Deeper than primes

[epix]

You contradict yourself.

Suppose there is only the quoted statement, but no reference to what is the nature of the contradiction. In that case what kind of contradiction does that statement refer to?

Since we are in the beginning of the whole evolutionary process, such a question can mess our head the "huh?" way. But that question doesn't stop the older guys who roam the Milky Way from answering . . .

Give a large group of college graduates a test that includes this question: Is the statement

exact = approximate

false or true?

The result shows that the vast majority opted for the false answer.

Then gather another large group of college graduates and give them a test that includes this question: Is the statement

1/3 = 0.333...

true or false?

According to the long division, the statement is true and that showed: all respondents agreed on the option that the statement was true.

But since 1/3 is a rational number written in exact format with 0.333... being the approximate equivalent, then exact = approximate, and both groups of the college graduates therefore significantly contradicted each other.

But how is this example particular to the statement "You contradict yourself"?

Well, "You contradict yourself" is 1 sentence comprising 3 words. It means that the following statement doesn't appear to be random at all:

You contradict yourself = 1/3

where the number 1/3 is the clue to the nature of a contradiction that Homo sapiens was asked to come up with.

By whom?

Let's answer by another question: 1 triangle has 3 vertex points. How are those three vertex points defined?

Let me see the triangle, and I tell you. Duh.

So what kind of triangle is it?

1) Triangle GOD
2) Triangle UFO
3) Triangle MAN

I think we must shoot it down and look inside.

 

[The Man]

You contradict yourself.

Since 0.000...1[base 10] is a non-strict value, it can't be the smallest existing value under [base 10], because being smallest is being strict value (for example 0 or 1), and 0.000...1[base 10] is a non-strict and therefore it is simultaneously smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0.

“smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0”, so it is smaller than itself? Also if it is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0” then it is your smallest “value > 0 that exists under [base 10]” by your own assertions. Talk about contradicting yourself.

You simply do not understand that the magnitude of existence of 1-dimensional space is stronger than the magnitude of existence of any given infinite collection of 0-dimensional spaces along it, such that the magnitude of any collection of 0-dimensional spaces is inaccessible w.r.t the magnitude of 1-dimensional space, where 0.000...1[base 10] is some example of this inaccessibility.

Doron once again this “inaccessibility” is entirely of your own making by having a non-zero infinitesimal and it restricts only you. Once again it simply means that your “1-dimensional space is” a collection of one dimensional spaces (your non-zero infinitesimal 0.000...1[base 10]”), so it is simply “0-dimensional spaces” that are “inaccessible” to you by your own self imposed limitation.

 

[doronshadmi]

“smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0”, so it is smaller than itself?

Once again it simply means that your “1-dimensional space is” a collection of one dimensional spaces

Again your reasoning collapses under the misleading reasoning of complete infinite collection, which is no more than your fantasy.

Once again The Man you can't distinguish between a complex like infinitely many segments and non-complex like 1-dimensional space, which exists independently of any collection of points or segments along it, simply because its magnitude of existence is stronger than any collection, including collection of segments, which do not have the completeness of 1-dimensional space.

This is exactly the reason of why a collection of segments like number 0.999...[base 10] < 1 by the infinitesimal number 0.000...1[base 10].

In general you do not understand the qualitative difference between the complex and the non-complex, because you are limited only to quantitative reasoning, whiteout the understanding that Quantity is the result of the link between different qualities (known as Non-locality and locality), and not vice versa.

 

[doronshadmi]

Since "...1" is real non-sense, then it must be a real number and therefore it is divisible, and if it is divisible, then the points of division do exist, and since the field of real numbers is infinitely divisible by definition, then there are infinitely many locations on that line segment that represents "...1" where you can point your finger on. End of proof.

There can be any cardinality of R members along ...1 and still ...1 exists exactly because the magnitude of existence of non-locality like ...1 is stronger than the cardinality of any collection of R members along it (R members are local-only things, where ...1 is a non-local thing).

 

[doronshadmi]

“smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0”, so it is smaller than itself? Also if it is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0” then it is your smallest “value > 0 that exists under [base 10]” by your own assertions. Talk about contradicting yourself.

