Exploring the Mystery of Prime Numbers

[matt grime]

Thanks for the input. If someone can help me with the above T/F statements then I would be very grateful and will be ready to close this thread (much to everyone's relief I am sure!). Obviously, I need to go study... : )

You can't answer them because it is purely a formal opinion of whether they are true or false. Your T/F questions seem purely ontological, if that's the word. In what sense do any mathematical objects exist?

 

[CRGreathouse]

You can't answer them because it is purely a formal opinion of whether they are true or false. Your T/F questions seem purely ontological, if that's the word. In what sense do any mathematical objects exist?

Heh. The only branch of philosophy I did ever get deeply into was ontology... maybe that's why I can take this discussion. As far as that goes, Matt, you're a formalist, yes?

 

[philiprdutton]

nesting

...3. All elements in the set {u*, S(u*), S(S(u*)), S(S(S(u*))), S(S(S(S(u*)))), S(S(S(S(S(u*))))), ...} are different.

Uhm... how do you handle "nestedness" without a reference point? This is why when I think about natural numbers and the operations I get so adamant about trying to figure out how the heck a reference point comes into play (if it actually does). In my humble lowly view, I believe there just absolutely has to be a reference point in there no matter whether you are using some set-theoretic notation or encoding the reference point in the axioms themselves...

Also, thanks so much for contributing your information thus far!

I must say my head hurts after trying to read it... but it is not you- it is me!

 

[philiprdutton]

hacking

They work in any order. It would make sense to have axiom 1 come before axiom 8, but this is not strictly necessary.

Axiom 9 is important for proofs but if left off, many problems could still be stated.
I leave off 2-5, as these simply define equality. You may amuse yourself by removing one or more of these, which effectively replaces equality with a certain kind of (possibly equivalence) relation.

If #1 is removed, the system is either null, unchanged, or unchanged except with the addition of finitely many ur-elements, which don't actually change things at all from a set-theoretic point of view. (They don't give it more expressive power.) Essentially all proofs are either nonconstructive or conditional.

If #6 is removed, the system may be unchanged or have only finitely many natural numbers -- perhaps only one.

I'm not quite sure what the effects of removing #7 would be. Could S be multivalued, or is it defined as a function? This may lead to natural numbers as an incomparable web rather than a chain. Perhaps Matt will lend his talents here...?

If #8 is removed there may be only finitely many numbers. If so, they may either end at an element (call it "infinity") that is its own successor, or may loop at some point. In either case there would be a finite chain of natural numbers, then a ring that functions like the integers modulo a constant.

If #9 is removed there may be inaccessible natural numbers (numbers not in {1, S(1), S(S(1)), ...}). Proofs become difficult.

Thank you for your analysis. I have begun to see many people resort back to a set-theoretic notation as in your case of the #1 axiom quoted above. A few times, lately, I have seen the set-theoretic articulation of the natural numbers. Every time I see such an articulation, I get the "shivers" because it looks so nested. When I think about "nesting" I can not understand how "nesting" could ever exist without a reference point.

The rest I am not able to comment on due to lack of understanding. However, I am deftly grateful for the valiant list you put forth. Honestly, I would love a poster of "what happens if you remove the axioms" as opposed to the axioms themselves (not afraid to admit that I love math posters!).

 

[CRGreathouse]

A few times, lately, I have seen the set-theoretic articulation of the natural numbers. Every time I see such an articulation, I get the "shivers" because it looks so nested. When I think about "nesting" I can not understand how "nesting" could ever exist without a reference point.

In set theory, the natural reference point is the null set, because once you know how to form sets ("set-builder notation") you don't need any further axioms to form the null set. It provides a concrete starting point too -- unlike the Peano "1", which could be anything, the nukll set is easy to grasp.

The rest I am not able to comment on due to lack of understanding. However, I am deftly grateful for the valiant list you put forth. Honestly, I would love a poster of "what happens if you remove the axioms" as opposed to the axioms themselves (not afraid to admit that I love math posters!).

