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Since you have gone down this route I think I will throw the Banach-Tarski paradox into the mix. A rather nice video on it can be found here on YouTube:If you divide an object into pieces, you can't just assume that you keep all the previous axioms without introducing contradictions, you have to show it. Your axioms ought to be written in terms of the most basic building blocks of the system.

[video=youtube;s86-Z-CbaHA]https://www.youtube.com/watch?v=s86-Z-CbaHA[/video]

I think I finally understand the mistake I have been making. I have spent the last several hours trying to find a solution to Chiro and Dan's last few post. I believe I might have done so...

A = any number

S = any set

Let the ordered pair (x,y) be described as follows...

((z1,z2),(z1,z2)) → (x,y) : ∀A in S

∀A ≠ 0: (z1,z2) = (A,A) = A

∀A = 0: (z1,z2) = (0,1) = 0

combined with the axiom

"Any A in binary operation of multiplication or division is only representing z1, or z2."

If this is worse not better, than I am forced to concede to the forum members. Only in order to continue my education in this regard. I sincerely hope however a few members might be willing to help me continue to "concrete" this idea.

Also I would ask permission from Dan, to post a new topic extolling the virtue's of the members of this forum, regarding this EXCELLENT peer review.

Sincerely Conway

Again we have confusion. I can see what you might be trying to reach for but you still aren't defining the "major players" all that well.Chiro, Dan, Archie

I think I finally understand the mistake I have been making. I have spent the last several hours trying to find a solution to Chiro and Dan's last few post. I believe I might have done so...

A = any number

S = any set

Let the ordered pair (x,y) be described as follows...

((z1,z2),(z1,z2)) → (x,y) : ∀A in S

∀A ≠ 0: (z1,z2) = (A,A) = A

∀A = 0: (z1,z2) = (0,1) = 0

combined with the axiom

"Any A in binary operation of multiplication or division is only representing z1, or z2."

For example: \(\displaystyle ( ( z_1 , z_2 ) , ( z_1, z_2 ) ) \to (x, y)\). What are z1 and z2? Would they be members of a set that you haven't mentioned yet? And how can \(\displaystyle ( z_1 , z_2 ) = x\) and \(\displaystyle ( z_1 , z_2 ) = y\) at the same time if x and y aren't the same?

Sure! We always like good feedback. But please put the thread in the Feedback Forum. (And you don't need my permission to post anything so long as it conforms to the Forum Rules below.)Also I would ask permission from Dan, to post a new topic extolling the virtue's of the members of this forum, regarding this EXCELLENT peer review.

-Dan

(z1,z2), are not members of any set...(they be considered pieces of the members of the set)

They are

z1=value

z2=space

"z1 and z2 for A, (any number) is "really" just the number given put into what appears to be an order pair"

z1,z2 for A = (A,A)

z1,z2 for x = (x,x)

z1,z2 for y = (y,y)

z1,z2 for 2 = (2,2)

z1,z2 for 3 = (3,3)

Any number has the same quantity of space, as it has value......except zero.

(_)= one ACTUAL space, no value = z2

(1)= one value, no space = z1

If I put these things together I get a number

(

If you "think" you may be seeing what I am reaching for... I will continue....please let me know if otherwise... so that I may adhere to my aforementioned promise directly!

It is the most basic building block of information [it contains attributes and a set is just a bunch of them].

If something is a piece of something then it is an element of a set.

Believe me - if you can't define the set then you can't define what you are trying to say in any specific or useful way.

But if (2, 2) is what we would typically call the number "2" why all the extra work? Not counting the 0 stuff it would seem that you are just adding complexity where none is needed. You could simply say that the number 2 has the properties of space and value and not worry about the whole formalism. Is there ever a situation where we would have, say (2, 3)? What would that mean?Dan

(z1,z2), are not members of any set...(they be considered pieces of the members of the set)

They are

z1=value

z2=space

"z1 and z2 for A, (any number) is "really" just the number given put into what appears to be an order pair"

z1,z2 for A = (A,A)

z1,z2 for x = (x,x)

z1,z2 for y = (y,y)

z1,z2 for 2 = (2,2)

z1,z2 for 3 = (3,3)

Any number has the same quantity of space, as it has value......except zero.

(_)= one ACTUAL space, no value = z2

(1)= one value, no space = z1

If I put these things together I get a number

(1) = 1

Listen, if there is ever a problem with your posts rest assured someone will mention it to you. Just go ahead and post and don't worry about any apologies.If you "think" you may be seeing what I am reaching for... I will continue....please let me know if otherwise... so that I may adhere to my aforementioned promise directly!

-Dan

In all fairness I know what a set is, I know what elements are. It is true that elements of a set are ALWAYS numbers? So then maybe you see the difficulty I am having. I am talking about pieces of numbers. Perhaps you missed that particular part where I was replying to Dan.....

(z1,z2), are not members of any set...(they may be considered pieces of the "members" of the set).....post #45

I am not aware of any set theory that currently "breaks" numbers up into their parts.....parts that are NOT numbers.

Dan

Understood.