Your limited reasoning simply can't grasp the notion of "permanently smaller" (in the case of infinite interpolation), or "permanently greater" (in the case of infinite extrapolation).

As can be seen, for example, in the case of 0.000...1[base 2] (which is the complement of 0.111...[base 2] to 1,
or in the case of 0.000...1[base 3] (which is the complement of 0.111...[base 3] to 1,

in both cases the infinitesimals 0.000...1 are smaller than the previous infinitely many segments,
which exist upon the scale levels of 0.111…[base 2] or 0.222…[base 3] AND greater than 0.

The existence of 0.000...1 is the result of the inaccessibility of the complex to the non-complex, which is an essential fact that can't be grasped by your complex-only\strict-only\local-only reasoning.

 

[The Man]

Again your reasoning collapses under the misleading reasoning of complete infinite collection, which is no more than your fantasy.

Again please show what member of such an infinite collection is not a member of that collection. Until you do the fantasies remain completely yours.

Once again The Man you can't distinguish between a complex like infinitely many segments and non-complex like 1-dimensional space, which exists independently of any collection of points or segments along it, simply because its magnitude of existence is stronger than any collection, including collection of segments, which do not have the completeness of 1-dimensional space.

Once again Doron you can’t distinguish between your fantasies and mathematics.

This is exactly the reason of why a collection of segments like number 0.999...[base 10] < 1 by the infinitesimal number 0.000...1[base 10].

In general you do not understand the qualitative difference between the complex and the non-complex, because you are limited only to quantitative reasoning, whiteout the understanding that Quantity is the result of the link between different qualities (known as Non-locality and locality), and not vice versa.

Specifically Doron you do not understand the difference between your self –inconsistent and generally inconsistent fantasies and the self-consistency and general consistency of mathematics.

Oh, by the way would you like to actually address those self-inconstant assertions of yours that I mentioned or will you continue to simply expound your self-inconsistent fantasies? Your history (and your next post) indicates it will be the latter.

Your limited reasoning simply can't grasp the notion of "permanently smaller" (in the case of infinite interpolation), or "permanently greater" (in the case of infinite extrapolation).

As can be seen, for example, in the case of 0.000...1[base 2] (which is the complement of 0.111...[base 2] to 1,
or in the case of 0.000...1[base 3] (which is the complement of 0.111...[base 3] to 1,
[qimg]http://farm3.static.flickr.com/2793/4318895416_e5d2042b0c_z.jpg?zz=1[/qimg]
in both cases the infinitesimals 0.000...1 are smaller than the previous infinitely many segments,
which exist upon the scale levels of 0.111…[base 2] or 0.222…[base 3] AND greater than 0.

The existence of 0.000...1 is the result of the inaccessibility of the complex to the non-complex, which is an essential fact that can't be grasped by your complex-only\strict-only\local-only reasoning.

Once again Doron that it “is a non-strict value” and is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0” was your assertion and would make it smaller than itself. Your diagrams and inane rambling do not change the self-inconsistency of that assertion. Would you like to actually address the self-inconsistency of that assertion or just continue to expound your self-inconsistent fantasies?

 

[epix]

Because of this fact 0.999…[base 10] < 1 by 0.000…1[base 10], where the expression "…1" of 0.000…1[base 10] is the uncovered 1-dimesional space w.r.t any cardinality of some collection of 0-dimensional spaces along it.

So 0.999... < 1. That's very nice, coz all numbers should be included in R and those numbers whose fractional part is made of one digit that repeats itself infinitely shouldn't make an exemption. So if we divide 0.999... by 3, we get another such number.

0.999... / 3 = 0.333...

When we divide, we may make a mistake, and so it's a good thing to make sure that we got the correct result. Since the opposite of division is multiplication, we multiply the result of the division with the dividend and should obtain the same divisor.