Heh, maybe I'll make one.

 

[philiprdutton]

null set = references point

In set theory, the natural reference point is the null set, because once you know how to form sets ("set-builder notation") you don't need any further axioms to form the null set. It provides a concrete starting point too -- unlike the Peano "1", which could be anything, the nukll set is easy to grasp.

So, the ability to talk of numbers and the operations on them, and the notion of prime, is essentially due to the power of NESTING? Is this all?

 

[phoenixthoth]

Uhm... how do you handle "nestedness" without a reference point? This is why when I think about natural numbers and the operations I get so adamant about trying to figure out how the heck a reference point comes into play (if it actually does). In my humble lowly view, I believe there just absolutely has to be a reference point in there no matter whether you are using some set-theoretic notation or encoding the reference point in the axioms themselves...

Also, thanks so much for contributing your information thus far!

I must say my head hurts after trying to read it... but it is not you- it is me!

I think I may be misunderstanding your main question here. If I do understand, then any selected element u* can be the reference point%, such as the set whose only element is the null set. That is a nice choice because its cardinality is what is "natural" to think of as "1."

As far as nestedness, I'm not sure what you mean so if it's basically the iterates of what I'm calling a successor function, they are no less "natural" than the first iterate.

Sorry if I misunderstood.

%so long as whatever function you're using for the successor does NOT have u*, the reference point, in its range, i.e., nothing's successor is u*.

btw, not that it matters much, my condition 3 on a successor function could be stated as something like no elements in the orbit of u* are periodic points of S (of any period). Or, perhaps not as compactly, as the following set is pairwise disjoint:
{{x} : x is in the orbit of u* under S}. Basically my three conditions are trying to generalize the essential characteristics a successor function would have. For example, u* could be the square root of 5, if U is taken to be the set of real numbers, and S(x) = x-1. What I'm thinking is the set
{sqrt(5), sqrt(5) -1, sqrt(5)-2, sqrt(5)-3, ...} is, in some sense, like the usual set of natural numbers (except in this case, the order R I defined would reverse the 'usual' order...I defined the order so that x <= y if y is in the orbit of the successor function applied to x.)

 

[philiprdutton]

nestedness in binary digit sequences

I think I may be misunderstanding your main question here. If I do understand, then any selected element u* can be the reference point%, such as the set whose only element is the null set. That is a nice choice because its cardinality is what is "natural" to think of as "1."

As far as nestedness, I'm not sure what you mean so if it's basically the iterates of what I'm calling a successor function, they are no less "natural" than the first iterate.

Sorry if I misunderstood.

%so long as whatever function you're using for the successor does NOT have u*, the reference point, in its range, i.e., nothing's successor is u*.

btw, not that it matters much, my condition 3 on a successor function could be stated as something like no elements in the orbit of u* are periodic points of S (of any period). Or, perhaps not as compactly, as the following set is pairwise disjoint:
{{x} : x is in the orbit of u* under S}. Basically my three conditions are trying to generalize the essential characteristics a successor function would have. For example, u* could be the square root of 5, if U is taken to be the set of real numbers, and S(x) = x-1. What I'm thinking is the set
{sqrt(5), sqrt(5) -1, sqrt(5)-2, sqrt(5)-3, ...} is, in some sense, like the usual set of natural numbers (except in this case, the order R I defined would reverse the 'usual' order...I defined the order so that x <= y if y is in the orbit of the successor function applied to x.)

What I meant by "nestedness" (or "nesting") is best described like:

linear: ...{}{}{}{}{}...
nested: nothing, then {}, then {{}}, then{ {}, {{}} } ... etc.A standard visualization of nesting can be seen from the output of, say, a Cellular Automaton like the "Rule 90":[/PLAIN]
http://mathworld.wolfram.com/Rule90.html[/URL]
Or, what can be seen with the bit sequences of successive binary numbers:
[URL="http://www.wolframscience.com/nksonline/page-117"]http://www.wolframscience.com/nksonline/page-117

 

[CRGreathouse]

So, the ability to talk of numbers and the operations on them, and the notion of prime, is essentially due to the power of NESTING?

