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I sincerly wish to only hear out others opnions. I have no desire to re-hash old topics of mine. With the moderators approval I ask the following, with out any intention of further reply.

Is there a diffrence between the space of Zero and the value of Zero?

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These terms (space of zero, value of zero) need to be precisely defined.

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Formally, no. The value of zero is defined as the empty set { } and a space is a structure defined as a set equipped with a relation over its elements, and as the empty set { } has no elements, the relation is empty too.

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These terms (space of zero, value of zero) need to be precisely defined.

I agree with mathematic's request for the terms to be defined carefully.

The value of the real number zero is the real number zero. I am not sure if you have some further meaning, but the value or evaluation of a real number is surely just that real number?

As the space, do you mean something like the span of an empty set of vectors?

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Formally, no. The value of zero is defined as the empty set { } and a space is a structure defined as a set equipped with a relation over its elements, and as the empty set { } has no elements, the relation is empty too.

First time I've seen that fundamental definition of sets and spaces distinguishing the two, pointed out here at SF.

There are some others who could also benefit from noting this.

+1

Edited by studiot
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The value of zero is defined as the empty set { }

You are describing some set theoretical construction of the natural numbers? Something like von Neumann construction?

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Ajb

I agree a more formal definition is needed, I shall not offer any so as to stay in "bounds".

Sato

Your just plain wrong. First off a set "please" verify, is defined as "any" collection of distinct objects....is zero an object then ? Is there more than one of them, so as to make a collection? Zero as a set does NOT fit the definition. Nor as you point out, does it contain any elements or any relations therein. All things required to be a set. If the "set" in question has no values or relationships, then there is not a "set" at all......by definition!

Thanks Sato for you time.

Studiot

You will notice that Sato did not distinguish the difference between the space and value of the "set" of zero. Sato claimed that the space of zero was a relationship of it's elements. Therefore there is not a difference between the two, only that to define one, you must have and measure the other. Which I can agree with, but it shows then that clearly there IS NOT a difference between the two. It's merely a matter of a "relationship" of the elements involved.

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Zero is often complicated.

Can you explain what you think the phrases " space of 1" and "value of 1" mean?

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Studiot

You will notice that Sato did not distinguish the difference between the space and value of the "set" of zero. Sato claimed that the space of zero was a relationship of it's elements. Therefore there is not a difference between the two, only that to define one, you must have and measure the other. Which I can agree with, but it shows then that clearly there IS NOT a difference between the two. It's merely a matter of a "relationship" of the elements involved.

I was congratulating Sato on his distinction between a set and a space.

A set is the more general object. Here is Simmons' version of the statement

Simmons: "Introduction to Topology and Modern Analysis"

The words set and space are often used in loose contrast to one and other. A set is merely an amorphous collection of elements, without coherence or form. When some kind of algebraic or geometric structure is imposed upon a set, so that the elements are organised into a systematic whole, then it becomes a space.

You should be aware that the words 'set', 'space' and 'relation' have special meaning in Mathematics.

Edited by studiot
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Your just plain wrong. First off a set "please" verify, is defined as "any" collection of distinct objects....is zero an object then ? Is there more than one of them, so as to make a collection? Zero as a set does NOT fit the definition.

No, zero is not an object (or element) in the set, it is the set without anything in (called the "empty set"). Even more informally, if you have no objects in your collection, then you have zero.

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No, zero is not an object (or element) in the set, it is the set without anything in (called the "empty set"). Even more informally, if you have no objects in your collection, then you have zero.

Hello Strange,

I think you have to be careful here as the terminology is confusing.

Is the introduction of set theory appropriate because zero is an element of some sets, as well as a set in its own right in some circumstances.

It is very easy to become confused between sets, elements, the relationships between sets and the relationships within sets (i.e. between elements) and properties of sets and properties of elements.

This is particularly so when the elements themselves are sets in their own right. Here the terminology is not universal. I like Simmons' approach to call sets of sets 'classes' when there is any doubt.

Anyway you are not strictly correct to say that zero 'is' a set.

Consider the following.

Let there be a set of elements. S, denoted {a, b, c , d} etc, equipped with a relation between elements, called multiplication, that produces a member of the set.

a * b = a member of the set

$If\;(a,b) \in S\quad then\quad (a*b) \in S$

Three possibilities arise

1) a * b = c where c is different from a or b

2) a * b = a for every a in the set

3) a * b = b for every a in the set

Result (2) makes the value of b one: 'b' is then known as the identity element

Result (3) makes the value of b zero: 'b' is then known as the zero element

But not all sets have an identity element or a zero element or both.

