a proper nerd grade treatise for non experts who are interested in maths, music and art. Really: maths, music and art from a mostly mathematical perspective. Hofstadter's writing style is very easy going and he is a master of clarity without complexity. I don't think you need any more Maths than you would get up to age 18 or so at school to understand the entire book and probably less. Even if you gloss the formal Maths the book still works.
        Maxatar 1y ago
        There's no shortage of blog posts on the topic, but here is one that is fairly rigorous but doesn't assume too much background knowledge: https://risingentropy.com/a-result-on-the-incompleteness-of-mathematics/ https://risingentropy.com/a-result-on-the-incompleteness-of-...
        ForOldHack 1y ago
        You could read Kurt Godels paper,but it's literally undecyperable. The book is one on the best reads ever. It will also teach you how to think in very very formal ways. It made Calculus half the class it was, and I breezed through finite math. Propositional Calulus will teach you to think in symbols you cannot even fathom. This alone is worth every minute reading the book. Every few years I reread it, and get a new sense of solving problems. The book can be divided into parts... But the whole...
    - overboard2 1y ago
      What do you mean by "not actually"? Edit: do you mean literally impossible?
      - Maxatar 1y ago
        I mean it's logically impossible to formally and specifically define the natural numbers without introducing a logical inconsistency. The best you can do is define a set that has all the properties of natural numbers but will also define things that aren't natural numbers as well. As an analogy you could imagine trying to define the set of all animals with a bunch of rules... "1. Animals have DNA, 2. Animals ingest organic matter. 3. Animals have a nervous system. 4. ... etc..." And this is true of all animals, but it will also be true of things that aren't animals as well, like slime molds which are not quite animals but very similar to them. Okay so you keep adding more rules to narrow down your definition and stamp out slime molds, but you find some other thing satisfy that definition... Now for animals maybe you can eventually have some very complex rule set that defines animals exactly and rules out all non-animals, but the principle is that this is not possible for natural numbers. We can have rules like "0" is a natural number. For every natural number N there is a successor to it N + 1. If N + 1 = M + 1 then N = M. There is no natural number Q such that Q + 1 = 0. Okay this is a good starting point... but just like with animals there are numbers that satisfy all of these rules but aren't natural numbers. You can keep adding more and more rules to try to stamp these numbers out, but no matter how hard, even if you add infinitely many rules, there will always be infinitely many numbers that satisfy your rules but aren't natural numbers. In particular what you really want to say is that a natural number is finite, but no matter how hard you try there is no formal way to actually capture the concept of what it means to be finite in general so you end up with these mutant numbers that satisfy all of your rules but have infinitely many digits, and these are called non-standard natural numbers. The reason non-standard natural numbers are a problem is because you might have a statement like "Every even integer greater than 2 can be written as the sum of two primes." and this statement might be true of the actual natural numbers but there might exist some freak mutant non-standard natural number for which it's not true. Unless your rules are able to stamp out these mutant non-standard natural numbers, then it is not possible to prove this statement, the statement becomes undecidable with respect to your rules. The only statements you can prove with respect to your rules are statements that are true of the real natural numbers as well as true of all the mutant natural numbers that your rules have not been able to stamp out. So it's in this sense that I mean that it's not possible to specifically define the natural numbers. Any definition you come up with will also apply to mutant numbers, and these mutant numbers can get in the way of you proving things that are in principle true about the actual natural numbers.
    - pja 1y ago
      > Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers. Are such objects not inevitably isomorphic to the natural numbers? Can you give an example of a formal definition that leads to something that obviously isn't the same as the naturals?
      - btilly 1y ago
        The Peano Axioms lead to both the standard model of arithmetic (the integers that we want), and nonstandard models. See https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet.... In that article you'll see references to "first order logic" and "second order logic". First order logic captures any possible finite chain of reasoning. Second order logic allows us to take logical steps that would require a potentially infinite amount of reasoning to do. Gödel's famous theorems were about the limitations of first order logic. While second order logic has no such limitations, it is also not something that humans can actually do. (We can reason about second order logic though.) Anyways a nonstandard model of arithmetic can have all sorts of bizarre things. Such as a proof that Peano Axioms lead to a contradiction. While it might seem that this leads to a contradiction in the Peano Axioms, it doesn't because the "proof" is (from our point of view) infinitely long, and so not really a proof at all! (This is also why logicians have to draw a very careful distinction between "these axioms prove" and "these axioms prove that they prove"...)
        pja 1y ago