No. Nesting is a convenient way to model the Peano successor function, but it isn't required -- another method could be used instead.

 

[philiprdutton]

No. Nesting is a convenient way to model the Peano successor function, but it isn't required -- another method could be used instead.

Just a clarification of my thoughts. Sure, there may be other methods besides "nesting" to model the Peano successor function. Actually, I was a bit more keen on the idea that there could be a successor function unlike the Peano successor function- one where there actually is not a reference point (like in the "counting/metronome" system I was trying to describe in earlier posts). Perhaps the thing that distinguishes the Peano successor function from some other successor function is that there might be a reference point in the Peano successor (as it is defined)? More precisely, are all successor functions based on a reference point? Is Peano's?

 

[CRGreathouse]

Actually, I was a bit more keen on the idea that there could be a successor function unlike the Peano successor function- one where there actually is not a reference point (like in the "counting/metronome" system I was trying to describe in earlier posts). Perhaps the thing that distinguishes the Peano successor function from some other successor function is that there might be a reference point in the Peano successor (as it is defined)? More precisely, are all successor functions based on a reference point? Is Peano's?

I don't get it. I did explain that if you take away the starting point ("1" or "0") you get the same system, one with ur-elements, or an empty collection. Do you mean something distinct from this? Perhaps you mean a system like this:

1. For each number n, S(n) exists.
2. For each number n, P(n) exists.
3. For each number n, S(P(n)) = n.

which could be a model of the integers instead of the natural numbers?

 

[philiprdutton]

somehow

I don't get it. I did explain that if you take away the starting point ("1" or "0") you get the same system, one with ur-elements, or an empty collection. Do you mean something distinct from this? Perhaps you mean a system like this:

1. For each number n, S(n) exists.
2. For each number n, P(n) exists.
3. For each number n, S(P(n)) = n.

which could be a model of the integers instead of the natural numbers?

Actually, I did absorb your explanation about taking away the starting point ("1" or "0") and getting the same system. So, I am wondering where exactly the "reference point" is (in the Peano system).

Do you mean something distinct from this?

Yes, I did mean something distinct from the explicit Peano axiom. For the pseudo-description of the system you give above, I don't see how there is a reference point to "1" or "0" or whatever you call it that starts the natural numbers. So, perhaps that is a candidate of what I was talking about when I was was saying:

Perhaps the thing that distinguishes the Peano successor function from some other successor function is that there might be a reference point in the Peano successor (as it is defined)? More precisely, are all successor functions based on a reference point? Is Peano's?

 

[CRGreathouse]

For the pseudo-description of the system you give above, I don't see how there is a reference point to "1" or "0" or whatever you call it that starts the natural numbers. So, perhaps that is a candidate of what I was talking about when I was was saying:

Well that system isn't strong enough to prove that there are numbers, but if at least one exists than you effectively have the integers. In fact, you may have more than one mutually-disjoint 'number lines' -- it's possible that you have 0 and 0', where 0'≠0, 0'≠S(0), 0'≠P(0), 0'≠S(S(0)), 0'≠P(P(0)), etc. you may even have infinitely many disjoint 'number lines'.

 

[philiprdutton]

Well that system isn't strong enough to prove that there are numbers, but if at least one exists than you effectively have the integers. In fact, you may have more than one mutually-disjoint 'number lines' -- it's possible that you have 0 and 0', where 0'≠0, 0'≠S(0), 0'≠P(0), 0'≠S(S(0)), 0'≠P(P(0)), etc. you may even have infinitely many disjoint 'number lines'.

