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I am studying the limit concept from mathematics using the classical theory of concepts. According to this theory a concept is;

"A structured mental representation which is characterised by a collection of necessary and sufficient conditions for its application"

The prime example is bachelor which have as nessesary condtions "is man" and "is unmarried".

The condtions I've come up with so far to characterise a "limit" is

Nessesary = "A number a sequence if approching", "The number we can get as close as we want to", "The number we can get as close as we want to if we move sufficently deep into the sequence", "The number we are guaranteed to be as close as we want to given we move sufficently deep into the sequence" and "The number we are guaranteed to be as close as we want to for the rest of the sequence after some element in it"

Sufficent="The number we are guaranteed to be as close as we want to for the rest of the sequence after some element in it"

I never really worked with this theory before, and so I have some questions

  1. Does this list of conditions have to unique?

  2. Have I applied the theory in a correct manner?

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  • The limit definition has two 3-place relations and quantifier dependence in its definition, classical theory of concepts will not get you there. This is why the limit concept had to wait until 19th century.
    – Conifold
    Dec 10, 2023 at 8:50
  • @Conifold why cant it handle these kind of ideas?
    – user21312
    Dec 10, 2023 at 9:01
  • Because the "necessary and sufficient conditions" it envisions come in the form of properties, like "man" and "unmarried". Definitions that quantify over variables in binary and higher order predicates cannot, in general, be reduced to stringing properties together by logical connectives. Monadic predicate calculus that underlies the classical theory is much weaker than full predicate calculus used in modern mathematics.
    – Conifold
    Dec 10, 2023 at 9:50
  • I suspect you and @Conifold mean different things by "classical". Conifold is thinking of two things that weren't discussed much before the 1800s: relations and concepts of concepts (like "all concepts that apply to themselves"). I believe this may confuse you if you are focusing on the classical structure of concepts, and are just assuming that structure can be easily transferred to relations and high-order concepts, so that you would include them in the "classical" theory. Dec 10, 2023 at 11:54
  • @DavidGudeman Thanks for this comment, this is what I mean by classical theory of concepts iep.utm.edu/classical-theory-of-concepts
    – user21312
    Dec 10, 2023 at 15:43

3 Answers 3

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The OP indicated in a comment that he is interested in broader notions of limit and not merely the one via the epsilon-delta definition. While the notion of limit formulated via epsilon-delta is inaccessible to the classical theory of concepts (CTC) because it relies heavily on alternating quantifier formulas not available in CTC, there is an equivalent definition that mitigates the dependence on alternating quantifiers and therefore is more accessible to CTC. For example, the limit of a function f(x) (as x tends to 0) equals 0 if and only if

for all nonzero infinitesimal x, one has f(x) ≈ 0.

The two crucial ingredients of this definition are (1) the notion of an infinitesimal, and (2) the relation of infinite proximity "≈" . Such an approach seems more accessible to CTC because it avoids complex logical formulas involving alternating quantifiers.

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  • I never understood why one cannot handle higher order logic .. do you mind elaborating?
    – user21312
    Dec 12, 2023 at 17:15
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    I am not sure what you are asking exactly. Can you elaborate? I didn't say anything about "higher-order logic". @user21312 Dec 12, 2023 at 17:54
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    @user21312, as far as alternating quantifiers are concerned, it is hard to imagine how one would fit them into the CTC framework. I wouldn't say it is a theorem that one cannot, because CTC is not a formal mathematical theory that one could prove a theorem about, but one doesn't even see where to start. Notice that the definition I gave can be adapted to a more general framework than metric spaces. In any topological space, you can define the notion of a halo (monad) of a point p, which captures the intuitive idea of "points at infinitesimal distance from p". ... Dec 13, 2023 at 10:45
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    ... This allows one to talk about the relation of infinite proximity in a general topological space. Dec 13, 2023 at 10:45
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    The criterion I provided is a necessary and sufficient condition for the existence of the limit. The case of continuity is similar (here you require that $f(0)=0$). Dec 14, 2023 at 7:36
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The limit concept is a basic concept from calculus.

In its most simple form for real numbers the definition is the following: A sequence (x_n)n∈N of real numbers converges to a real number x_0, if for all ε>0 exists an index N∈N such that for all n ≥ N

|x_n−x_0| < ε

The crucial part of this definition is to capture

  • “as close as we want”: For arbitrary ε > 0 the distance |x_n−x_0| < ε
  • “we move sufficiently deep into the sequence”: There exists an index N such that for all indices n ≥ N

The above definition of the limit concept states the necessary conditions which all together are also sufficient.

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  • Thanks! Good idea to seperate these two clearly I didnt do that in my list. Are you familiar with the classical theory of concepts?
    – user21312
    Dec 10, 2023 at 9:19
  • @user21312 I am not familiar with a formal "theory of concepts". Possibly you can add a reference to your question; thanks.
    – Jo Wehler
    Dec 10, 2023 at 9:31
  • I added a good summary I found
    – user21312
    Dec 10, 2023 at 10:39
  • @user21312 After reading half of the paper from your link I am a bit sceptical that the paper helps to form clear and useful concepts. Useful concepts are those concepts, which qualify as a tool to build propositions and to design theories, which solve the problems of the field under investigation. – The main achievement when creating the limit-concept was to formalize the two intuitive conceptions by two clear expressions. The usefulness of the resulting concept is shown by its application as the basis of calculus.
    – Jo Wehler
    Dec 10, 2023 at 12:41
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The concept of 'limit' has become more sophisticated since the early days of calculus. I think the approach I was taught came from Kronecker. This is much the same argument as Jo Wheler's reply, but I thought I would flesh it out with an example...

Consider y = x.cos(1/x). What is the y value for x = 0?

The cosine must between -1 and +1, so y is zero. However, consider what this function does as it goes to zero. Between zero and any point you chose, the gradient will go above and below any chosen finite value an infinite number of times. If you had an unknown function with these properties you might be feeling doubts at this point. Does the point at the origin properly exist at all?

A proof that supposes an infinite number of iterations is a proof that never finishes. However, if we can sandwich our function between two limits (in this case y = +x and y = -x) then we can believe in a limit if we can show these two limits may come to within some arbitrarily small value of each other in a finite number of steps. This proof does not have to work for all x but just a small range either side of the limit we seek.

But what if we want the limit as x goes to infinity? Suppose we were solving for Y = cos(X)/X at positive infinity. This is the same as our original curve with X = 1/x. You can make a substitution to a problem where you can approach the limit from both sides.

Following Jo Wehler's comment:

I recently went to a 50th anniversary at my college; it has been a long while since I studied. I don't claim x.cos(1/x) is mathematically defined at x=0. If you write the solution as 0.cos(1/0) then it does not look mathematically healthy at all. If the reader thinks "it's zero", then "it looks like zero", "it could be anything", and finally "it looks like zero because the limits on either side go toward zero", then I have been understood. Whether it is zero is left as an exercise for the reader.

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  • The function f(x):=x*cos(1/x) is not defined for x=0. But the "limit f(x) for x to 0" exists and equals 0. For x to infinity the limit of the same function does not exist, because cos(1/x) converges to cos(0)=1 and x is unbounded.
    – Jo Wehler
    Dec 10, 2023 at 12:51

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