IF a/b = c THEN c*b = a

When we wish to obtain the result of the division in the approximate form, we use an algorithm called "the long division" that sometimes produces bunch of digits after the decimal point, such as

1/3 = 0.333333333333333333333333333333...

which is usually expressed as 0.333.... This tells us that the 3s repeats infinitely. But did we divide right? To make sure, let's do the multiplication check:

IF
1/3 = 0.333...
THEN
0.333... * 3 = 0.999...



I think we made a mistake in that long division, but I can't find it. Can you give it a try, Doron? Give it your best, otherwise

In mathematics, the repeating decimal 0.999... which may also be written as 0.9, or 0.(9), denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

and your claim that "0.999... < 1" will become Turbulent Nonsense of Unusual Desire residing within the pink walls of Organic Mathematics, the heir designated to Traditional Mathematics.

So remember once again: You contradict yourself is 1 sentence divided into 3 words, and so "you contradict yourself" = 1/3.

 

[doronshadmi]

Again please show what member of such an infinite collection is not a member of that collection. Until you do the fantasies remain completely yours.

We are not talking about any specific member of a given collection, but about the fact that the magnitude of existence of any given infinite collection (where a collection is a complex thing) is smaller than the magnitude of existence of non-complex AND non-local thing.

As a result the magnitude of existence of non-complex AND non-local thing is inaccessible to any magnitude of existence of complex of infinitely many things.

It is very simple to understand this notion, as follows:

...____________... is the non-complex 1-dimensional space, which exists independently of any sub-things along it, whether these sub-things are collection of 0-dimensional spaces or collection of segments, where each segment has the magnitude of existence of a complex that is inaccessible to the magnitude of existence of ...____________... , which is the non-complex 1-dimensional space.

Before we start to research the difference between the existing complex and the existing non-complex, we define our framework that exists between two opposite totalities, which are Emptiness that is defined as “that has no predecessor”, and Fullness that is defined as “that has no successor”.

According to this framework, Cardinality is the measurement of the magnitude of existence of a given thing.

This framework is notated as:

∞(

)

where ∞( ) represents Fullness, and the unmarked space between ( ) represents Emptiness.

Any given dimensional space that is defined by an integer, is a non-complex thing, and it exists independently of any other non-complex or complex dimensional spaces w.r.t it, as follows (this is an example, which uses 0-dimensional space and 1-dimensional space, without a loss of generality):

a) ...____________... is the non-complex 1-dimensional space.

b) . is the non-complex 0-dimensional space.

c) The cardinality of the complex result of (a) AND (b), is smaller than the cardinality of (a).

d) Under (c), (a) is connectivity and (b) is isolation, where connectivity or isolation are qualitative distinct properties.

e) By using the qualitative distinct properties of non-complex (a) and non-complex (b) under complex (a) AND (b), any complex (a) AND (b) has cardinality that is smaller than the cardinality of non-complex (a) ,such that (a) is the minimal case that has simultaneous existence at an beyond any arbitrary given (b), which is a property that no (b) has w.r.t to (a) or another (b).

f) No infinite amount of segments ( where a segment is complex result of (a) AND (b) ) has the cardinality of (a), simply because (a) exists independently and beyond any collection of sub-elements along it, for example:

...___________.________.___________... and it is easily and unconditionally understood that the cardinality of (a) is beyond (is greater than) the cardinality of any collection of (b) or segments (b)_________(b), along it.

g) This fact is unchanged, whether complex (a) AND (b) converges ( has the tendency to be (b) ) or diverges ( has the tendency to be (a) ).

Would you like to actually address the self-inconsistency of that assertion or just continue to expound your self-inconsistent fantasies?

Would you able to distinguish between Emptiness, Fullness, (a), (b) and (b)_________(b) , or just continue to expound your self-inconsistent fantasies?

 

[doronshadmi]

EDIT:

Once again Doron that it “is a non-strict value” and is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0” was your assertion and would make it smaller than itself.

Once again The Man, your reasoning is unable to grasp the inability of a segment to be a point (which is resulted by infinitely many smaller segments that are naturally > 0), and since we are dealing with infinitely many segments, then no one of them is the final and smallest segment > 0.