It may not be strong enough to have numbers. However, it's core structure has some similarity with the core structure (or form) of that which exists with a number system- specifically, that "philosophically" speaking, you can view a number as just a "thing" that happens to have a "successor" and a "predecessor" (and which is also a "successor" and a "predecessor"). If you think about numbers in terms of graph theory, a number is a node with two edges. The edges connect to other nodes... and so forth. Each edge is used by the successor and predecessor functions (whatever those are). So, in this case, should one define the number with an edge-centric definition or a node centric definition? It is hard to think about which way to define a number. Perhaps you define it in terms of BOTH.

This is kind of what I feel happens in number systems. Now, with this graph theory experiment, you can not define a number until you define a reference point.

still being resolved:

1) Are all successor functions based on a reference point? Is Peano's?
2) Can nesting exist without a reference point?
3) Do any alternate formalizations of the natural numbers effectively use nesting? (such as what might be considered a set-theoretric formalization where one attempts to use {},{{}},{{{}}},.. where {} is the empty set.)
4) What are some examples of formal systems which actually use a reference point (field,border,group,whatever)? Does Peano's system use a reference point?​

 

[CRGreathouse]

It may not be strong enough to have numbers. However, it's core structure has some similarity with the core structure (or form) of that which exists with a number system- specifically, that "philosophically" speaking, you can view a number as just a "thing" that happens to have a "successor" and a "predecessor" (and which is also a "successor" and a "predecessor").

You misunderstand. If there is "something" there, I'm calling it a number. (You may have another term for it.) The system isn't strong enough to show that there is anything there at all.

If you think about numbers in terms of graph theory, a number is a node with two edges. The edges connect to other nodes... and so forth. Each edge is used by the successor and predecessor functions (whatever those are). So, in this case, should one define the number with an edge-centric definition or a node centric definition? It is hard to think about which way to define a number. Perhaps you define it in terms of BOTH.

Of course this leads to the possibility of generalizing the concept of number my using multivalued functions for S and P, graph-theoretically allowing for more than two edges. (To keep the graph theory sound, remember that the underlying structure is a digraph not a graph.)

I can't see how defining numbers in terms of edges instead of points changes anything.

Does Peano's system use a reference point?

Huh? Of course it does, "1 is a natural number". Do I misunderstand the question?

3) Do any alternate formalizations of the natural numbers effectively use nesting? (such as what might be considered a set-theoretric formalization where one attempts to use {},{{}},{{{}}},.. where {} is the empty set.)

Hmm... I had a thought that might let you express yourself better with the mathematicians here. Perhaps by "nesting" you mean "recursion"? If not, explain what you mean by nesting again (and what its relationship is to recursion: is one a subset of the other or are they disjoint?).

 

[philiprdutton]

recursion = nesting?

Huh? Of course it does, "1 is a natural number". Do I misunderstand the question?

You are talking about the reference point in terms of the numbers that the system gives you. I am talking about the system's reference point in terms of the construction of the formal system. Where specifically is the reference point defined? Not, "how can I define the reference point in terms of the objects the system creates."

In other words, why is "1 is a natural number" the reference point? We pegged this question before and ended up in totally different formalizations of the natural numbers. This is why, when looking at the Peano axioms I do not see an explicit axiom that defines the reference point. You said already you can take out the "0/1 is a natural number" axiom and still have a working system. So, maybe the reference point is somehow already hardwired into the Peano successor function? This is why I asked what makes the Peano' successor function special in comparision to some other successor function? Is the Peano successor function equipped already with a reference point?

Can you tell me what the Peano successor function is seperately from the Peano axioms? Did Peano axioms create the successor function?

Also, yes it is possible that nesting and recursion are the same phenomenon. I will have to think about it a while longer however. Is the Peano successor function simply recursion? I can't understand what recursion would be if there was not a reference point.

 

[philiprdutton]

side note on graph theory

If you think about numbers in terms of graph theory, a number is a node with two edges. The edges connect to other nodes... and so forth. Each edge is used by the successor and predecessor functions (whatever those are). So, in this case, should one define the number with an edge-centric definition or a node centric definition? It is hard to think about which way to define a number. Perhaps you define it in terms of BOTH.

