Infinite number of elements in some sets but no infinitely large number

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jamesadrian

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I would like to get some confirmation that our concept of infinity includes sets that have an infinite number of elements but does not include the idea that there is any particular number that is infinitely large. Is this true? Is there a proof? Is there a consensus?

Thank you for your help.

Jim Adrian
 
jamesadrian said:
I would like to get some confirmation that our concept of infinity includes sets that have an infinite number of elements but does not include the idea that there is any particular number that is infinitely large. Is this true? Is there a proof? Is there a consensus?
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There is a very simple answer if one understands that any real number is one of two kinds: rational or irrational.
The notation \([0,1]\) stands for the set of all real numbers that are greater than or equal to zero or less than or equal to one. Everyone of those numbers is either rational or irrational. There are infinitely many rationals in \([0,1]\) and there are infinitely many irrationals in \([0,1]\). Yet there are infinitely more irrational than there are rational. And to your question about a proof the answer is yes: Cantor's diagonal argument.
In the set \([0,1]=\{x: 0\le x\le1\}\) has a largest element, \(1\), and a least, \(0\), both are rational,
But the set \([0,1]\) has no largest irrational nor least irrational.
Again every number in the set \([0,1]\) is finite.
 
Infinity is the concept of something that has no bounds or limits. There is no such actual thing: it is purely conceptual. The concept of infinity does appear naturally in many ways, however. For example: in how many different ways could you divide a length into smaller lengths? There is no limit on this question, as you could continue dividing into arbitrarily small pieces, so the answer is said to be "infinity".

A set--that is, some category or group of elements--can have a definition that leads to it "containing" an infinite quantity. Such a set cannot be written down in its entirety, since by definition there is no bound on its "contents", making it impossible to complete the task. Things like "integers above zero" or "real numbers between zero and one" describe numeric sets with unlimited "contents". I keep putting "contain" in quotes here because, as before, it is purely conceptual: there is no box you could put all the distinct numbers in.

It is commonly said that there are different "sizes" of infinity, which leads to grammatical arguments and muddies the waters a bit. For instance, between the elements of an infinitely large countable set, there exist additional infinite uncountable sets. It is often stated that uncountable infinities are "larger" and "more numerous" than countable infinities. Strictly speaking, any infinity is effectively the same because they are all without bounds.

In mathematics, infinity commonly appears in the context of limits. Some examples:
  • If you begin with [MATH]1[/MATH] and continue to divide by [MATH]2[/MATH] (forming a geometric series), you get a sequence that begins with [MATH]1, \frac{1}{2}, \frac{1}{4}[/MATH] and so-on. If you add these up, you get some number. For three terms, as shown here, the number is [MATH]1 + \frac{1}{2} + \frac{1}{4} = 1\frac{3}{4}[/MATH]. What number will you get if you continue this sum for an infinite number of terms? Each time you add a new term, you cover half of the remaining difference towards [MATH]2[/MATH], so doing it "for infinity" will get you to [MATH]2[/MATH].
  • The tangent function is undefined at intervals of [MATH]\pi[/MATH] beginning at [MATH]\frac{\pi}{2}[/MATH] because the ratio of the corresponding triangles' side lengths--opposite divided by adjacent--produces a division by zero. If you look at a graph of the tangent function, you can see that the function rapidly ascends towards positive infinity on the left side of these places, and rapidly ascends from negative infinity on the right. You can get "infinitely close" to these asymptotes, approaching "an infinity" depending on which side you're on, but there is no defined value exactly on them.
 
I seems that while sets can be infinite, numbers are never infinitely large. I hope I have that right.

There is a related matter I wish to bring up:

According to mathematicians, an infinitesimal is a quantity that is indefinitely small and approaches zero in some process. This does not require the concept of infinity. If infinity is to exist in math at all, it must be added as an axiom. Infinity can never otherwise be established, whether in the form of infinitely large sets, or infinitely large numbers, or infinitely small numbers. Because any ordered set of numbers is unbounded (we can always correctly define a number that is larger or smaller than some number currently under consideration), an infinitesimal can be used to express a limit without being characterized as infinitely small. It needs only to be arbitrarily small or indefinitely small. Numbers which are indefinitely small are sufficient to formally define limits. The idea that calculus cannot be developed without infinity is a popular but incorrect myth, despite its long history in math education.

Does anybody agree with this?

Thank you for your help.


Jim Adrian
 
You do not need the concepts of either infinity or infinitesimals to provide a basis for calculus. You just need the concept of limit. Whether those concepts are helpful expository devices is a different question.
 
jamesadrian said:
I seems that while sets can be infinite, numbers are never infinitely large. I hope I have that right.
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No, this is not true at all. While some infinite sets may have an upper and lower bound (like the numbers between 0 and 1) not all infinite sets have these bounds. The integers for example do not have an upper bound nor a lower bound as they go from negative infinity to positive infinity. The integers are ...-3,-2,-1,0,1,2,3,...
 
I would be very interested in whether anybody here understands that an infinitesimal needs only be indefinitely small and does not need to be characterized as infinitely small.

Also, is it yet common knowledge that infinity must be added as an axiom for their to be infinite sets or infinite anything in the system?

Thank you for your help.


Jim Adrian
 
See https://en.wikipedia.org/wiki/Axiom_of_infinity for the history.

Ask any Ph.D. mathematician in a college math department.

They all know, but most wait for you to ask unless you are at the level where you are trying to create an axiomatic system your self. Most math professors prefer to presume that infinity is a part of nature. I have had extensive conversations with them as an older student.