In other words, your weak reasoning forces notions that are taken from finite collection of segments, on a collection of infinitely many ever smaller segments.


By the way, the "signature" of non-locality in nature is shown by experiments like http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.3316v1.pdf .

 

[The Man]

We are not talking about any specific member of a given collection, but about the fact that the magnitude of existence of any given infinite collection (where a collection is a complex thing) is smaller than the magnitude of existence of non-complex AND non-local thing.

No Doron we were specifically talking about your assertion of what you claim is “the misleading reasoning of complete infinite collection”. So again please show what member of such an infinite collection is not a member of that collection. Until you do the fantasies remain completely yours.

So are you now claiming an infinite collection is complete?

As a result the magnitude of existence of non-complex AND non-local thing is inaccessible to any magnitude of existence of complex of infinitely many things.

As you have failed to define your “magnitude of existence” other then erroneously associating it to cardinality, your “magnitude of existence” remains simply inaccessible even to you.

It is very simple to understand this notion, as follows:

...____________... is the non-complex 1-dimensional space, which exists independently of any sub-things along it, whether these sub-things are collection of 0-dimensional spaces or collection of segments, where each segment has the magnitude of existence of a complex that is inaccessible to the magnitude of existence of ...____________... , which is the non-complex 1-dimensional space.

Once again a space is not independent of its subspaces.

Before we start to research the difference between the existing complex and the existing non-complex, we define our framework that exists between two opposite totalities, which are Emptiness that is defined as “that has no predecessor”, and Fullness that is defined as “that has no successor”.

According to this framework, Cardinality is the measurement of the magnitude of existence of a given thing.

Nope, again cardinality has a specific meaning and it is not “the measurement of the magnitude of existence of a given thing” and your simple assertion of “Emptiness that is defined as “that has no predecessor”, and Fullness that is defined as “that has no successor”” doesn’t change that. In fact your purported “framework” mentioned above has absolutely nothing to do with cardinality as by your own assertions it is specifically based on the lack of a predecessor or a successor.

This framework is notated as:

∞(

)

where ∞( ) represents Fullness, and the unmarked space between ( ) represents Emptiness.

You can note your nonsense “framework” any way you want. It still doesn’t make it about anything other then the lacking of a of a predecessor or a successor nor does it make cardinality “the measurement of the magnitude of existence of a given thing.”

Any given dimensional space that is defined by an integer, is a non-complex thing, and it exists independently of any other non-complex or complex dimensional spaces w.r.t it, as follows (this is an example, which uses 0-dimensional space and 1-dimensional space, without a loss of generality):

Once again Doron the “integer” you are referring to represents the number of coordinates needed to define a point in that “given dimensional space” so “Any given dimensional space” is certainly not independent of the points that define the dimensionality of that space.

a) ...____________... is the non-complex 1-dimensional space.

b) . is the non-complex 0-dimensional space.

c) The cardinality of the complex result of (a) AND (b), is smaller than the cardinality of (a).

Are you claiming that the cardinality of a set containing a line is greater that the cardinality of a set containing two rays? If so your going to have to show how you arrive at that obviously erroneous result.

d) Under (c), (a) is connectivity….

“Connectivity”? of what or to what? Remember “1-dimensional space, which exists independently of any sub-things along it, whether these sub-things are collection of 0-dimensional spaces or collection of segments”. You’ve got nothing to ‘connect’ by your own self imposed limitation.

…and (b) is isolation, where connectivity or isolation are qualitative distinct properties.

e) By using the qualitative distinct properties of non-complex (a) and non-complex (b) under complex (a) AND (b), any complex (a) AND (b) has cardinality that is smaller than the cardinality of non-complex (a) ,such that (a) is the minimal case that has simultaneous existence at an beyond any arbitrary given (b), which is a property that no (b) has w.r.t to (a) or another (b).

Um Doron, you do you do understand that connected spaces are not isolated, don’t you? That is the difference between discrete and continuous spaces.

f) No infinite amount of segments ( where a segment is complex result of (a) AND (b) ) has the cardinality of (a), simply because (a) exists independently and beyond any collection of sub-elements along it, for example:

...___________.________.___________... and it is easily and unconditionally understood that the cardinality of (a) is beyond (is greater than) the cardinality of any collection of (b) or segments (b)_________(b), along it.