Of course this leads to the possibility of generalizing the concept of number my using multivalued functions for S and P, graph-theoretically allowing for more than two edges. (To keep the graph theory sound, remember that the underlying structure is a digraph not a graph.)

Using the graph theory construction you eventually have to have a reference node before you can begin to say anything about what node represents a given number. That is why I brought it up. So, somehow, a graph theoretician would specify this reference. Likewise, with the Peano axioms, when/where in the construction of the system is the reference defined? Or maybe it isn't because a reference point is already implicitly given in the successor function he was using. Is the Peano successor function recursion? Does raw and pure recursion have a reference point?

 

[CRGreathouse]

You are talking about the reference point in terms of the numbers that the system gives you. I am talking about the system's reference point in terms of the construction of the formal system. Where specifically is the reference point defined? Not, "how can I define the reference point in terms of the objects the system creates."

But "1" could be anything in the Peano axioms -- unlike, say, the standard set-theoretic "1", which is {{}}. The Peano "1" exists only because there's an axiom that says it does, which doesn't tell us anything about it.

In other words, why is "1 is a natural number" the reference point? We pegged this question before and ended up in totally different formalizations of the natural numbers.

Qua?

This is why, when looking at the Peano axioms I do not see an explicit axiom that defines the reference point. You said already you can take out the "0/1 is a natural number" axiom and still have a working system.

I don't know that I said that. Without that axiom you could easily have no numbers, in which case you can't use the successor operation (because it applies only to numbers) or induction (because it requires numbers, and specifically 1). In fact in that case no axiom has any meaning at all.

That is, in every system you can construct that has no numbers, the Peano axioms are true. *Any* numberless system at all.

Can you tell me what the Peano successor function is seperately from the Peano axioms?

Sure, it gives you the "next" number. The axioms define just what that means, but this is the philosophical meaning.

Also, yes it is possible that nesting and recursion are the same phenomenon. I will have to think about it a while longer however. Is the Peano successor function simply recursion? I can't understand what recursion would be if there was not a reference point.

The successor function can be applied to something that has the successor function applied to it, which is a recursive use of the function. Is "nesting" just any such use of a function, or is it specific in some way to the successor?

 

[CRGreathouse]

Using the graph theory construction you eventually have to have a reference node before you can begin to say anything about what node represents a given number. That is why I brought it up. So, somehow, a graph theoretician would specify this reference. Likewise, with the Peano axioms, when/where in the construction of the system is the reference defined? Or maybe it isn't because a reference point is already implicitly given in the successor function he was using. Is the Peano successor function recursion? Does raw and pure recursion have a reference point?

Recursion has to work on something, so if by reference point you mean "something", then yes. If you mean "a distinguished point that is the unique 'beginning' of the numbers" then no, you don't need that.

Why do you say the graph needs a reference point? Perhaps I simply don't understand your neologisms.

The Peano successor function isn't recursion, but you can use it recursively.

 

[philiprdutton]

graph reference point.

Why do you say the graph needs a reference point? Perhaps I simply don't understand your neologisms.

Well basically here is the graph we talked about:
(in vertical form- my horizontal ascii graph didn't work right.):

.
.
.
\
0 << node
/
0 << node
\
0 << node
/
0 << node
\
.
.
.It is a crude representation but anyway, where is the node which represents "zero" or "one" ? I was saying that you can not look at any nodes and talk about what they are (in terms of numbers) until after you define the reference point (or reference node).

 

[CRGreathouse]

It is a crude representation but anyway, where is the node which represents "zero" or "one" ? I was saying that you can not look at any nodes and talk about what they are (in terms of numbers) until after you define the reference point (or reference node).

But the same could be said for set theory, right?

{}
{{}}
{{}, {{}}}
{{}, {{}}, {{}, {{}}}}
{{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}
. . .
S(n) = n U {n}

 

[philiprdutton]

yes

But the same could be said for set theory, right?

