# Whole prime number

1. Jul 29, 2007

### CRGreathouse

Still not following. What do you mean when you say you "equate the two systems"?

When you're working with the pure Peano axioms, there's noting you can say about time, space, or other complexity. If you choose a particular model of the Peano axioms, then you can talk about it.

2. Jul 29, 2007

### CRGreathouse

It's done a lot, sure. A system A can be shown to be consistent relative to a (stronger) system B by constructing a model of A inside B. Kelley-Morse set theory can model ZFC (and prove its consistency too, showing that KM is actually stronger).

Which work? Godel made several major contributions to mathematics... do you mean his Incompleteness Theorem, or perhaps his earlier Completeness Theorem?

3. Jul 29, 2007

### philiprdutton

Godel's proof

I was referring to the one which states that "for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms." (copied from wikipedia)

4. Jul 29, 2007

### CRGreathouse

That's the Incompleteness Theorem. I don't think his proof used the mapping technique you mention.

5. Jul 29, 2007

### philiprdutton

You said "consistent"... does that mean "equivalent?"

6. Jul 29, 2007

### philiprdutton

godel mapping

Well he used some kind of mathematical mapping.

7. Jul 29, 2007

### philiprdutton

which proof?

Sorry I am talking about the more famous of the two.

I think maybe his Godel number function is the "mapping" I am referring to. I thought it was a generic mathematics technique.

8. Jul 29, 2007

### CRGreathouse

No. In fact those two are inequivalent -- and you should know the reason, since you just posted it: the Incompleteness Theorem. No sufficiently strong theory* can prove its own consistency, so since KM proves ZFC to be consistent the two can't be equal (unless one is inconsistent, in which case they're both equal to the theory "for all p, p" in which everything is true).

* Any theory containing Peano arithmetic is strong enough.

9. Jul 29, 2007

### CRGreathouse

The Incompleteness Theorem is the more famous of the two, and it's the one you quoted. The Completeness Theorem is the one that essentially says that in first-order logic, provability <--> truth.

10. Jul 29, 2007

### philiprdutton

Godel encoding

Godel encoding was used by Godel as follows: "Gödel used a system of Gödel numbering based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic he was dealing with."

Can I assign each natural number from the Peano system to the counting systems' statements which we have been talking about here? I want each step of the counting algorithm output to be assigned the associated natural number.

(sorry human language is making it difficult to be formal and to keep all my terms in proper context. Obviously, there is no existing association when I said "associated natural number" but you know what I mean...)

Last edited: Jul 29, 2007
11. Jul 29, 2007

### philiprdutton

enlightening...

Okay, sorry for my confusion thus far. I really wanted to get close to the idea of showing that the counting system and the peano system are both using the "number line" in a "synchronized" fashion.

If each system "creates" a "number line"... and, each "number line" has the same form, then I want to equate the two systems on that basis.

12. Jul 29, 2007

### CRGreathouse

The only statements you have in the counting system are the natural numbers and whatever your underlying logic allows with that:

"1 is in N"
"1 is in N and 7 is in N"
"(1 is in N and 7 is in N) or 6 is not in N"

You can certainly give a Godel numbering to your counting system's statements, but I don't understand to what end you are doing that. Also, do you mean statements or just theorems? Are you including false ones like "1 is not in N"?

Also, what algorithm do you mean?

13. Jul 29, 2007

### CRGreathouse

Still not getting it. Let me break this down and you can help me through what I don't get.

1. Each system creates a number line.
1a. Your counting scheme creates a number line.
1b. Peano arithmetic creates a number line.
2. The Peano number line has the same form as your counting scheme's number line.
3. If two number lines have the same form, they are equivalent in some sense.

What's a number line? That is, what properties does something need before you'll call it that? Surely any sensible definition will make 1b true, but some could make 1a false.

What do you mean when you say "form"? I would think this means the two share certain properties, but which?

In what sense do you want to equate the systems? Usually this would mean that systems which fit certain properties can prove a certain collection of facts about their members, but which?

14. Jul 29, 2007

### philiprdutton

form

I do not know what a number line is nor "WHEN" it gets created in relation to either system. That is why I asked about what "comes first" in Peano: the number line that we all were taught as kids or the axioms. Also, I casually referred to Peano in terms of whether or not he was biased by the notion of "number line." Maybe for fun we could talk about a "counting line" since each system can at least produce or use one. Whenever I talk of number line I am refering to that "form" which has become so damn intuitive that I can't prevent it from affecting my thinking about math in general.