        All of these models appear to contain infinitely sized objects that are explicitly named / manipulable within the model, which makes them extensions of the Peano numbers though, or else they add other, extra axioms to the Peano model. If you (for example) extend Peano numbers with extra axioms that state things like “hey, here are some hyperreals” or “this Goedel sentence is explicitly defined to be true (or false)” it’s unsurprising that you can end up in some weird places.
    - john-h-k 1y ago
      Can you elaborate on this? My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable. You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable. Im not sure how undecidable <=> “will define things that are similar to natural numbers but are not” but maybe I am missing something
      - Maxatar 1y ago
        Yeah for sure. If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisfies S and B satisfies not S. So which one is the standard natural numbers, is it A or B? Either A or B will be an example of something that satisfies your definition of natural numbers and yet is not the natural numbers.
    - sam_ezeh 1y ago
      >Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you can find the proof here [1] if you're interested [1]: https://builds.openlogicproject.org/content/second-order-logic/metatheory/second-order-arithmetic.pdf https://builds.openlogicproject.org/content/second-order-log...
      - Maxatar 1y ago
        This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.
        sam_ezeh 1y ago
        > So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. Can you explain what you mean here? Full semantics for second-order logic has a unique interpretation i.e. the standard natural numbers
        Maxatar 1y ago
        Interpretation under full second‑order logic is not intrinsic to the logic itself but is always supplied by a richer meta‑theory, usually set theory/ZF. The sentence "All subsets of N" has no standalone meaning in second-order logic, it must be defined inside of the meta-theory, which in turn relies on its own meta‑theory, and so on ad infinitum. Thus, although full second order Peano axioms are categorical, second order logic by itself never delivers a self‑contained model of the natural numbers. Any actual interpretation of the natural numbers in second order logic requires an infinite regress of background theories.
    - LudwigNagasena 1y ago
      Second-order arithmetic formalizes both natural and real numbers. Though it has a host of issues with inference.
  - drdeca 1y ago
    The previous poster didn’t describe the natural numbers as trivial. Rather, described a case as trivial. Specifically, the case of the divisor being 1.
- seanhunter 1y ago
  Exactly. This is similar to the case of how the zero function provides a trivial solution to almost every differential equation.
JadeNB 1y ago
> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently explaining that we declare that the empty set is not a set, to avoid complicating theorems, proofs, and exposition by the endless repetition of "non-empty set." (I agree that this is different, because there's no interesting "unique factorization theorem" for sets, but I can still imagine things developing this way. And, indeed, there are complications caused by allowing the empty set in a model of a structure, and someone determined to do so can make themselves pointlessly unpopular by asking "but have you considered the empty manifold?" and similar questions. See also https://mathoverflow.net/questions/45951/interesting-examples-of-vacuous-void-entities https://mathoverflow.net/questions/45951/interesting-example....)
- tux3 1y ago
  That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth. That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!
  - JadeNB 1y ago
    > That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable! "Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me. > That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth. In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.
tikhonj 1y ago
And the reason we'd have to constantly exclude 1 is that it behaves in a qualitatively different way than prime numbers—and understand what this means and why that's the case is the real insight here.
jordigh 1y ago
To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-odd-prime https://math.stackexchange.com/questions/1177104/what-is-an-...
- kordlessagain 1y ago
  The concept of “one” holds a dual role. It represents a countable unit: something you can put in a bowl and also stands for indivisibility itself. When you divide any quantity by an indivisible unit, you’re simply counting how many of those indivisibles fit within it. Then comes 2: the first number that is divisible, but only by itself and the indivisible one. That’s what makes it prime. A prime is a number divisible only by itself and by 1, the indivisible origin of all counting.
  - aleph_minus_one 1y ago
    > Then comes 2: the first number that is divisible, but only by itself and the indivisible one. This does hold in the ring Z. In the ring Z[i], 2 = (1+i)*(1-i), and the two factors are prime elements.
- brennopost 1y ago
  It's actually the least odd prime
- chrismcb 1y ago
  It isn't odd at all! And that I'm being pendantic. But you can't say it is very odd, and then I'm the next sentence day "for every odd prime..."
- arcastroe 1y ago
  "2 is the only even prime number. Therefore, it's the oddest of them all!"
- bluepnume 1y ago
  It's hardly odd. "Even" just means "divisible by 2" "2 is the only prime that is divisible by 2" "3 is the only prime that is divisible by 3" "5 is the only prime that is divisible by 5" ... "N is the only prime that is divisible by N"
mystified5016 1y ago
Seems that if we must add all these conditions to make the definition of prime consistent, maybe we shouldn't consider it prime?