{}
{{}}
{{}, {{}}}
{{}, {{}}, {{}, {{}}}}
{{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}
. . .
S(n) = {n, {n}}

I think the set theory version has a reference point: The empty set.

 

[CRGreathouse]

Yes. That is exactly my point.

In both cases, there's no need for a special element. *Any* element would suffice. Without the axiom "1 is a number", you don't know that there are any numbers (or nodes). If, instead, you have as an axiom "A number exists", then you have a nonconstructive system that may have non-numbers preceding numbers (unless you have a P operator). Without that nonconstructive axiom, you may have no numbers at all, a system with non-numbers that eventually become numbers with enough uses of the S operator, or a system just like the Peano one.

So, to answer your question about when the reference point is defined, I'll let you pick which of the three situations you'll allow.

 

[philiprdutton]

In both cases, there's no need for a special element. *Any* element would suffice. Without the axiom "1 is a number", you don't know that there are any numbers (or nodes). If, instead, you have as an axiom "A number exists", then you have a nonconstructive system that may have non-numbers preceding numbers (unless you have a P operator). Without that nonconstructive axiom, you may have no numbers at all, a system with non-numbers that eventually become numbers with enough uses of the S operator, or a system just like the Peano one.

So, to answer your question about when the reference point is defined, I'll let you pick which of the three situations you'll allow.

Actually I do think the set theory version has a reference point: the empty set. The "0" (or "1") of the set theory version is the empty set. All other sets include at least one empty set, therefore they are not empty. Just that first one. In this case, I am not required to speak of numbers at all in order to talk about this reference point.

 

[CRGreathouse]

Actually I do think the set theory version has a reference point: the empty set. The "0" (or "1") of the set theory version is the empty set. All other sets include at least one empty set, therefore they are not empty. Just that first one. In this case, I am not required to speak of numbers at all in order to talk about this reference point.

Yes, set theory has a special point you can pick. But it doesn't need to be that special element for the Peano axioms to hold -- you could even choose {{{}}} as your element "1", even though it's not on the standard list of ordinal sets. You just need somewhere to start.

 

[philiprdutton]

falling out

Yes, set theory has a special point you can pick. But it doesn't need to be that special element for the Peano axioms to hold -- you could even choose {{{}}} as your element "1", even though it's not on the standard list of ordinal sets. You just need somewhere to start.

I really think that the nested set theory version makes the reference point sort of fall out naturally. You don't need to pick. Think about it, if, as you suggested, you choose {{{}}} as your element "1", then were will you "store" this piece of information? With the nested set theory version, you don't even need to encode this information because the empty set is a natural boundry or reference point... all because of the nested nature of the setup. So, back to Peano, is the reference explicitly stated or is it implicitly defined? I have to go back through a few posts here to find the first attempt at this answer.

 

[CRGreathouse]

I really think that the nested set theory version makes the reference point sort of fall out naturally. You don't need to pick. Think about it, if, as you suggested, you choose {{{}}} as your element "1", then were will you "store" this piece of information? With the nested set theory version, you don't even need to encode this information because the empty set is a natural boundry or reference point... all because of the nested nature of the setup. So, back to Peano, is the reference explicitly stated or is it implicitly defined? I have to go back through a few posts here to find the first attempt at this answer.

I agree that the empty set is natural; I just wanted to make clear that it isn't needed -- any set, even one that isn't an ordinal, will work.

If I understand correctly, it is impossible to answer your question for Peano. I can answer for set theory because it is a model of Peano arithmetic, but I can only answer for this and other models -- in some models of Peano arithmetic it's explicitly defined, while in others it's "natural".

 

[philiprdutton]

I agree that the empty set is natural; I just wanted to make clear that it isn't needed -- any set, even one that isn't an ordinal, will work.