Yes. I meant that the two systems "store results" in the same "form." A linear form with "points." I want to equate the two systems in terms of how their counting features use the form. Then I would allow the peano system to fully express itself and reveal the notion of prime... but then I would be able to say that you can have your prime but not in terms of the counting features only. If you can not have the prime in terms of the counting features only then that invariably says something about not having "prime" in terms of the FORM that each systems "use" (or "create").

My hope is that I can find a simple way to prevent the millions of people who know of the "prime" numbers from attributing the notion of "prime" to the "place in the form in which that number happens to reside."

(note: in my interpretation, a number can not reside anywhere until you have defined a way to talk about that number in terms of where you stopped on the counting line in order to "arrive" there.

Last edited: Jul 29, 2007
15. Jul 29, 2007

### CRGreathouse

I would argue that your counting system can't actually count, and thus isn't a "counting line" as such. That's why knowing what you mean by form is so important to me.

But Peano arithmetic has many more true results than your counting system, and I still don't know what "form" is.

But the counting system of yours isn't linear, is it?

Truly, I understand almost none of the key words in this sentence: "equate", "counting features", and "form".

As I understand it, your counting system does not have a way to "talk about that number in terms of where you stopped on the counting line in order to 'arrive' there".

16. Jul 29, 2007

### philiprdutton

number line vrs. counting line

Here is something that I wrote which might give you insight into the madness going on in my head. :)

Everyone has some understanding of the number line. I do not know if people just simply remember what they have been taught in grade school or if they intuitively have this uncanny understanding of the number line. Somewhere in between we humans know how to count using the number line. My question is about counting. Can you count without knowing numbers? If I ask you to count to 100 you can easily do this.

What if I tell you to do the same thing again but do not use the base 10 decimal system. In fact don't use any number based system. Can you count now? Sure you can. But you will soon loose track of where you are. You will know not if you are getting close to the original number that I requested you to count to. You will not know if you have passed this number.

In this context, we have a new phenomenon. The number line is basically still there but we do not have any more reason to call it a number line. Let us call it a "counting line."

the above is from this post

With this above line of thinking, I arrived at the point where I had to use "da,da,da,da,...,da" as a way to describe what happens after you abandon all the number BASED systems.

Last edited: Jul 29, 2007
17. Jul 29, 2007

### philiprdutton

iterate

Sure, you can count using the counting system. I just never said you could interpret each position as a number in the sense of what can be done once you define a number base system.

Yes, indeed you can count with the counting system. My definition of count will have to become something like: "take another algorithmic step"

I am interested in a system that lets me move forward in the "line" and I don't care at this point about whether or not you can label each position. I know this system is going to be almost useless for most people. But if you remember that the Peano system is basically able to simulate this counting system then you can not deny that lots of stuff in mathematics is related to a counting system... .it might just be harder to recognize that fact since there are so many other things you can do with Peano like the fancy multiplication or addition.

18. Jul 29, 2007

### CRGreathouse

You're getting into linguistics now!*

First of all, counting is not a natural thing, and there are people who do not count (most famously the Piraha of South America). Babies and animals can spot the difference between 1, 2, 3, 4-5, 6-9, and so forth, but more particular nuances are generally the area of counting which is a human construct.

But even people who can't count can use tally sticks to record and compare numbers. Essentially every truly ancient civilization used them in some form or other: notches carved into pieces of wood or whatever was convenient. (The Inca used knots in ropes instead.)

But even people who can't count and don't use devices like tally sticks, abacuses, or the like can compare numbers by setting up bijections. Imagine you want to compare the number of sheep I have to the number you have. Just pass one of yours and one of mine through a gate until one of us has none left. If we both have none left we had the same number; otherwise the one with more left has more.

This works even if, like the Piraha, you have no abstract concept of "number".

* Fortunately I've picked that up as a hobby (having read a few textbooks on the subject recommended to me by my friend who has a degree in the field).

19. Jul 29, 2007

### CRGreathouse

In the systems I give, you may suppose modus ponens is the only underlying logic.