by JJMcJ 1y ago

One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".

reaperman 1y ago

Yes, it's more of a convention where we assume language like "...ignoring the trivial case of 1 being an obvious factor of every integer." It's not interesting or meaningful, so we ignore it for most cases.

gerdesj 1y ago
I'm no expert but: "...ignoring the trivial case of 1 being an obvious factor of every integer." I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8) Then you have to define what factor means ...
- Maxatar 1y ago
  Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.
  - stouset 1y ago
    Do you have a link to where I could learn more about this?
    - gerdesj 1y ago
      You might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach That's the GEB I mentioned above.
      - pinkmuffinere 1y ago
        I know this is a great book, it’s been on my to-read list for about 5 years. But I never get to it. Is there not another (shorter) discussion I could read on this? Even an academic paper would be acceptable.
        gerdesj 1y ago
        As I said above, I'm not an expert. However, I read GEB on a whim when bored at school and I think it still informs my thinking 35 years later. Move GEB up the reading list right now! The edition I initially read was hard bound and was quite worn. I bought and read it again about 20 years ago and found more treasures. It is a proper nerd grade treatise for non experts who are interested in maths, music and art. Really: maths, music and art from a mostly mathematical perspective. Hofstadter's writing style is very easy going and he is a master of clarity without complexity. I don't think you need any more Maths than you would get up to age 18 or so at school to understand the entire book and probably less. Even if you gloss the formal Maths the book still works.
        Maxatar 1y ago
        There's no shortage of blog posts on the topic, but here is one that is fairly rigorous but doesn't assume too much background knowledge: https://risingentropy.com/a-result-on-the-incompleteness-of-mathematics/ https://risingentropy.com/a-result-on-the-incompleteness-of-...
        ForOldHack 1y ago
        You could read Kurt Godels paper,but it's literally undecyperable. The book is one on the best reads ever. It will also teach you how to think in very very formal ways. It made Calculus half the class it was, and I breezed through finite math. Propositional Calulus will teach you to think in symbols you cannot even fathom. This alone is worth every minute reading the book. Every few years I reread it, and get a new sense of solving problems. The book can be divided into parts... But the whole...
  - overboard2 1y ago
    What do you mean by "not actually"? Edit: do you mean literally impossible?
    - Maxatar 1y ago
      I mean it's logically impossible to formally and specifically define the natural numbers without introducing a logical inconsistency. The best you can do is define a set that has all the properties of natural numbers but will also define things that aren't natural numbers as well. As an analogy you could imagine trying to define the set of all animals with a bunch of rules... "1. Animals have DNA, 2. Animals ingest organic matter. 3. Animals have a nervous system. 4. ... etc..." And this is true of all animals, but it will also be true of things that aren't animals as well, like slime molds which are not quite animals but very similar to them. Okay so you keep adding more rules to narrow down your definition and stamp out slime molds, but you find some other thing satisfy that definition... Now for animals maybe you can eventually have some very complex rule set that defines animals exactly and rules out all non-animals, but the principle is that this is not possible for natural numbers. We can have rules like "0" is a natural number. For every natural number N there is a successor to it N + 1. If N + 1 = M + 1 then N = M. There is no natural number Q such that Q + 1 = 0. Okay this is a good starting point... but just like with animals there are numbers that satisfy all of these rules but aren't natural numbers. You can keep adding more and more rules to try to stamp these numbers out, but no matter how hard, even if you add infinitely many rules, there will always be infinitely many numbers that satisfy your rules but aren't natural numbers. In particular what you really want to say is that a natural number is finite, but no matter how hard you try there is no formal way to actually capture the concept of what it means to be finite in general so you end up with these mutant numbers that satisfy all of your rules but have infinitely many digits, and these are called non-standard natural numbers. The reason non-standard natural numbers are a problem is because you might have a statement like "Every even integer greater than 2 can be written as the sum of two primes." and this statement might be true of the actual natural numbers but there might exist some freak mutant non-standard natural number for which it's not true. Unless your rules are able to stamp out these mutant non-standard natural numbers, then it is not possible to prove this statement, the statement becomes undecidable with respect to your rules. The only statements you can prove with respect to your rules are statements that are true of the real natural numbers as well as true of all the mutant natural numbers that your rules have not been able to stamp out. So it's in this sense that I mean that it's not possible to specifically define the natural numbers. Any definition you come up with will also apply to mutant numbers, and these mutant numbers can get in the way of you proving things that are in principle true about the actual natural numbers.