If I understand correctly, it is impossible to answer your question for Peano. I can answer for set theory because it is a model of Peano arithmetic, but I can only answer for this and other models -- in some models of Peano arithmetic it's explicitly defined, while in others it's "natural".

So, Uhm, are we essentially saying that there are no models of Peano arithmetic that do not have said reference point?



PS: I had no idea that set theory was based on Peano arithmetic. I guess Cantor created it after the Peano axioms?

 

[Hurkyl]

So, Uhm, are we essentially saying that there are no models of Peano arithmetic that do not have said reference point?

Yes -- the existence of an initial element is explicitly stated as an axiom.

 

[Hurkyl]

PS: I had no idea that set theory was based on Peano arithmetic. I guess Cantor created it after the Peano axioms?

He misspoke -- he meant that the finite ordinals are a model of Peano's axioms. Set theory was based on logic; a set, intuitively, is an object that represents the class of all "things" satisfying some condition.

 

[philiprdutton]

initial element

Yes -- the existence of an initial element is explicitly stated as an axiom.

Is there a particular "word" in the literature that refers to this "initial element" (or reference point)?? I feel it is so crucial yet so hard to talk about.

 

[Hurkyl]

Is there a particular "word" in the literature that refers to this "initial element" (or reference point)?? I feel it is so crucial yet so hard to talk about.

Yes: typically, one goes so far as to choose a single character to represent it. '0' and '1' are common choices, though I'm sure I've seen authors use other symbols like 'a', 'i', 'e', or even '$\epsilon$' if they are worried about a conflict of notation, or simply to reduce the possibility of confusing the reader.

For flavor, I will use the symbol 'v' in this post. I will use 'S' for the successor function.

One of Peano's axioms states that $v \neq Sx$, no matter what x is.

Incidentally, a minimalist might not even give the initial element a name -- they would adjust the above axiom to assert that there exists some object with that property.


The reason one might call v the 'initial element' is that it's traditional to define a total ordering on the natural numbers (traditionally called '<') such that x < Sx for any x. One can then prove that v is, in fact, the smallest element relative to this particular total ordering.

 

[philiprdutton]

ahh i see

Yes: typically, one goes so far as to choose a single character to represent it. '0' and '1' are common choices, though I'm sure I've seen authors use other symbols like 'a', 'i', 'e', or even '$\epsilon$' if they are worried about a conflict of notation, or simply to reduce the possibility of confusing the reader.

For flavor, I will use the symbol 'v' in this post. I will use 'S' for the successor function.

One of Peano's axioms states that $v \neq Sx$, no matter what x is.

Incidentally, a minimalist might not even give the initial element a name -- they would adjust the above axiom to assert that there exists some object with that property.The reason one might call v the 'initial element' is that it's traditional to define a total ordering on the natural numbers (traditionally called '<') such that x < Sx for any x. One can then prove that v is, in fact, the smallest element relative to this particular total ordering.

Thank you for the clear explanation. This very feature of number systems is what I have been thinking about much lately. In particular, I wanted to explore the idea that "primality" can perhaps alternatively be studied from the perspective of the "reference" feature in a system. The majority of people tend to study primes in terms of well defined arithmetic operations or distribution properties, etc. What I am saying is that the whole "family" of systems which use a reference point quite possibly exhibit similar behavior. Perhaps that "family" of systems will be very broad. This I do not know because I am not a professional mathematician and I have not properly surveyed the systems.

I feel that primes have been so clearly defined (for thousands of years perhaps). The definition in the literature is so clear. The idea that numbers can be broken down into various primes is also clear. Do not forget that everything that can be done with the Peano system is in terms of the reference point. Therefore, if "prime" is such a basic feature, then it probably shows up in other mathematic systems which use a reference point (possibly in combination with "nesting"/recursion). If those systems do not define numbers, then we must view the "primality notion" without traditional "institutionalization" of numbers. I am quite ambitious and confident that the "effect" is going to be there in many systems.