No Doron a set containing just a line has a cardinality of 1 while a set containing an “infinite amount of segments” or even an infinite number of points in just one of those line segments has a, well, infinite cardinality. After all this time it still just comes down to your deliberate misrepresentation of cardinality as your “magnitude of existence” .

g) This fact is unchanged, whether complex (a) AND (b) converges ( has the tendency to be (b) ) or diverges ( has the tendency to be (a) ).


Would you able to distinguish between Emptiness, Fullness, (a), (b) and (b)_________(b) , or just continue to expound your self-inconsistent fantasies?

“Would you able to distinguish between” your “magnitude of existence” and cardinality? Evidently not. By all means please show any “self-inconsistent fantasies” of mine that I have expounded on this thread. So are going address those self-inconstant assertions of yours that I mentioned or will you continue to simply expound your self-inconsistent fantasies and just repeat what others say to you? Once again your history indicates that it will simply be the latter.

 

[The Man]

EDIT:


Once again The Man, your reasoning is unable to grasp the inability of a segment to be a point (which is resulted by infinitely many smaller segments that are naturally > 0),

Who ever claimed a segment could be a zero dimensional point?

and since we are dealing with infinitely many segments, then no one of them is the final and smallest segment > 0.

So you’ve got a segment that is smaller than your “infinitesimal number 0.000...1[base 10].” Once again by your own assertions it is “smaller than any strict or non-strict value > 0 that exists under [base 10] AND it is also > 0”, so it is just smaller than itself and any other segment. Thus it remains your smallest segment even with your self-contradictory assertions.

In other words, your weak reasoning forces notions that are taken from finite collection of segments, on a collection of infinitely many ever smaller segments.

In the exact words I used above your self-contradictory assertions force your “infinitesimal number 0.000...1[base 10].” to simply be smaller than itself and your smallest segment.

By the way, the "signature" of non-locality in nature is shown by experiments like http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.3316v1.pdf .

As explained to you before Doron in physics ‘non-local’ has a very specific meaning related to space like separations. It has nothing to do with your “"signature" of non-locality” which is evidently you just spouting self-contradictory nonsense.

 

[doronshadmi]

EDIT:

No Doron we were specifically talking about your assertion of what you claim is “the misleading reasoning of complete infinite collection”. So again please show what member of such an infinite collection is not a member of that collection. Until you do the fantasies remain completely yours.

So are you now claiming an infinite collection is complete?

No The Man, we are specifically talking about your inability to grasp that any given complex (and a non-empty set is a complex) can't reach the magnitude of existence of the non-local AND non-complex, which is not a set and not a member of a set (for example: 1-dimensional space, if its non-local property is considered).

Any member of a given set belongs to a complex and since the cardinality of a complex is smaller than the cardinality of the non-local AND non-complex (which, again, it is not a set) we do not need to show some missing member of some complex, because our comparison is between the concept of non-empty set, which is no more than a complex, and the concept of non-locality AND non-complex, which its exitence is stonger than any given complex. Your complex-only reasoning simply can't comprehend Emptiness, Fullness, (a), (b) or (b)______(b), as shown in http://www.internationalskeptics.com/forums/showpost.php?p=6564133&postcount=12348.

Are you claiming that the cardinality of a set containing a line is greater that the cardinality of a set containing two rays? If so your going to have to show how you arrive at that obviously erroneous result.

Are you claiming that you are unable to get the simple notion that a line (if its non-local property is considered) is not a set and not a member of a set, but it is a non-local and non-complex mathematical object?

In the exact words I used above your self-contradictory assertions force your “infinitesimal number 0.000...1[base 10].” to simply be smaller than itself and your smallest segment.

No The Man, you simply can't get the simple fact that a segment can't be reduced into a point exactly because of the existence of incomplete collection of smaller segments upon infinitely many scale levels, where 0.000…1 is the ever smaller AND > 0 segement. No segment is smaller than itself exactly because the collection of ever smaller segments is infinite, and no one of its members is a point.