  - pja 1y ago
    > Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers. Are such objects not inevitably isomorphic to the natural numbers? Can you give an example of a formal definition that leads to something that obviously isn't the same as the naturals?
    - btilly 1y ago
      The Peano Axioms lead to both the standard model of arithmetic (the integers that we want), and nonstandard models. See https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic https://en.wikipedia.org/wiki/Non-standard_model_of_arithmet.... In that article you'll see references to "first order logic" and "second order logic". First order logic captures any possible finite chain of reasoning. Second order logic allows us to take logical steps that would require a potentially infinite amount of reasoning to do. Gödel's famous theorems were about the limitations of first order logic. While second order logic has no such limitations, it is also not something that humans can actually do. (We can reason about second order logic though.) Anyways a nonstandard model of arithmetic can have all sorts of bizarre things. Such as a proof that Peano Axioms lead to a contradiction. While it might seem that this leads to a contradiction in the Peano Axioms, it doesn't because the "proof" is (from our point of view) infinitely long, and so not really a proof at all! (This is also why logicians have to draw a very careful distinction between "these axioms prove" and "these axioms prove that they prove"...)
      - pja 1y ago
        All of these models appear to contain infinitely sized objects that are explicitly named / manipulable within the model, which makes them extensions of the Peano numbers though, or else they add other, extra axioms to the Peano model. If you (for example) extend Peano numbers with extra axioms that state things like “hey, here are some hyperreals” or “this Goedel sentence is explicitly defined to be true (or false)” it’s unsurprising that you can end up in some weird places.
  - john-h-k 1y ago
    Can you elaborate on this? My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable. You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable. Im not sure how undecidable <=> “will define things that are similar to natural numbers but are not” but maybe I am missing something
    - Maxatar 1y ago
      Yeah for sure. If a sentence S is undecidable from your axioms for the natural numbers then there are two models A and B satisfying those axioms where A satisfies S and B satisfies not S. So which one is the standard natural numbers, is it A or B? Either A or B will be an example of something that satisfies your definition of natural numbers and yet is not the natural numbers.
  - sam_ezeh 1y ago
    >Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you can find the proof here [1] if you're interested [1]: https://builds.openlogicproject.org/content/second-order-logic/metatheory/second-order-arithmetic.pdf https://builds.openlogicproject.org/content/second-order-log...
    - Maxatar 1y ago
      This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.
      - sam_ezeh 1y ago
        > So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. Can you explain what you mean here? Full semantics for second-order logic has a unique interpretation i.e. the standard natural numbers
        Maxatar 1y ago
        Interpretation under full second‑order logic is not intrinsic to the logic itself but is always supplied by a richer meta‑theory, usually set theory/ZF. The sentence "All subsets of N" has no standalone meaning in second-order logic, it must be defined inside of the meta-theory, which in turn relies on its own meta‑theory, and so on ad infinitum. Thus, although full second order Peano axioms are categorical, second order logic by itself never delivers a self‑contained model of the natural numbers. Any actual interpretation of the natural numbers in second order logic requires an infinite regress of background theories.
  - LudwigNagasena 1y ago
    Second-order arithmetic formalizes both natural and real numbers. Though it has a host of issues with inference.
- drdeca 1y ago
  The previous poster didn’t describe the natural numbers as trivial. Rather, described a case as trivial. Specifically, the case of the divisor being 1.
seanhunter 1y ago
Exactly. This is similar to the case of how the zero function provides a trivial solution to almost every differential equation.