"But if (2, 2) is what we would typically call the number "2" why all the extra work?"

Because in reality (2,2) as a representation of 2 is poor.

The reality is as follows

1

is composed of

(_) one quantity of space

as well as

(1) one quantity of value

if and only if I place the value into the space do I get a number

(

This is imperative for reason relating ONLY to multiplication and divison....so as you said forgetting zero.....

2 x 3

The above expression does not have "numbers", they are z1 and z2, I may label either symbol as value or space....(commutative property)

so that....

2(as value,z1) x 3(as space,z2).......yields the following action

I take three spaces.....

(_,_,_)

I take the value of two.....

(2)

I put the value into the spaces and then add

(

or

3(as value,z1) x 2(as space,z2)........yields the following actions

(_,_) I take two spaces

(3) I take the value of three

I put the value into the space and then add

(

No. The set of all Disney elephants contains Dumbo, some other elephants, and no numbers.

Your parts are, mathematically, numbers. You might want them to represent something else, but unless you introduce some rules to the contrary, they appear to function exactly as numbers.I am talking about pieces of numbers... I am not aware of any set theory that currently "breaks" numbers up into their parts.....parts that are NOT numbers.

What you have thus far described (not in these terms exactly, although you were closing in on it) is that you have a set of ordered pairs \(\displaystyle (z_1,z_2) \in \mathbb N \times \mathbb N\). You call \(\displaystyle z_1\) the "space" and \(\displaystyle z_2\) the "value" (although these terms mean nothing mathematically: their behaviour is defined by other statements). You all have implicitly defined a bijection \(\displaystyle \mathbb N \times \mathbb N \mapsto \mathbb N\) defined by \(\displaystyle (a,a) \mapsto a\).

That's all well and good, but very thin. What happens next needs some formalisation. It reads as a veryad hocconstruction that looks unlikely to stand up to any rigorous interrogation. You say that, for multiplication \(\displaystyle (a,a) \times (b,b) = (a, ab)\): taking the space from \(\displaystyle (a,a)\) and then putting \(\displaystyle a\) copies of value \(\displaystyle b\) into those spaces. (I note here that you have absolutely used the fact that both the space and the value are functioning exactly as numbers). However, as noted before, we now have something that isn't in the definition of the set. You also claim that we have a map \(\displaystyle (a,ab) \mapsto ab\) which stops the mapping from being a bijection because we have \(\displaystyle (ab,ab) \mapsto ab\) already. Essentially, we seem to be heading to a place where the "space" is completely immaterial to the mapping. You have also stated that, due to commutativity of multiplication we have \(\displaystyle (b,b) \times (a,a) = (b, ab)\) and that this also maps \(\displaystyle (b,ab) \mapsto b\) further damaging the bijection idea and further illustrating that (up to now) the "space" is irrelevant.

None of this stops you from continuing to define this system, although it does make me question the point to it all.

Archie

Excellent post!

Lol...your statement regarding elephants was NOT fare....though entirely accurate........

If 1 = elephant.....then elephant is a number...........What are the pieces of an elephant?.....

It's "dimensions" , or space....

It's "being", or value.....

neither of which are described as elements in the set......

Do you at least see my point here....

I have withdrawn my claims on mapping .....as I do NOT believe it is possible....or applicable....as I pointed out in a reply on mapping that came directly from Chiro.

quote

(I note here that you have absolutely used the fact that both the space and the value are functioning exactly as numbers).

I have expressed fiercely the exact opposite.(except when combined) It is impossible to separate space and value phisically...it can only be done abstractly......hence my attempt to do so with "symbolic numbers"

If you will....

Let's set aside all set theory

Let's set aside all expressions involving zero

Do you think/agree the following makes sense....

1

is composed of

(_) one quantity of space (sometimes large or small)

as well as

(1) one quantity of value (sometimes elephant or 1)

if and only if I place the value into the space do I get a number

(__1__) = 1

If you Archie, Chiro, Dan do not find the above to be logical...

Then I yield here and now....

Excellent post!

Lol...your statement regarding elephants was NOT fare....though entirely accurate........

If 1 = elephant.....then elephant is a number...........What are the pieces of an elephant?.....

It's "dimensions" , or space....

It's "being", or value.....

neither of which are described as elements in the set......

Do you at least see my point here....

I have withdrawn my claims on mapping .....as I do NOT believe it is possible....or applicable....as I pointed out in a reply on mapping that came directly from Chiro.

quote

(I note here that you have absolutely used the fact that both the space and the value are functioning exactly as numbers).

I have expressed fiercely the exact opposite.(except when combined) It is impossible to separate space and value phisically...it can only be done abstractly......hence my attempt to do so with "symbolic numbers"

If you will....

Let's set aside all set theory

Let's set aside all expressions involving zero

Do you think/agree the following makes sense....

1

is composed of

(_) one quantity of space (sometimes large or small)

as well as

(1) one quantity of value (sometimes elephant or 1)

if and only if I place the value into the space do I get a number

(

If you Archie, Chiro, Dan do not find the above to be logical...

Then I yield here and now....

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