Please show us how the limit concept is rigorous

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In summary, the limit concept is a rigorous definition for determining the convergence or divergence of a sequence x_n indexed by the natural numbers to a limit x. This means that for any given margin of error e, there exists a point m in the natural numbers where all points after m lie within the interval of [0,e]. This is known as the invariant state and is the reason why we call it a rigorous definition. The limit of a sequence is what the number gets close to, and for real numbers, the limit of 1/x is infinity while the value of 1/0 is undefined
  • #141
Ah, paradigm shifts, automatic 40 point penalty on the crackpot index. Organic, I have forgotten more about this than you will ever learn, stop patronizing please.
 
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  • #142
automatic 40 point penalty on the crackpot index
1) patronizing? Who gives the points here, me?

2) Please show us a mathematical branch that define organic numbers as I show in my previous post.

3) Once you asked me to explain how N is not a complete collection by show n which is not in N.

My answer is very simple: natural numbers do not exists because of the existence of N, but because of the axioms that define them, N is only the name of the container that its content is infinitely many elements that can never be completed.
 
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  • #143
How can we demonstrate organic numbers, as you've never defined them?
 
  • #144
Here they are, in colors, to help you to understand their structures:

Let x be a general notation for a singleton.

When a finite collection of singletons have the same color, we mean that all singletons are identical, or have the maximum symmetry-degree.

When each singleton has its own unique color, we mean that each singleton in the finite collection is unique, or the collection has the minimum symmetry-degree.

Multiplication can be operated only among identical singletons, where addition is operated among unique singletons.

Each natural number is used as some given quantity, where in this given quantity we can order several different sets, that have the same quantity of singletons, but they are different by their symmetrical degrees.

In more formal way, within the same quantity we can define all possible degrees, which existing between a multiset and a "normal" set, where the complete multiset and the complete "normal" set are included too.

If we give an example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n >= 1 forms:
Code:
[b]1[/b]
(+1) = [COLOR=Black]{x}[/COLOR]

[COLOR=Blue][b]2[/b][/COLOR]
(1*2)    = [COLOR=Blue]{x,x}[/COLOR]
((+1)+1) = [COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR]

[COLOR=DarkGreen][b]3[/b][/COLOR]
(1*3)        = [COLOR=Darkgreen]{x,x,x}[/COLOR]
((1*2)+1)    = [COLOR=Darkgreen]{[COLOR=Blue]{x,x}[/COLOR],x}[/COLOR]
(((+1)+1)+1) = [COLOR=Darkgreen]{[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR],x}[/COLOR]

[COLOR=Magenta][b]4[/b][/COLOR]
(1*4)               = [COLOR=Magenta]{x,x,x,x}[/COLOR] <------------- Maximum symmetry-degree, 
((1*2)+1*2)         = [COLOR=Magenta]{[COLOR=Blue]{x,x}[/COLOR],x,x}[/COLOR]              Minimum information's 
(((+1)+1)+1*2)      = [COLOR=Magenta]{[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR],x,x}[/COLOR]            clarity-degree
((1*2)+(1*2))       = [COLOR=Magenta]{[COLOR=Blue]{x,x}[/COLOR],[COLOR=Blue]{x,x}[/COLOR]}[/COLOR]            (no uniqueness) 
(((+1)+1)+(1*2))    = [COLOR=Magenta]{[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR],[COLOR=Blue]{x,x}[/COLOR]}[/COLOR]
(((+1)+1)+((+1)+1)) = [COLOR=Magenta]{[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR],[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR]}[/COLOR]
((1*3)+1)           = [COLOR=Magenta]{[COLOR=Darkgreen]{x,x,x}[/COLOR],x}[/COLOR]
(((1*2)+1)+1)       = [COLOR=Magenta]{[COLOR=Darkgreen]{[COLOR=Blue]{x,x}[/COLOR],x}[/COLOR],x}[/COLOR]
((((+1)+1)+1)+1)    = [COLOR=Magenta]{[COLOR=Darkgreen]{[COLOR=Blue]{[COLOR=Black]{x}[/COLOR],x}[/COLOR],x}[/COLOR],x}[/COLOR] <------ Minimum symmetry-degree,
                                              Maximum information's  
                                              clarity-degree                                            
                                              (uniqueness)
[COLOR=Red][b]5[/b][/COLOR]
[COLOR=Red]...[/COLOR]

Can you give me an address of some mathematical brach that researches these kind of relations between multisets and "normal" sets?

Thank you,

Organic
 
  • #145
You're just defining certain types of partition functions. Why should we do your research? Do I patronize you? Tell you things you already know? Strange you don't seem to know lots of things. I'm certainly not polite to you, but that's because you don't earn respect from me, not that you'd want to obviously.
 
  • #146
You're just defining certain types of partition functions
Have you seen before any use of these partition functions as I do?
 
  • #147
Yes and no. I have not seen people assign words to things as you do, but then as you never explain what any of those words mean that is highly irrelevant. You don't actually do anything with the things you write down as you admit yourself (and that scores you 50 more points on the crackpot index as well).
 
  • #148
<h2>1. What is the limit concept?</h2><p>The limit concept is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as the input gets closer and closer to a specific value.</p><h2>2. How is the limit concept used in calculus?</h2><p>The limit concept is used to define important concepts in calculus, such as continuity, derivatives, and integrals. It allows us to analyze the behavior of functions and make predictions about their values at specific points.</p><h2>3. What does it mean for a limit to be rigorous?</h2><p>A rigorous limit is one that is well-defined and can be proven to exist using mathematical principles. This means that the limit is not simply an approximation, but rather a precise value that can be calculated and proven to be correct.</p><h2>4. How is the limit concept proven to be rigorous?</h2><p>The limit concept is proven to be rigorous through the use of mathematical definitions and theorems. These definitions and theorems provide a framework for understanding the concept and proving its validity.</p><h2>5. Can you provide an example of a rigorous limit?</h2><p>One example of a rigorous limit is the limit of the function f(x) = x^2 as x approaches 2. This limit can be proven to exist and have the value of 4 using the formal definition of a limit and the properties of limits.</p>

1. What is the limit concept?

The limit concept is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as the input gets closer and closer to a specific value.

2. How is the limit concept used in calculus?

The limit concept is used to define important concepts in calculus, such as continuity, derivatives, and integrals. It allows us to analyze the behavior of functions and make predictions about their values at specific points.

3. What does it mean for a limit to be rigorous?

A rigorous limit is one that is well-defined and can be proven to exist using mathematical principles. This means that the limit is not simply an approximation, but rather a precise value that can be calculated and proven to be correct.

4. How is the limit concept proven to be rigorous?

The limit concept is proven to be rigorous through the use of mathematical definitions and theorems. These definitions and theorems provide a framework for understanding the concept and proving its validity.

5. Can you provide an example of a rigorous limit?

One example of a rigorous limit is the limit of the function f(x) = x^2 as x approaches 2. This limit can be proven to exist and have the value of 4 using the formal definition of a limit and the properties of limits.

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