whole prime number

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.

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.

philiprdutton

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.

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.

philiprdutton

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... {{{}, {{{}}}}, {{}}}

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.

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?

philiprdutton

No. I am saying unary is the slowest of them all because unary is essentially a system where you have to count (my def) "out loud" in order to express the point where you are in the counting line.

visually:

the number of decimal 10 is viewed as:

10
but in unary it is:
..........

You have some correct points about relative speeds which I missed but still, nothing is slower than unary. Unary = counting (my def)

CRGreathouse

Well the standard form for decimal 10 in unary would be 1111111111, but that's beside the point. Of course both could be written with different symbols, but that's just a simple replacement issue.

I suppose one could construct systems which are slower than unary...

I presume the equal sign above means "is a kind of"?

Where were you going with this?

philiprdutton

Yes my equal sign is "a kind of."

Now, we can see that both the counting system and the Peano system are unary speed systems (for practical human purposes). Essentially, the Peano system at it's CORE has a counting system (my def).

So, before you can build a Peano system you must have the counting system.

The complement (as in set theory) of the counting system within the peano system is what causes the notion of "prime"... NOT the counting system.

That is where I am trying to go.

Visually take two concentric circles. The inner circle is the counting system which is a sub feature of the Peano system. The outer circle is the whole Peano system. The complement of the inner circle is what creates the ability to talk about primes.

CRGreathouse

That's a claim, but what you want is a proof or an example. What axioms can you remove from Peano arithmetic so it can still count but not talk about primes? I may have actually given an example of this earlier on the thread...

philiprdutton

Actually, I did not originally care about removing pieces from the Peano system. I was originally thinking about this in terms of how to build up from scratch a basic system that did not support primes but had some commonality with a system like Peano. But now that you mention it, it would be a great exercise to see how much must be removed from Peano in order that notion of "prime" can not be supported.

We essentially are talking about two directions (top down or bottom up approaches) with the same goal: to explore "when" the notion of "prime" is added to a system (which in our discussion has been Peano.

dodo

While you create new systems, try to add to them a very desirable property than Peano's have: the ability to express very big (or infinite) objects in a finite, even very compact, space of symbols. This is what makes better a system like "1 is a number, Sa is a number if a is" than "foo is a number, foo foo is a number, foo foo foo is a number, ...", or than "1 is a number, S1 is a number, SS1 is a number, SSS1 is a number, ...".

CRGreathouse

My basic answer would be when you make a system strong enough to support a discrete chain/total order, you essentially have a way to talk about divisibility and thus primality.

dodo

I have some confusion here. The relation < defines a total order on R, yet that doesn't make R isomorphic to N. Without that isomorphism, you get 7 is not a prime because it is divisible by 7/2.

philiprdutton

I am not interested in building a system with "very desirable property than Peano's have." I have a specific reason why I am limiting the functionality of the counting system.

dodo

No problem; as long as you don't count over 100, you won't spend too much paper.

But CR has a point up there. Divisibility is actually a very basic concept, that can come from addition, not necessarily multiplication. For instance, define numbers with dots, x, xx, xxx, xxxx..., and addition as concatenation. Then define divisibility by these two axioms:
1) Every number divides itself, f.i. xxx divides xxx.
2) If b divides a, then b divides a+b. That is, if xx divides xx, then xx divides xxxx, and also xx divides xxxxxx ...