Edited by studiot
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I think you have to be careful here as the terminology is confusing.

You are right, but I was only trying to clarify the definition of zero as the empty set (as used by Sato) rather than a member of a set, for the benefit of someone who is not familiar with formal mathematics terminology. (I hope you haven't caused even more confusion with your clarification!)

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You are right, but I was only trying to clarify the definition of zero as the empty set (as used by Sato) rather than a member of a set, for the benefit of someone who is not familiar with formal mathematics terminology. (I hope you haven't caused even more confusion with your clarification!)

I hope so too.

If I didn't think conway had started sprinkling the red dust around, undoing my encouragement for Sato, I would point out to him that case (3) is probably the zero he is looking for.

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If something (zero), is not a member of a set, and it IS the "set" of "emptiness"...........then there is NO set! Nothing is not the absence of something...because "it" or "nothing" doesn't exist.. There is no such thing as the absence of something. Zero therefore is NOT a set or an "empty" set......it is not a set at all....empty or otherwise. I do not use your point methods.....the "voting", or red and green "dust" is a little silly....I will however balance out all votes to zero where they actually belong until people like Studiot start throwing around the green pixie sugar. In any case no one other than Sato actually tried to give an opinion (on topic) here so TRULY thank you Sato.......lock this thread down boys and move on.

Studiot

If case 3 is what I am looking for, and then zero IS a member of the set...then zero is an element yes.....? So then if zero is an element then there MUST be something more to note on regards to the space and value contained in the element zero. Specifically how it "relates" to any other element in the set given.....so if yo don't want some red dust.......give an answer for the original question using your support from post 11.

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If something (zero), is not a member of a set, and it IS the "set" of "emptiness"...........then there is NO set!

Being able to have a set with no members (often represented as {} ) is fundamental to set theory and therefore nearly all of mathematics.

Nothing is not the absence of something...because "it" or "nothing" doesn't exist..

What is it then?

There is no such thing as the absence of something.

Of course there is. I have no plutonium in my house. There is an absence of plutonium in my house.

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lock this thread down boys and move on.

!

Moderator Note

Done, because trying to correct misunderstandings is nearly impossible when one insists on redefining familiar terms.

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You are describing some set theoretical construction of the natural numbers? Something like von Neumann construction?

A value, as used in the original question, usually corresponds to a cardinal, and the cardinals up through ω correspond with von Neumann's ordinals (so that ω = $\mathbb{N}$). Past that though, the cardinals do not align with the ordinals, except for a few which are defined so.

First time I've seen that fundamental definition of sets and spaces distinguishing the two, pointed out here at SF.

There are some others who could also benefit from noting this.

+1

...

Anyway you are not strictly correct to say that zero 'is' a set.

Consider the following.

Let there be a set of elements. S, denoted {a, b, c , d} etc, equipped with a relation between elements, called multiplication, that produces a member of the set.

a * b = a member of the set

$If\;(a,b) \in S\quad then\quad (a*b) \in S$

Three possibilities arise

1) a * b = c where c is different from a or b

2) a * b = a for every a in the set

3) a * b = b for every a in the set

Result (2) makes the value of b one: 'b' is then known as the identity element

Result (3) makes the value of b zero: 'b' is then known as the zero element

But not all sets have an identity element or a zero element or both.

Thank you for the kind words!

In set theory we usually do associate the empty set with the value 0, incrementing perpetually with the successor usually defined by adjunction of sets, $S(x) := x \cup \{x\}$. After the naturals, this usually stops and the power set operation is used to define greater and greater transfinite values. In type-theoretic formalizations we do similar by defining zero as a single variable symbol, our basic unit, and successively applying a successor function to it. I think it is similar in category theory with 0 as a distinguished object with successors and predecessors related by morphisms (functions), but ajb could probably expound that more accurately, maybe with some category theoretic connection between values and spaces.

I am not so sure the algebra example elucidates too much, as the element having the zero (absorbing) property in a given algebraic structures doesn't necessarily have the characteristics of the value zero.