Jim Adrian
 
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jamesadrian said:
I would be very interested in whether anybody here understands that an infinitesimal needs only be indefinitely small and does not need to be characterized as infinitely small. Also, is it yet common knowledge that infinity must be added as an axiom for their to be infinite sets or infinite anything in the system?
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jamesadrian said:
See https://en.wikipedia.org/wiki/Axiom_of_infinity for the history.
Ask anybody in a college math department.
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Although I am now retired, I was a university director of a division of mathematical sciences: mathematics, applied mathematics, mathematics education, and computer sciences. Now having been part of a national effort in non-standard analysis as part of calculus reform in the 1980's I assure that your statement "an infinitesimal needs only be indefinitely small" is misleading. Did you download the text material from the link posted in reply #7? If you are really interested is a rigorous study then study that material.
 
aka,

Thank you for this message.

I have downloaded and looked over this document.

I notice that its conclusions rely on axioms. These conclusions, perhaps you agree, could not be generated without those axioms and instead from definitions alone.

I disagree with the conclusions. We are far apart.

Might it be possible to construct a system that requires infinitesimals to be infinitely small rather than indefinitely small to allow a calculus and yet not involved axioms?

I prefer a mathematics that assumes nothing.

Thank you for your help.


Jim Adrian
 
pka,

The link you gave me also mentions real numbers. In a system that has no axioms, those real number that are not rational number would instead be processes. For instance, pi = 3.14159265 . . would be a process resulting in a rational number, taken out a far you you need it to be. As far as I can see, a system without axioms would not have well-defined numbers beyond rational numbers except for imaginary numbers. This does not seem to prevent limits or calculous because the rational numbers are unbounded and can represent numbers that are indefinitely small.

Jim Adrian
 
jamesadrian said:
Pka,
I have downloaded and looked over this document.
I notice that its conclusions rely on axioms. These conclusions, perhaps you agree, could not be generated without those axioms and instead from definitions alone.
I disagree with the conclusions. We are far apart.
Might it be possible to construct a system that requires infinitesimals to be infinitely small rather than indefinitely small to allow a calculus and yet not involved axioms?
I prefer a mathematics that assumes nothing.
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There is no mathematics that does not assume axioms.
Now one of my favorite books is Stanislas Debaene's NUMBER SENSE, how the mind creates mathematics. That is a book for anyone who is interested in the nature of the foundations of mathematics. Now if you are really interest in the applications of infinitesimals in mathematics get the book by Martin Davis, APPLIED NONSTANDARD ANALYSIS.
But, the best critique of infinitesimals is written by George Berkeley a bishop in the Church of England in the 17th century. His most memorable description is: "infinitesimals are the souls of dearly departed numbers".
 
Pka,

I agree that there is, as of yet, no math without axioms.

Thank you for these references.


Jim Adrian
 
Pka,

I have left my contact information and a little about myself in "account details."

Your messages have been informed and courteous. I will be including in my website provisional definitions leading to the definition of the derivative and more. The links to these web pages are not advertised or sent to my subscribers. They are for feedback and criticism from people who are knowledgeable and interested.

I hope that you will be among those who will contact me.

Very sincerely,

James Adrian
1-585-360-6037
121 Laura Drive
Rochester, NY 14626
jim@futurebeacon.com

Future Beacon

James Adrian
www.futurebeacon.com
 
pka said:
George Berkeley a bishop in the Church of England in the 17th century. His most memorable description is: "infinitesimals are the souls of dearly departed numbers".
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Nice to see a reference to that interesting man.

Nit picking to follow.

He was not a bishop in the 17th century. He was only 15 when the 17th century ended. And the quotation has been “improved” by someone familiar with modern religious speech.

The actual quotation is

And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

Citation: paragraph 35 of The Analyst
 
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The whole world is taught that there is no math without axioms. I am appalled.

Why don't people try it?

I suggest trying to define these terms first:

number, procedure, element, collection, set, ordered set, one, two, zero, numeral, and one-to-one correspondence.

You need the starting point of realizing that our perceptions include perceptions that are quantitative in nature. The undefined terms can be the starting point for formal definitions.

You can tell the difference between a single thing and a pair of things as surely as you can tell night from day or red from blue. Children reason and communicate with such perceptions as soon as they learn which words other people use to refer to them. We all have the ability to perceive a single thing, a pair of things, and many things. These terms are given meaning through experience in the world. Terms this basic cannot be defined by more basic terms. The terms any, some, more than, less than, at least, no more than, at most, and few refer to perceptions, not to formal definitions. These and related meanings are the starting point for mathematical definitions. These terms refer to empirical observations.

Here's one I believe works:

A number is the name of an amount or a quantity.

The math you create might be very different from mine. Here is what I am up to:

The terms axiom and postulate are each synonyms of the term assumption. The aim of this effort is to define a mathematics that includes calculus and complex numbers, and does not use assumptions as the foundation of that mathematics.

Rational Mathematics is a system of scientific definitions in which rational numbers play a dominant role. The number line consists of rational numbers together with processes that produce rational numbers. It does not include transcendental numbers or other irrational numbers except as names for processes. This system contains no concept of an infinitesimal.

Jim Adrian
jim@futurebeacon.com
 
In the text material I wrote for my own classes here is a rather standard definition I used.
I set, \(\mathcal{A}\) is said to be infinite if and only if \((\exists f)[f:\mathcal{A}\to\mathcal{A}]\) that is an injection that is not a surjection.
That is there is a one-to-one function from \(\mathcal{A}\) to \(\mathcal{A}\) but that function is not onto \(\mathcal{A}\).
In other words, some element of \(\mathcal{A}\) is not in the range of \(f\).
 
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