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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#141

Post by mathgeek »

JadeNB wrote: post #89
mathgeek wrote: post #77

Earlier quoted context omitted.

> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why. Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends…

> commonly denoted as N_0 and N_1 Oh my, it had never occurred to me that one could disagree, not just about whether the natural numbers include 0 or don't, but also about how to denote "natural numbers with 0" and "natural numbers without." Personally, I'm a fan of Z_{\ge 0} and Z_{> 0}, which are a little ugly but which any mathematician, regardless of their preferred conventions, can read and understand without fu…

Yep, lots of ways to denote these sets. It’s not a disagreement but rather a preference (although certainly some folks will gladly disagree).

Re: Is 1 Prime, and Does It Matter?

#142

Post by mathgeek »

ForOldHack wrote: post #133
mathgeek wrote: post #77

Earlier quoted context omitted.

> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why. Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends…

Number theory includes zero as the identity element for addition, much as 1 is the identity element for multiplication. I am totally assuming you knew this already.

For the sake of making an easy transition to the monoid, yes. Personally a fan.

Re: Is 1 Prime, and Does It Matter?

#143

Post by seanhunter »

reaperman wrote: post #6
JJMcJ wrote: post #3

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".

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.

Exactly. This is similar to the case of how the zero function provides a trivial solution to almost every differential equation.

Re: Is 1 Prime, and Does It Matter?

#144

Post by JadeNB »

chongli wrote: post #140
JadeNB wrote: post #115

Earlier quoted context omitted.

> I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty. It's not necessarily a problem that the empty set cannot be a group. (Although the only reason that it cannot is a definition, and, similarly, the definition of a field requires two distinct elements, which hasn't stopped some people from positing that it is a pr…

If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set. In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial s…

> If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set.

You certainly can do that, but it's not the only way. Even in this universe, I would expect to show that concrete sets A and B are disjoint by showing x ∈ A → x ∉ B, which makes perfect sense even without an empty set.

> In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial set containing only the special element. What a clumsy hack that would be!

Rather, in this alternate universe, intersection is partially defined. Again, even in this universe, we're used to accepting some operations being partial!

Re: Is 1 Prime, and Does It Matter?

#145

Post by chongli »

JadeNB wrote: post #144
chongli wrote: post #140

Earlier quoted context omitted.

If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set. In an alternative axiomatization (without the empty set) you’re going to need to create some special element which belongs to every set and then your definition of disjoint sets is that their intersection is equal to the trivial s…

> If you don’t allow the empty set to be a set then you break the basic operations of set theory. For example, to show two sets are disjoint you compare their intersection with the empty set. You certainly can do that, but it's not the only way. Even in this universe, I would expect to show that concrete sets A and B are disjoint by showing x ∈ A → x ∉ B, which makes perfect sense even without an empty set. > In an a…

Rather, in this alternate universe, intersection is partially defined.

Yes, but then topology becomes a very tedious exercise because so many proofs rely on the fact that the empty set is contained in every topology, that the empty set is both closed and open, and that intersections frequently yield the empty set. With partially defined intersection you're forced to specially handle every case where two sets might be disjoint.

Re: Is 1 Prime, and Does It Matter?

#146

Post by latexr »

bluepnume wrote: post #102
jordigh wrote: post #50

Earlier quoted context omitted.

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-...

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"

Your explanation is true of every prime. I’m pretty sure GP just meant that “2 is the only prime with the additional characteristic of being an even number”. So it’s odd (read “interesting”) in that sense, like if it would be if (for example) any number were to be the sole prime composed of exactly X digits.

Re: Is 1 Prime, and Does It Matter?

#147

Post by implements »

wesselbindt wrote: post #4

If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course

Empty product and also 0! (Which is extra unintuitive because 0 isn’t empty ie the products of 0 are 0)

Re: Is 1 Prime, and Does It Matter?

#148

Post by btilly »

pja wrote: post #138
btilly wrote: post #59

Earlier quoted context omitted.

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_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 potentia…

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)”…

We are able to recognize that they are nonstandard because they contain numbers that we recognize are infinite. But there is absolutely no statement that can be made from within the model from which it could be discovered that those numbers are infinite.

Furthermore, it is possible to construct nonstandard models such that every statement that is true in our model, remains true in that one, and ditto for every statement that is false. They really look identical to our model, except that we know from construction that they aren't. This fact is what makes the transfer principle work in nonstandard analysis, and the ultrapower construction shows how to do it.

(My snark about NSA is that we shouldn't need the axiom of choice to find the derivative of x^2. But I do find it an interesting approach to know about.)

Re: Is 1 Prime, and Does It Matter?

#149

Post by Maxatar »

pja wrote: post #138
btilly wrote: post #59

Earlier quoted context omitted.

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_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 potentia…

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)”…

No additional axioms are needed for the existence of these models. On the contrary additional axioms are needed in order to eliminate them, and even still no amount of axioms can eliminate all of these extensions without introducing an inconsistency.

Re: Is 1 Prime, and Does It Matter?

#150

Post by mystified5016 »

JJMcJ wrote: post #3

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".

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