Consider the system:
Axiom 1. A

You can take as many algorithmic steps as you like with this system:
1. A (1)
2. A (1)
3. A (1)
4. A (1)
. . .

Thus it lets you count in your terminology. Perhaps you mean taking steps that are essentially different from those before?

Consider this system:
Axiom 1. A
Axiom 2. A --> B
Axiom 3. B --> A

We can take as many algorithmic steps as you like:
1. A (1)
2. A --> B (2)
3. B (MP)
4. B --> A (3)
5. A (MP)
. . .

Alternatively:

Axiom 1. A
Axiom 2. For all x, x --> x.

1. A (1)
2. A --> A (2)
3. (A --> A) --> (A --> A) (2)
4. A --> (A --> A) (MP)
. . .

Plenty of algorithmic steps, but there's no real way to count with this one. For a more concrete system, consider forming sets:

{}
{{}}
{{}, {{}}}
{{}, {{}}, {{{}}}}
{{}, {{{}}}}
{{{}, {{{}}}}, {{}}}

Sets that are subsets of others can be said to be smaller, but some sets are incomparable -- neither is smaller. This doesn't make a "number line" so much as a web.

I don't think the Peano axioms simulate arithmetic; I think they define how something has to act to be arithmetic.

I see set theory as the basis for mathematics more than counting, but I'm sure a counting system could be used as an alternate basis. My field (number theory) would find that particularly natural.

20. Jul 29, 2007

### philiprdutton

Here is another thought:

What is faster? Counting in binary or counting in decimal? Neither. You get there at the same rate.

Who talks about numbers faster? A people who communicate about numbers only using the binary system or a people group who communicate about numbers only using base 10. They both use the same language. If you have to listen to one of these people speak out loud as they count then who takes the longest at each number when using their own number base to communicate?

Now, if you do not use any number based system when "counting out loud" you are just going to have to make a noise over and over... "buh,buh,buh,buh,buh....buh."

What is the slowest possible way for a human to count out loud? By not using a number based system to describe what point the count is currently at. They are still counting. Just not describing it with fancy short cuts. So, number systems are basically short cuts. They are an encoding which prevents people from having to "count" when exchanging numbers verbally, on paper, or whatever.

Last edited: Jul 29, 2007
21. Jul 29, 2007

### philiprdutton

counting: my definition

YES!! That is what I mean when I say "count." I am sorry I had to keep perverting the standard meaning of "count" but I felt it necessary to push the thinking as far as possible down this course of study using that term.

22. Jul 29, 2007

### CRGreathouse

All of those are number systems. You're saying that decimal is just as fast as binary, but unary is slower. I would say that unary is slower than binary for numbers greater than 1, binary is slower than decimal for numbers greater than 1, ternary is slower than decimal for numbers greater than 2, hextal is slower than decimal for numbers greater than 1295, decimal is slower than hexadecimal for numbers greater than 99999, and so on.

23. Jul 29, 2007

### philiprdutton

web

Considering your web system: Sure I can count (my def) with it:

{}
{{}}
{{}, {{}}}
{{}, {{}}, {{{}}}}
{{}, {{{}}}}
{{{}, {{{}}}}, {{}}}

is simply as follows:

step da { }
step da,da {{}}
step da,da,da {{}, {{}}}
step da,da,da,da {{}, {{}}, {{{}}}}
step da,da,da,da,da {{}, {{{}}}}
etc... {{{}, {{{}}}}, {{}}}

Last edited: Jul 29, 2007
24. Jul 29, 2007

### CRGreathouse

So consider this system.

Axiom 1. A point exists.
Axiom 2. From any point, you may draw a 1-unit arrow down and to the left. The end of the arrow is a point.
Axiom 3. From any point, you may draw a 1-unit arrow down and to the right. The end of the arrow is a point.

The metalogic of the system is that two diagrams are equal iff they have the same arrow structure, and one diagram is larger than another iff the first contains all the arrows of the second but the two are not equal.

So "/\" > "/" > "" and "/\" > "\" > "", but not ("/" > "\") and not ("\" > "/"). The system can make many different theorems ("diagrams" in its own terminology) but they don't work like the natural numbers, or any sensible number line at all.

25. Jul 29, 2007

### CRGreathouse

But someone else could use those axioms and come up with theorems in a different order. You don't want different things to be equal to each other, do you?