JadeNB 1y ago

> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently explaining that we declare that the empty set is not a set, to avoid complicating theorems, proofs, and exposition by the endless repetition of "non-empty set." (I agree that this is different, because there's no interesting "unique factorization theorem" for sets, but I can still imagine things developing this way. And, indeed, there are complications caused by allowing the empty set in a model of a structure, and someone determined to do so can make themselves pointlessly unpopular by asking "but have you considered the empty manifold?" and similar questions. See also https://mathoverflow.net/questions/45951/interesting-examples-of-vacuous-void-entities https://mathoverflow.net/questions/45951/interesting-example....)

tux3 1y ago
That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth. That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!
- JadeNB 1y ago
  > That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable! "Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me. > That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth. In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.

tikhonj 1y ago

And the reason we'd have to constantly exclude 1 is that it behaves in a qualitatively different way than prime numbers—and understand what this means and why that's the case is the real insight here.

jordigh 1y ago

To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-odd-prime https://math.stackexchange.com/questions/1177104/what-is-an-...

kordlessagain 1y ago
The concept of “one” holds a dual role. It represents a countable unit: something you can put in a bowl and also stands for indivisibility itself. When you divide any quantity by an indivisible unit, you’re simply counting how many of those indivisibles fit within it. Then comes 2: the first number that is divisible, but only by itself and the indivisible one. That’s what makes it prime. A prime is a number divisible only by itself and by 1, the indivisible origin of all counting.
- aleph_minus_one 1y ago
  > Then comes 2: the first number that is divisible, but only by itself and the indivisible one. This does hold in the ring Z. In the ring Z[i], 2 = (1+i)*(1-i), and the two factors are prime elements.
brennopost 1y ago
It's actually the least odd prime
chrismcb 1y ago
It isn't odd at all! And that I'm being pendantic. But you can't say it is very odd, and then I'm the next sentence day "for every odd prime..."
arcastroe 1y ago
"2 is the only even prime number. Therefore, it's the oddest of them all!"
bluepnume 1y ago
It's hardly odd. "Even" just means "divisible by 2" "2 is the only prime that is divisible by 2" "3 is the only prime that is divisible by 3" "5 is the only prime that is divisible by 5" ... "N is the only prime that is divisible by N"

mystified5016 1y ago

Seems that if we must add all these conditions to make the definition of prime consistent, maybe we shouldn't consider it prime?