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fresh_42

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You know it when there is a theorem that proves it. Differential manifolds are too manifold to finally categorize them, Abelian groups and ordered fields are known.

You often need additional properties to get a chance for a classification, e.g. finite simple groups instead of all finite groups. Unclassified examples are nilpotent or solvable objects, groups as well as rings and algebras. There are too many of them, so any classification will probably - if at all - be done over additional constraints.

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So a complete classification of some mathematical objects may sometimes be impossible? Because there are counterexamples? The same is for generalisations of theorems or definitions of things in math? Or perhaps when having a definition of math object and someone changes it a little and gets another example then this may lead to a generalisation? The same can be applied to theorems, but theorems must not have a counterexample and must be proved?You know it when there is a theorem that proves it. Differential manifolds are too manifold to finally categorize them, Abelian groups and ordered fields are known.

You often need additional properties to get a chance for a classification, e.g. finite simple groups instead of all finite groups. Unclassified examples are nilpotent or solvable objects, groups as well as rings and algebras. There are too many of them, so any classification will probably - if at all - be done over additional constraints.

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fresh_42

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No, because there are too many of a kind. E.g. if you want to classify all stars, then you will have a problem. But you can introduce additional properties like the main sequence or certain sorts like pulsars, and concentrate on them. The same is true in mathematics. "Classify all nilpotent finite groups!" is not doable for there are too many of them. You will need additional properties as e.g. the nilpotency class (degree), or those with a given center.So a complete classification of some mathematical objects may sometimes be impossible? Because there are counterexamples?

Yes. In the example of nilpotent groups you could require a certain kind of automorphism group, anything that breaks down the task into portions that can be handled.The same is for generalisations of theorems or definitions of things in math? Or perhaps when having a definition of math object and someone changes it a little and gets another example then this may lead to a generalisation?

Yes.The same can be applied to theorems, but theorems must not have a counterexample and must be proved?

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I do not understand how with a theorem a person shows that he has all types for classification and none is left out of it. Could you tell me?No, because there are too many of a kind. E.g. if you want to classify all stars, then you will have a problem. But you can introduce additional properties like the main sequence or certain sorts like pulsars, and concentrate on them. The same is true in mathematics. "Classify all nilpotent finite groups!" is not doable for there are too many of them. You will need additional properties as e.g. the nilpotency class (degree), or those with a given center.

Yes. In the example of nilpotent groups you could require a certain kind of automorphism group, anything that breaks down the task into portions that can be handled.

Yes.

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fresh_42

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If ##G## is a finite, simple group, then ##G## is one of the following groups:

- cyclic groups of prime order
- alternating groups ##A_n## with ##n>4##
- groups of Lie type over a finite field
- one of 26 sporadic (known) groups

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For them to show that there are no other possibilities they theorized there is one and reached contradiction?

If ##G## is a finite, simple group, then ##G## is one of the following groups:

This is a complete classification for finite simple groups. The proof of the theorem has of course many steps and is historically spread over decades. You have to show that these groups are all finite and simple, and that there are no other possibilities.

- cyclic groups of prime order
- alternating groups ##A_n## with ##n>4##
- groups of Lie type over a finite field
- one of 26 sporadic (known) groups

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fresh_42

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They have found the 26 sporadic ones by assuming there is one, and finally that there is no room for a 27th. The proof is pretty complicated. The biggest of them hasFor them to show that there are no other possibilities they theorized there is one and reached contradiction?

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 elements.

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Office_Shredder

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Every integer is either even or odd.

Every integer is either zero, positive, or negative.

Every group of four elements is either ##\mathbb{Z}_4## or ##\mathbb{Z}_2 \times \mathbb{Z}_2##.

Every abelian group is either the direct sum of cyclic groups, or is infinite.

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For the first and the second you wrote, to show that no more is of another kind, they theorised another element and it lead to the already proven kinds? For the third theorem, perhaps it can not be generalised? I think it can not. I have not tried to prove it but i think counterexamples perhaps can be shown that contradict more general cases.

Every integer is either even or odd.

Every integer is either zero, positive, or negative.

Every group of four elements is either ##\mathbb{Z}_4## or ##\mathbb{Z}_2 \times \mathbb{Z}_2##.

Every abelian group is either the direct sum of cyclic groups, or is infinite.

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Mark44

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As @Office_Shredder mentioned, some of the statements he posted are very trivial. It's very easy to show directly that any integer is either evenly divisible by 2 or it is not. The same is true for showing that any integer falls into exactly one of the three possible divisions: negative, zero, or positive.For the first and the second you wrote, to show that no more is of another kind, they theorised another element and it lead to the already proven kinds?

Generalized in what way? Groups of larger size?For the third theorem, perhaps it can not be generalised? I think it can not.

You first need to state what you mean by "generalizations."I have not tried to prove it but i think counterexamples perhaps can be shown that contradict more general cases.

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One generalisation could be a group of 4n elements where n is natural number.As @Office_Shredder mentioned, some of the statements he posted are very trivial. It's very easy to show directly that any integer is either evenly divisible by 2 or it is not. The same is true for showing that any integer falls into exactly one of the three possible divisions: negative, zero, or positive.

Generalized in what way? Groups of larger size?