The reason one might call v the 'initial element' is that it's traditional to define a total ordering on the natural numbers (traditionally called '<') such that x < Sx for any x. One can then prove that v is, in fact, the smallest element relative to this particular total ordering.

I clearly see now that any attempt to define the natural numbers with some particular formal system will require the ordering to be "installed" (hence the reference point). Would I be correct in asserting that the set theoretic version of the natural numbers just encodes the ordering into the nesting (recursion- where {} is the reference point). My understanding is that a set's elements can not be ordered. But, if a set can contain a set, then you can take advantage of that and place your "ordering" in that feature.

Anyway, some questions:

-Do most of the standard formal systems use a reference point?
-What standard systems do not?
-Where does the set theoretic version of Peano encode the impose the ordering? (or by what feature of set theory)? I just want to make sure I totally understand this.Thanks a million!

 

[Hurkyl]

There is something called a pointed set. It is simply a set with a 'distinguished' element. Such an object has no structure whatsoever, aside from the fact that one of its elements is 'distinguished'. The simplest way to define such a thing is

A pointed set is a pair (S, x), where S is a set and x is an element of S.

If you wanted a formal system instead, then the theory of a pointed set is presented simply by specifying that the language has a constant symbol. (usually, '*' is used) In particular, no axioms are given -- this theory consists only of tautologies.

Pointed topological spaces are a useful object for some purposes. As the name suggests, it's simply a topological space with a distinguished point. A familiar example would be a Euclidean line or plane with a specified origin.

But these things don't really have a notion of "primeness".



Incidentally, Peano's axioms don't yield a notion of primeness either -- there are lots of ways to put an ordering or an algebraic structure on an object satisfying Peano's axioms. For example, one can define addition recursively by:

a + v = v
a + S(b) = S(a + b)

(often, one would use the symbol '0' instead of 'v' if one intends to use this definition of addition)

or, one might define addition by

a + v = S(v)
a + S(b) = S(a + b)

(and one would typically use the symbol '1' instead of 'v')

One could even define addition so that v+v is undefined, and

S(a) + v = a
v + S(b) = b
a + S(b) = S(a + b)

(one might use the symbol '-1' instead of 'v' if one were to adopt this definition)

For the above examples I was using S as if it were an "add one" operation -- but one could define an addition operation in an entirely different way, if one so desires!

The point is that primeness doesn't automatically make sense -- it is only meaningful relative to an algebraic structure.



The (usual) ordering on the ordinal numbers is indeed given by containment: $\alpha < \beta$ if and only if $\alpha \subseteq \beta$. And for the usual set-theoretic model of the natural numbers, this ordering does agree with the usual ordering on the natural numbers.

Incidentally, the main practical reason for studying ordinal numbers is that they are very useful for analyzing and proving things -- the practical content of the fact the finite ordinals model the natural numbers is that it allows us to transfer our expertise with natural numbers into a set-theoretic context.



Probably the most pervasive notion of "primeness" in mathematics is that of a prime element in a lattice. For example, the notion of primeness you're familiar with -- primeness of integers -- is a special case of this. You can organize the positive integers into a lattice by defining $a \leq b$ if and only if a divides b. Or equivalently, by defining $a \vee b = lcm(a, b)$ and $a \wedge b = gcd(a, b)$.

 

[philiprdutton]

okay

Thanks for the quick information. I am grateful.

The point is that primeness doesn't automatically make sense -- it is only meaningful relative to an algebraic structure.

Yes, I agree.

I had one more question about natural number systems which use a reference point and recursion("or nesting"). In the Peano system the recursion seems to be progressing in what I call the "forward" direction on the number line. This seems pretty obvious due to the fact that their is a function which happens to be called "successor" function. Now, this question is a bit hard to verbalize but I will give it a shot:

How does the succession "stop?"

I can see how in each step of the succession, the system might just start over at the reference object and start to step again using the successor function. But I do not see how it can know to stop as might be required when performing operations like addition/multiplication.