As a result 0.999…[base 10] < 1 by the ever smaller segment 0.000…1[base 10] of the infinite (and therefore incomplete) collection of segments.

As explained to you before Doron in physics ‘non-local’ has a very specific meaning related to space like separations. It has nothing to do with your “"signature" of non-locality” which is evidently you just spouting self-contradictory nonsense.

As explained to you before The Man, your fundamental inability to distinguish between a complex thing like collection and a non-complex thing like the non-locality of 1-dimansional space, put you under the category of meaningless replies about this interesting subject.

“Connectivity”? of what or to what? Remember “1-dimensional space, which exists independently of any sub-things along it, whether these sub-things are collection of 0-dimensional spaces or collection of segments”. You’ve got nothing to ‘connect’ by your own self imposed limitation.

Now we see that you can't distinguish between total connectivity ( which is at least 1-dimensional space self state, notated as (a) ) and connectivity under complex (a) AND (b).

Also you do not distinguish between total isolation ( which is at least 0-dimensional space self state, notated as (b) ) and isolation under complex (a) AND (b).

Once again a space is not independent of its subspaces.

Once again you can't distinguish between the complex and the non-complex.

No Doron a set containing just a line has a cardinality of 1

No The Man, a line is not a set, not a member of a set (if its non-local property is considered), a collection or any other complex thing.

Once again a space is not independent of its subspaces.

Wrong once again, for example 1-dimensional space exists whether there are 0-dimansional spaces or segments along it, or not.



The Man, your two last posts trivially demonstrate that you simply have no clue of what you are trying to criticize.

 

[epix]

Once again a space is not independent of its subspaces.

That all depends . . .

I remember when I transferred to Ridgemont High, my math teacher drew a cylinder on the board, gave it the height h = 4 feet and asked me to compute the volume of the cylinder. So I asked him to reveal the area of the circle that made the bottom of the cylinder, but the teacher became curious why I needed to know that. I told him that the volume of 3-dim object depends on the area of its 2-dim component where the area depends on the length of its 1-dim component. The teacher look perplexed, so did the class.

The teacher asked me over to the board to explain in detail, and so I chalked in what I learned when I attended Angus Young High: V = pi*r²*h.

But the math teacher said that the formula was what the traditional math claimed and that it was inferior to the new look at 3-dim objects that something called "Organic Mathematics," which he taught at Ridgemont High, offered. In general, any multidimensional space is independent of its subspaces and so it suffices to give only one parameter to compute the volume of the cylinder.

So I asked how the volume was computed according to OM, but the teacher said that we would learn the conceptual framework for the derivation of the formula in the spring. It never came to that, coz he OD'd in the restroom shortly thereafter and the new teacher taught only the traditional math.

Maybe Doron would know how to compute the volume of the cylinder according to OM. My guess is that the cylinder must be filled with Fullness and emptied with Emptiness into a ten-gallon fish tank so an estimate could be made first.

 

[jsfisher]

It is very simply, the expression "2" is not the same as the expression "{2}".

Your reading comprehension has failed you, doron. That isn't what I said. Nevertheless, here's a repeat of some of your more interesting claims:

• 2 is not a member of the set {2}.
• A set is equal to the union of its members.
• 1/4 and 0.25 different numbers.
• Sets, maps, and functions are all the same thing.
• Geometry is a very weak branch of Mathematics.
• The construct A if B is equivalent to A only if B.
• There is no such thing as a number in standard Mathematics.

 

[epix]

Wrong once again, for example 1-dimensional space exists whether there are 0-dimansional spaces or segments along it, or not.

If such a space exists, then show an example of it -- just draw it.

 

[doronshadmi]

Your reading comprehension has failed you, doron. That isn't what I said. Nevertheless, here's a repeat of some of your more interesting claims:

• 2 is not a member of the set {2}.
• A set is equal to the union of its members.
• 1/4 and 0.25 different numbers.
• Sets, maps, and functions are all the same thing.
• Geometry is a very weak branch of Mathematics.
• The construct A if B is equivalent to A only if B.
• There is no such thing as a number in standard Mathematics.