You first need to state what you mean by "generalizations."

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This is now getting ridiculous. Office_Shredder simply gave some examples of very simple classifications. It was not meant to be an invitation to extend any of them to be more general.One generalisation could be a group of 4n elements where n is natural number.

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I am sorry for that.This is now getting ridiculous. Office_Shredder simply gave some examples of very simple classifications. It was not meant to be an invitation to extend any of them to be more general.

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Yes, i understand now that the first one is trivial but i am not sure if my proof is correct, because i have a problem sometimes when i am trying to prove a statement in math until now.

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Office_Shredder

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Why don't you post your proof then?

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mathwonk

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i recommend you read a proof that classifies all compact topological surfaces. as i recall vaguely, they all arise, starting from the sphere, by adding on handles and crosscaps.

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pbuk

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No the arrogance is yours. The dictionary was not created for you, it was created for everyone. And when the majority of people type 'crosscaps' they have made a mistake. All you have to do is add the word to your local configuration of the dictionary.The programmer who did this is not an imbecile, rather he is probably very bright, but perhaps arrogant. I.e. he thinks he knows what I mean to say better than I do. This is annoying.

A dictionary that included every technical term from every field would be useless: almost any pronouncable combination of letters means something to someone!

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jbriggs444

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No. It is a handwave, not a proof. What is missing is a demonstration that {1,2} union {3,4} union {5,6} and so on is equal to the set of all natural numbers. As it is, we have a handwave rather than a proof.

A good rule of thumb is that if you want to prove something about every integer, then mathematical induction is the appropriate tool.

Side note: It is very common for someone unfamiliar with mathematics to look at the word "**induction**" and immediately think "oh yes, I know what that is: 'deduction' is when you use a general principle and arrive at a conclusion about a specific case. 'induction' on the other hand is when you start with a bunch of specific cases and use them to guess at a general rule. Induction can never be used for a true proof because it is not 100% reliable".

That is not what mathematical induction is. Mathematical induction is a valid tool for deductive reasoning. It is 100% reliable.

An argument by mathematical induction takes a standard form:1. Show that a particular property holds for the first natural number.

2. Show that whenever that property holds for a natural number n, it necessarily holds for n+1.

3. Conclude that the property holds for all natural numbers.

An inductive proof for the case at hand is simple. The property we will use is "is either odd or even". We will use your definition for odd: "Is equal to 2k-1 for some natural number k" and even: "Is equal to 2k for some natural number k"

1. The first natural number is either odd or even.

Proof: 1 is the first natural number. For natural number k=1, 1=2k-1

2. If n is a natural number that is either odd or even, n+1 is also either odd or even.

Proof:

Case 1: If n is odd then it is equal to 2k-1 for some natural number k. It follows that n+1 is equal to (2k-1)+1 = 2k. So n+1 is even.

Case 2: If n is even then it is equal to 2k-1 for some natural number k. It follows that n+1 is equal to 2k+1 = 2(k+1)-2. Since k+1 is a natural number, n+1 is odd.

3. By mathematical induction, we conclude that every natural number is either even or odd.

I leave it as an exercise for the interested reader (if any) to prove that no natural number is both even and odd.

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Thank you. I knew about mathematical induction, i was not sure if i should use it. If we have n=2k-1 for odds and n=2k for evens then a number can not be even and odd because 2k≠2k-1 because 0≠1. If we assume they are equal it leads to a contradiction.I leave it as an exercise for the interested reader (if any) to prove that no natural number is both even and odd.

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jbriggs444

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No. That proof does not work.Thank you. I knew about mathematical induction, i was not sure if i should use it. If we have n=2k-1 for odds and n=2k for evens then a number can not be even and odd because 2k≠2k-1 because 0≠1. If we assume they are equal it leads to a contradiction.

You have not exhibited a guarantee that a "k" that exists to demonstrate that n=2k is even is the same "k" that exists to demonstrate that n=2k-1 is odd. [Which indicates that you also have some opportunities for better education about variable naming, scopes and quantifiers].

Let me provide you with an axiom to make your work easier: 1 is not even. Or, if you prefer: 0 is not odd.

[It is not easy to prove that 1 is not even. Especially if you do not know what axioms or lemmas you have to work with]

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Let us try. So, Suppose we have a=2b-1, c=2d, (i) if b=2e, d=2f, then if a=b=>2b-1=2d=>2(d+b)=1=>2(2e+2f)=1=>4(e+f)=1 but this is a contradiction because e and f must be odd integers and 2k-1+2l-1=2((k+l)-1) so 4(e+f)=1=>4((2(k+l)-1)=1 which means that 4 times an odd number is equal to 1, contradiction. Similar proofs are for the cases where b is an odd number and d an even , and b is an even number and d is an odd.No. That proof does not work.

You have not exhibited a guarantee that a "k" that exists to demonstrate that n=2k is even is the same "k" that exists to demonstrate that n=2k-1 is odd. [Which indicates that you also have some opportunities for better education about variable naming, scopes and quantifiers].

Let me provide you with an axiom to make your work easier: 1 is not even. Or, if you prefer: 0 is not odd.

[It is not easy to prove that 1 is not even. Especially if you do not know what axioms or lemmas you have to work with]

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