If something (zero), is not a member of a set, and it IS the "set" of "emptiness"...........then there is NO set! Nothing is not the absence of something...because "it" or "nothing" doesn't exist.. There is no such thing as the absence of something. Zero therefore is NOT a set or an "empty" set......it is not a set at all....empty or otherwise. I do not use your point methods.....the "voting", or red and green "dust" is a little silly....I will however balance out all votes to zero where they actually belong until people like Studiot start throwing around the green pixie sugar. In any case no one other than Sato actually tried to give an opinion (on topic) here so TRULY thank you Sato.......lock this thread down boys and move on.

You are misunderstanding the definition of a set. It was taken intuitively from the idea of "a container of things", so that you can abstract an empty box to the empty set. A set within a set is more clearly read as a box within a box than as a collection within a collection. More so your problem is that you haven't put any effort into learning the basic notions of set theory, while sitting here trying to attack ideas that are established upon those very notions.

Thousands of people have thought long and hard about set theory, most within the last century and a half; there were heated debates in the earlier years when the foundations were being laid, intuitions thrown aside in favor of formal consistency, and its founder (George Cantor) attacked into depression by contemporaries. Many of them, some of which surely took their work to heart, died with the comfort that they'd cleared some of the path for those curious in the future, and you, who have access to the internet and enough time to start and engage in discussions like these, do them so much respect as to not put any effort into learning the ideas that they worked so hard to develop.

I asked Phi to reopen this thread so I could communicate this to you, that if you care even a drop about these curiosities on the idea of a "collection", "value", and "space", and appreciate at all the time others will to put in to help you here, you'd take a look into an introduction to set theory, then the same for topology and geometry in further interest.

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Sato

I took my defention for "set" from factual sources. Is the following definition wrong? "A set is a distinct collection of objects". Agreed I may be entrley misunderstanding this, but I am not disregarding them or makeing them up. I have not done a diservice to cantor or any others. Not by my recollection. In any case the box that is ment to hold the elements is not the set. It is the elements that are the set. Please copy and paste this "misrepesention" of notation that you claim I have made. I will try to rectify it. I don't recall useing any notation at all as of yet. But if you will....

The set of zero is {}.

The "brackets" are not the set....the set is within the brackets....there is nothing in the brackets....therefore there is NO set.

Lastly... if I did not really care about these "abstract" "nonprofit" "theoretical" ideas....then I would not be on this forum at all. You do me a disservice here.

Thank you for your willingness to discuss this idea.

Sato

"Set Theory is the true study of inﬁnity".........Ha ha the very first sentence! Your sir or madam....can NOT approach zero on ANY number line by means of current number group definitions "real's, rationales, etc.".....therefore zero can not be a set of any kind! If zero is not a set then what is the nature of its space and value.......................?.................

Side Note For AJB

As the measure of space and time is relative to your perspective, so also is multiplication and division relative to your perspective.

Edited by conway
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Side Note For AJB

As the measure of space and time is relative to your perspective, so also is multiplication and division relative to your perspective.

I think we have discussed this before... special and general relativity have no bearing on the fundamental properties of rings and fields.

A value, as used in the original question, usually corresponds to a cardinal, and the cardinals up through ω correspond with von Neumann's ordinals (so that ω = $\mathbb{N}$). Past that though, the cardinals do not align with the ordinals, except for a few which are defined so.

Okay.

I think it is similar in category theory with 0 as a distinguished object with successors and predecessors related by morphisms (functions), but ajb could probably expound that more accurately, maybe with some category theoretic connection between values and spaces.

This is not something I am familiar with. I know that the cardinal numbers capture the basic structures of sets in a categorical sense, but I do not know the details.

Edited by ajb
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Sato

I took my defention for "set" from factual sources. Is the following definition wrong? "A set is a distinct collection of objects".

It is an informal definition, not a mathematical one, so in that sense it is wrong.

This is where you are showing your disrespect to the hundreds of mathematicians who have worked to clarify these concepts (and the millions of students who have worked to understand them) - by dismissing them out of hand because their work does not jibe with your intuition and you are too lazy to learn.

"Set Theory is the true study of inﬁnity".........Ha ha the very first sentence!

Did you stop at that point? Or did you make an attempt to learn something?

Did you get as far as page 9:

Our naive intuition about sets is wrong here. Not every collection of

numbers with a description is a set. In fact it would be better to stay away

from using languages like English to describe set

Edited by Strange
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