Let us correct it:

• Number 2 (notated as "2") and number 2 as a member of a singleton set (notated as "{2}") is not the same expression.
  
• Set {{a},{b}} is the result of the union between set {{a}} and set {{b}}, where the members of {{a},{b}} are the members of set {{a}} and set {{b}}.
  
• From a structural point of view 1/4 is a single AND strict location along segment 0_______1, where 0.25[base 10] is a non-single AND strict location along 0_______1.
  
• A comparison between objects of one or more collections is based on the same principle, which is the linkage between non-locality and locality.
  
• Any mathematical branch that does not deal with the non-finite is weaker than any mathematical branch that deals with the non-finite.
  
• In this philosophical thread, "only if" and "if and only if" are not always used by their formal mathematical meaning.
  
• The concept of Number under Mathematics is understood according to its type. The type can be Natural number, Whole number, Rational number, Algebraic number, Irrational number, Transcendental number, Complex number, p-adic number etc ...

 

[zooterkin]

• From a structural point of view 1/4 is a single AND strict location along segment 0_______1, where 0.25[base 10] is a non-single AND strict location along 0_______1.
  

You mean 0.25 represents more than one point? If not, what do you mean by non-single?

(And why the need to note that 0.25 is base 10, but not for 1/4?)

 

[doronshadmi]

No Doron a set containing just a line has a cardinality of 1

As I said, your reasoning is limited to Quantity, by avoiding any reasoning which deals with the foundation that enables Quantity, in the first place.

The amount of a given collection is defined by the linkage among Non-locality and Locality.

The simplest result (without a loss of generality) of this linkage is done between 1-dimensional space, known as Line, and 0-dimensional spaces, known as Points.

One of the notations of Non-locality is done by "{" and "}", where the cardinality is determined by the number of the distinct objects that are defined between "{" and "}" (and in this post we shell use the traditional notion of the concept of Cardinality).

This notion is equivalent to the amount of the objects that are defined along a 1-dimensional space, as follows:

If there are no objects along the 1-dimensional space, than the cardinality is 0.

If there are objects along the 1-dimensional space, then the cardinality is the number of distinct objects along the 1-dimensional space.

Be aware of the following facts:

1) "{" "}" or ...________... exists (or used, if you wish) even if there are no objects (the cardinality is 0).

2) Any cardinality > 0 is a result of the linkage between "{" "}" or ...________..., where "{" "}" or ...________... is the non-local property, which enables the membership between distinct objects that are local w.r.t "{" "}" or ...________... (they belong XOR not belong to "{" "}" or ...________...), and this membership is exactly the non-local property of "{" "}" or ...________... w.r.t to the distinct objects, which enables the existence of non-empty collections, and the value of the Cardinality of non-empty collections.

3) It is a simple fact that any given amount of distinct objects that are determined by the linkage between the distinct objects and "{" "}" or ...________..., does not have the non-local property of "{" "}" or ...________... w.r.t the collection of such objects, because being non-local is logically be at AND beyond the collected objects (and in this case the cardinality of the non-local is greater than the cardinality of the collected objects), whether "beyond" enables the infinite interpolation or infinite extrapolation of a given collection of distinct objects w.r.t "{" "}" or ...________...

 

[doronshadmi]

You mean 0.25 represents more than one point? If not, what do you mean by non-single?

(And why the need to note that 0.25 is base 10, but not for 1/4?)

From a structural view 0.25 has more than a one location along 0_______1, where 1/4 has exactly one location along 0_______1.

(And why the need to note that 0.25 is base 10, but not for 1/4?)

Good question!

0.25[base 100] is not 0.25[base 10], so we need do determine also the base value.

1/4, 1/3 ,1, Pi etc ... do not depend on any base value (unless the x of expression 1/x is determined by some base value) , and they have exactly one location along ..._________...

 

[zooterkin]

From a structural view 0.25 has more than a one location along 0_______1

And what are those locations, exactly?