Math Is Fun Forum

Phrzby Phil

Member

From: Richmond, VA

Registered: 2022-03-29

Posts: 50

Arithmetic of Infinity

I see numerous (and in my opinion repetitive and meaningless) posts about doing various arithmetic with infinity: ∞.

The theory of transfinite arithmetic was initiated mainly by Georg Cantor around the 1890's.

The theory explores in a definitive and axiomatic way the various (differently sized) infinite sets of different types of numbers (e.g., Natural Numbers, Rationals, Irrationals, Transcendentals).

This is too involved to say more here, but a good starting point are these articles:

https://en.wikipedia.org/wiki/Aleph_number

https://en.wikipedia.org/wiki/Transfinite_number

https://en.wikipedia.org/wiki/Continuum_hypothesis

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World Peace Thru Frisbee

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Arithmetic of Infinity

I see numerous (and in my opinion repetitive and meaningless) posts about doing various arithmetic with infinity: ∞.

The theory of transfinite arithmetic was initiated mainly by Georg Cantor around the 1890's.

The theory explores in a definitive and axiomatic way the various (differently sized) infinite sets of different types of numbers (e.g., Natural Numbers, Rationals, Irrationals, Transcendentals).

This is too involved to say more here, but a good starting point are these articles:

https://en.wikipedia.org/wiki/Aleph_number

https://en.wikipedia.org/wiki/Transfinite_number

https://en.wikipedia.org/wiki/Continuum_hypothesis

Why are you INDIRECTLY taking a shot at my posts about infinity? My posts are either textbook problems or online problems. I don't make up MEANINGLESS and REPETITIVE math questions. If you feel that way or find yourself bored with my posts, the solution is quite simple===>SKIP MY POSTS AND DON'T WASTE YOUR PRECIOUS TIME.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Arithmetic of Infinity

nycguitarguy:  Unnecessary response.  Please don't do this.  As this wasn't even in 'your' thread you could have "skipped this post and avoided wasting your time.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Phrzby Phil

Member

From: Richmond, VA

Registered: 2022-03-29

Posts: 50

Re: Arithmetic of Infinity

Well, I do apologize.  I was expecting a thanks for pointing forum members to what I consider a most interesting field.

In short: two sets are considered to have the same size (or cardinality) if they can be put into a 1-1 relationship.

E.g., the naturals N = {1,2,3,...} would seem naively to be twice as big as the evens E = {2,4,6,...}.  However, as they can be matched up 1-1: (1,2), (2,4), (3,6), and so on, they have the same size.  Georg Cantor called this smallest transfinite size by the first Hebrew letter aleph, with a subscript zero, read as aleph-null.

It turns out that many infinite sets are in fact aleph-null sets.

Take the rationals Q = {p/q where p and q are integers}.  Although there are an infinite number of rationals between any two integers, Q is in fact aleph-null.  Here's a listing that contains all rationals (some may be duplicated): {1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 2/2, -2/2, 1/3, -1/3, 2/3, -2/3, etc.}. You should get the point here.  Since they can all be listed, they match 1-1 with N, hence aleph-null (also called countably infinite).

BUT - the reals are > aleph-null.

Proof by reductio ad absurdum.  Assume you have a list of all reals.  Then here's a real not in your list: Construct it by creating a number that differs from the nth number in your list in its nth position.  Since this new number is not in your list, you did not in fact provide such a list, hence it cannot be done.  Even if you add that number to the list, we'll just do this all over again.

Here's the really cool part: In 1900 or so David Hilbert created his famous list of 23 problems for the next century.  First on the list: the Continuum Problem: is the cardinality of the reals aleph-one, i.e., the very next transfinite size?

In 1963 Paul Cohen proved (get ready for this) - You can have it either way.  That is, the statement "size of the reals is aleph-one" and its negation are independent of the generally accepted axioms of set theory, ZFC (Zermelo-Fraenkel with Axiom of Choice).

Last edited by Phrzby Phil (2024-05-11 02:29:58)

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World Peace Thru Frisbee

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Arithmetic of Infinity

nycguitarguy:  Unnecessary response.  Please don't do this.  As this wasn't even in 'your' thread you could have "skipped this post and avoided wasting your time.

Bob

Sorry Bob but he was clearly talking about me.

1. My posts are NOT my own questions.

2. Questions come from textbooks and online math sites.

3. I was offended by his remarks concerning infinity math problems.

4. Do you know another member who is bombarding this math site with annoying, repetitive math questions? No, right? It is clearly me.

5. This is a math forum. Anything concerning mathematics should not be listed as meaningless and repetitive. A true math lover enjoys and welcomes all types of questions and concepts.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Arithmetic of Infinity

Well, I do apologize.  I was expecting a thanks for pointing forum members to what I consider a most interesting field.

In short: two sets are considered to have the same size (or cardinality) if they can be put into a 1-1 relationship.

E.g., the naturals N = {1,2,3,...} would seem naively to be twice as big as the evens E = {2,4,6,...}.  However, as they can be matched up 1-1: (1,2), (2,4), (3,6), and so on, they have the same size.  Georg Cantor called this smallest transfinite size by the first Hebrew letter aleph, with a subscript zero, read as aleph-null.

It turns out that many infinite sets are in fact aleph-null sets.

Take the rationals Q = {p/q where p and q are integers}.  Although there are an infinite number of rationals between any two integers, Q is in fact aleph-null.  Here's a listing that contains all rationals (some may be duplicated): {1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 2/2, -2/2, 1/3, -1/3, 2/3, -2/3, etc.}. You should get the point here.  Since they can all be listed, they match 1-1 with N, hence aleph-null (also called countably infinite).

BUT - the reals are > aleph-null.

Proof by reductio ad absurdum.  Assume you have a list of all reals.  Then here's a real not in your list: Construct it by creating a number that differs from the nth number in your list in its nth position.  Since this new number is not in your list, you did not in fact provide such a list, hence it cannot be done.  Even if you add that number to the list, we'll just do this all over again.

Here's the really cool part: In 1900 or so David Hilbert created his famous list of 23 problems for the next century.  First on the list: the Continuum Problem: is the cardinality of the reals aleph-one, i.e., the very next transfinite size?

In 1963 Paul Cohen proved (get ready for this) - You can have it either way.  That is, the statement "size of the reals is aleph-one" and its negation are independent of the generally accepted axioms of set theory, ZFC (Zermelo-Fraenkel with Axiom of Choice).

I do accept your apology. Now my 5 points below.

1. Math posts are NOT my own questions.

2. Questions come from textbooks and online math sites.

3. I was offended by your remarks concerning infinity math problems.

4. Do you know another member who is bombarding this math site with annoying, repetitive math questions? No, right? It is clearly me.

5. This is a math forum. Anything concerning mathematics should not be listed as "meaningless and repetitive". A true math lover enjoys and welcomes all types of questions and concepts.